From f1643f0bda0e2ab964fccb9c0192a94a0a18ceff Mon Sep 17 00:00:00 2001 From: Stephen Mildenhall Date: Sun, 15 Jun 2025 12:10:48 +0100 Subject: [PATCH] 3.2.0 integration of tikz, better column width More TeX snippets. --- README.md | 221 +- docs/index.rst | 7 +- greater_tables/__init__.py | 2 +- greater_tables/cli.py | 16 +- greater_tables/data/blog.csv | 3354 +++++ greater_tables/data/book.csv | 5849 ++++++++ greater_tables/data/notes.csv | 3592 +++++ greater_tables/data/tex_list.csv | 22567 +++++++++++++++++++++-------- greater_tables/data/tex_list.py | 39 +- greater_tables/gtcore.py | 651 +- greater_tables/testdf.py | 21 +- greater_tables/tex_svg.py | 59 +- 12 files changed, 30257 insertions(+), 6121 deletions(-) create mode 100644 greater_tables/data/blog.csv create mode 100644 greater_tables/data/book.csv create mode 100644 greater_tables/data/notes.csv diff --git a/README.md b/README.md index 465bd80..f089bd2 100644 --- a/README.md +++ b/README.md @@ -1,17 +1,9 @@ -sort out the variety of readmes... this is the main one +# `greater_tables` Project -https://shields.io/badges/read-the-docs +![GitHub commit activity](https://img.shields.io/github/commit-activity/y/mynl/greater_tables_project) -# v 3.0 update - -* config files -* unified col width and info dataframe -* de-texing -* cli for config and writeout a csv etc. -* tests - -# TODO +## TODO * Ratio cols with multi index columns * % in tex output - never allow comments? @@ -19,109 +11,14 @@ https://shields.io/badges/read-the-docs * ?Option to hide index * Bring over the roll your own logger -*** + +## Greater Tables -# from GPT +Display graphics vs. data tables +no colors, sparklines, shading, ... -Absolutely—here’s a structured summary of everything we’ve covered, organized by topic. - ---- - -## 🧱 Project Structure & Philosophy - -* Your `GreaterTables` class formats a `pandas.DataFrame` to **HTML, text, or LaTeX**. -* The class is **immutable**: formatting is fixed at construction time, like a pure value object. -* You avoid branchy, incremental APIs (like `ggplot`) and prefer creating fresh objects. -* You wanted a way to handle growing config complexity—→ led to a **YAML config + Pydantic schema** design. - ---- - -## 📁 Project Layout - -``` -greater_tables_project/ -├── greater_tables/ -│ ├── __init__.py -│ ├── gtconfig.py ← config model + loader -│ ├── gtcore.py ← GreaterTables class -│ └── defaults/ -│ └── config_template.yaml -├── tests/ -├── pyproject.toml -``` - -* `GTConfigModel` = schema + default source of truth -* `GTConfig` = singleton loader and validator -* `config_template.yaml` = editable fallback + documentation base - ---- - -## 🔧 Config Management - -* All defaults and types are declared in `GTConfigModel` (Pydantic). -* Config is **loaded from YAML**, validated by `GTConfigModel`. -* You can **generate** a valid config file from the model using `.model_dump() → YAML`. -* Singleton pattern (`GTConfig.__new__`) caches the config at runtime. - -### Helpers - -* `GTConfig().get(overrides=...)` gives a safe, override-able config -* `write_template(path)` writes a default config YAML for user to edit - ---- - -## 🛠 Git Workflow (Solo Dev, Linear) - -* Use **tags** (`git tag v0.2.0`) to label stable versions -* Use **`git reset --hard `** to roll back and discard later commits -* Avoid branches entirely—keep a **single linear history** -* Tags let you bounce around safely, with names instead of hashes -* Releases on GitHub are tags + metadata, optional for publishing - ---- - -## ⚙️ CLI Tool - -* Built with `click`, with subcommands: - - * `gt render data.csv --format html` - * `gt write-template` -* Reads any Pandas-supported file (`.csv`, `.feather`, `.pkl`, etc.) -* Outputs to console or to file -* Uses current config by default, or override with `--config path.yaml` - ---- - -## 🧠 Design Principles You’re Following - -| Principle | Your Approach | -| ---------------------------- | ------------------------------------------- | -| Immutability | `GT(df, config)` is fixed once created | -| Separation of concerns | `GTConfigModel` holds defaults/types | -| Config as code/documentation | `config_template.yaml` generated from model | -| CLI-first mindset | `click` used to expose functionality | -| Linear Git workflow | Tags for rollback, no branches | - ---- - -Let me know if you want me to generate: - -* a Markdown doc for contributors -* a `.bat` script to roll back to a tag -* test scaffolding or release automation - -You're in great shape. Gum-level perfection achieved. - - - -*** - -# OLD - -# Greater Tables - -Creating presentation quality tables from pandas dataframes is frustrating. It is hard to left-align text and right-align numbers using pandas `display` or `df.to_html`. The `great_tables` package does a really nice job with pandas and polars dataframes but does not support indexes or TeX output. +Creating presentation quality tables is difficult. It is hard to left-align text and right-align numbers using pandas `display` or `df.to_html`. The `great_tables` package does a really nice job with pandas and polars dataframes but does not support indexes or TeX output. This package provides consistent HTML and TeX table output with flexible type-based formatting, and table rules. Neither output relies on the pandas `to_html` or `to_latex` functions. TeX output uses Tikz tables for very tight control over layout and grid lines. The package is designed for use in Jupyter Lab notebooks Quarto documents. @@ -137,9 +34,19 @@ been maintaining a set of macros called reusable, extensible actuarial tools) in VBA and Python since the late 1990s, and call all my macro packages "GREAT". +## Documentation + +![](https://img.shields.io/readthedocs/greater_tables_project) + +Available on +[readthedocs](https://greater-tables-project.readthedocs.io/en/latest). + + ## Installation -``` python +![](https://img.shields.io/pypi/format/greater_tables) + +```python pip install greater-tables ``` @@ -148,7 +55,7 @@ pip install greater-tables The following example shows quite a hard table. It is formatted using the `sGT` class, which is a subclass of `GT` with a few defaults set. -``` {.python .cell-code} +```python import pandas as pd import numpy as np from greater_tables import sGT @@ -196,21 +103,37 @@ The output illustrates: More coming soon. -## Documentation -![](https://img.shields.io/readthedocs/greater_tables_project) +## History -Available on -[readthedocs](https://greater-tables-project.readthedocs.io/en/latest). +3.2.0 +------- +* Added more tex snippets! +* Refactored tikz and column width behavior -## Versions +3.1.0 +------- +* adjustments for auto format +* rearranged gtcore order of methods 3.0.0 ------- +* config files / pydantic config input +* unified col width and info dataframe +* de-texing +* cli for config and writeout a csv etc. + +* testdf suite +* Automated TeX to SVG + 2.0.0 ------ +* **v2.0.0** solid release old-style, all-argument GT +* Better column widths +* Custom text output +* Rich table output 1.1.1 ------- * Added logo, updated docs. @@ -251,3 +174,65 @@ Early development + + +## 📁 Project Layout + +``` +greater_tables_project/ +| LICENSE +| pyproject.toml +| README.md +| ++---dist +| ++---docs +| | books.bib +| | conf.py +| | greater_tables.data.rst +| | greater_tables.rst +| | index.rst +| | library.bib +| | make.bat +| | Makefile +| | modules.rst +| | start-server.bat +| | style.csl +| | ++---greater_tables +| | __init__.py +| | cli.py +| | gtconfig.py +| | gtcore.py +| | gtenums.py +| | gtformats.py +| | hasher.py +| | testdf.py +| | tex_svg.py +| | +| +---data +| | | __init__.py +| | | tex_list.csv +| | | tex_list.py +| | | words-12.md +| ++---greater_tables.egg-info +| ++---img +| hard-html.png +| hard-tex.png +``` + + + +## 🧠 Design Principles You’re Following + +| Principle | Your Approach | +| ---------------------------- | ------------------------------------------- | +| Immutability | `GT(df, config)` is fixed once created | +| Separation of concerns | `GTConfigModel` holds defaults/types | +| Config as code/documentation | `config_template.yaml` generated from model | +| CLI-first mindset | `click` used to expose functionality | +| Linear Git workflow | Tags for rollback, no branches | + + diff --git a/docs/index.rst b/docs/index.rst index c5b1ccd..ade6926 100644 --- a/docs/index.rst +++ b/docs/index.rst @@ -14,11 +14,8 @@ Welcome to greater_tables's documentation! greater_tables.data -Introduction -============ - -.. mdinclude:: ../README.md - +.. include:: ../README.md + :parser: myst_parser.sphinx_ Other ======= diff --git a/greater_tables/__init__.py b/greater_tables/__init__.py index d57b160..907ee4c 100644 --- a/greater_tables/__init__.py +++ b/greater_tables/__init__.py @@ -1,4 +1,4 @@ -__version__ = '3.1.0' +__version__ = '3.2.0' __project__ = 'greater_tables' __author__ = 'Stephen J Mildenhall' diff --git a/greater_tables/cli.py b/greater_tables/cli.py index d691f95..eb1e18b 100644 --- a/greater_tables/cli.py +++ b/greater_tables/cli.py @@ -1,18 +1,20 @@ import click import pandas as pd from pathlib import Path -from .gtconfig import GTConfig, write_template -from .gtcore import GreaterTables +from .gtconfig import GTConfigModel, write_template +from .gtcore import GT + @click.group() def cli(): """Greater Tables CLI tool""" pass + @cli.command() @click.argument("input_file", type=click.Path(exists=True)) @click.option("--output", "-o", type=click.Path(), help="Write rendered output to file") -@click.option("--format", "-f", type=click.Choice(["html", "text", "latex"]), default="html") +@click.option("--format", "-f", type=click.Choice(["html", "text", "latex", "svg", "pdf"]), default="html") @click.option("--config", type=click.Path(), help="Path to a YAML config file") def render(input_file, output, format, config): """Render a table from a data file.""" @@ -28,8 +30,8 @@ def render(input_file, output, format, config): else: raise click.UsageError(f"Unsupported extension: {ext}") - cfg = GTConfig(Path(config) if config else None).get() - gt = GreaterTables(df, config=cfg) + cfg = GTConfigModel(Path(config) if config else None).get() + gt = GT(df, config=cfg) rendered = ( gt.render_html() if format == "html" @@ -37,11 +39,15 @@ def render(input_file, output, format, config): else gt.render_latex() ) + if format in ('svg', 'pdf'): + print('more work to do!!') + if output: Path(output).write_text(rendered, encoding="utf-8") else: print(rendered) + @cli.command() @click.argument("path", type=click.Path(), default="config.yaml") def write_template(path): diff --git a/greater_tables/data/blog.csv b/greater_tables/data/blog.csv new file mode 100644 index 0000000..4aacb42 --- /dev/null +++ b/greater_tables/data/blog.csv @@ -0,0 +1,3354 @@ +,expr +0,$E[M_\tau] \ge 1$ +1,$\text{VaR}_{0.99}$ +2,$1/(24\times 60\times 60) = 0.000011574074074\dots$ +3,$\alpha=2$ +4,$\kappa_{T_\nu}(y)$ +5,$J(1)$ +6,$Y_1=X_1 + \dots + X_n$ +7,$\mathsf{Pr}(X\in A)=0$ +8,"$(\Omega,\F,\P)$" +9,$\alpha > 2$ +10,$\alpha=-\infty$ +11,"$\mathsf{E}[\mathsf{CP}(\lambda, X)]=\lambda\mathsf{E}[X]$" +12,$\beta=\dfrac{c_1-c_2}{c_1+c_2}$ +13,$p<0$ +14,$0 < \alpha_1=\alpha^{-1} < 1$ +15,$N':=kNT$ +16,$V_2 = V_1 / 12$ +17,$f_n$ +18,"$\Gamma(S,T)=\{ (S(\omega), T(\omega)\mid \omega\in\Omega \}$" +19,$\theta = -1/\mu$ +20,"$\mathsf{CP}_1(\mu, X)$" +21,$p-1=28$ +22,"$(E,\mathsf{E}E)$" +23,$u_i \ge 0$ +24,$m_Y(s)=\mathsf E[Y\mid S=s]$ +25,$A_t$ +26,$0.9999\dots=0.9 \times (1 + 0.1 + 0.001 +\cdots) = 0.9 \times (1 - 0.1)^{-1} =1$ +27,$0\le k < 2^m$ +28,$2***52 \le f 2^N < 2**53$ +29,$D=\sum_{i\in I} D_i$ +30,$T(y)$ +31,$Z$ +32,$1/V(\mu)$ +33,$g(s) = s^\alpha$ +34,$Mg$ +35,$K(t):=\log\mathsf{E}[e^{tY}]=\kappa(t+\theta)-\kappa(\theta)$ +36,$R^S=g^{kS}$ +37,$\P(B\mid\A)(\cdot)$ +38,$ is the MGF of a gamma with shape $ +39,$\mathsf{Pr}(X>x)$ +40,$\beta=1$ +41,$\Gamma\not\in\F\otimes\B$ +42,$\theta(\mu)=\arctan(\mu)$ +43,"$s_0, s_1, s_2$" +44,$-3$ +45,$N_a$ +46,$1+\mu^2-\sqrt{1+\mu^2}$ +47,$r_2=$ +48,$C > 0$ +49,$\int_0^1 xj(x)dx<\infty$ +50,$A = \bigcup_{n=1}^{\infty} A_n$ +51,$\tilde\Theta_1$ +52,$00$ +120,$-\alpha < -2$ +121,$e^{-\theta x}$ +122,$λ^*(N) = 1$ +123,$L(\nu)=\sum_{1\le i\le \nu} X_i$ +124,$x$ +125,$\sigma(\A\otimes\B)$ +126,"$X_1, \dots, X_n$" +127,$\A\otimes \B$ +128,$n=\sum_i n_i$ +129,$(1-p)(1-\alpha)=(p-1)(\alpha-1)= -1$ +130,$x_1+y_1 \le x_2+y_1\le x_2+y_2$ +131,"$\mu^t P^x_t f(x,t)$" +132,$\alpha$ +133,$\nu(M\setminus f(L))=0$ +134,$X_1$ +135,$j(x)=\alpha x^{-\alpha-1}$ +136,$\sigma=0$ +137,$\sum_i a_i=a$ +138,$\beta$ +139,$g(T(X)\mid \lambda)$ +140,$y\theta-\kappa(\theta)$ +141,$N:=(1-\alpha)M$ +142,$\tau(\theta)=\mu$ +143,$\nu=f\mu$ +144,$r \approx 1.496 \times 10^{11} \ \text{m}$ +145,$\kappa_X(\theta)=\lambda(e^\theta-1)$ +146,$m\to\infty$ +147,$\int X_n=1$ +148,$\psi=1_{A\times B}$ +149,$(3) \rightarrow (9 = 9) \rightarrow (27 = 4) \rightarrow (12 = 12) \rightarrow (36 = 13) \rightarrow (39 = 16) \rightarrow (48 = 2) \rightarrow (6 = 6) \rightarrow (18 = 18) \rightarrow (54 = 8) \rightarrow (24 = 1)$ +150,$\mu=19.005$ +151,$\mathsf E[XY] \not=\mathsf E[X]\mathsf E[Y]$ +152,$p=0.831588$ +153,$\forall X\exists U[\forall Y\forall x(x\in Y \wedge Y \in X)\rightarrow x\in U]$ +154,$\theta=\log(\mu/(n-\mu))$ +155,"$1_A:\Omega\to\{0,1\}$" +156,$Q=T\P$ +157,$\sup$ +158,$\text{ш}$ +159,$X=x$ +160,$100 - 10^5$ +161,$c(y)e^{\theta y}$ +162,$dy/y = kdx / kx=dx/x$ +163,"$\displaystyle\int_{T^{-1}(B)} X \,d\P$" +164,$c_1 + c_2 >0$ +165,$x\le 0$ +166,$\G\subset \F$ +167,$a=5$ +168,"$[0,0] \succeq [\epsilon, \epsilon]$" +169,$Y_t=X_1+\cdots + X_{N(t)}$ +170,$\mathsf{E}[X\mid\G](ω)=\mathsf{E}[X]$ +171,$\kappa(\theta)=e^\theta$ +172,"$178.7 billion of expenses. Commissions and brokerage accounted for 25.1 percent and claim adjustment services for 13.5 percent of the total. Taxes licenses and fees were 6.3 percent. However, their remaining expense items are broken out by expense category, such as employee salaries and benefits or advertising, rather than insurer value-add function. They also reported a cost of capital of 13 percent, applied to equity capital of $" +173,$\mathsf{E}[\mathsf{E}[X\mid \G_1]\mid \G_2]=\mathsf{E}[X\mid \G_1]$ +174,$\kappa'(\theta)=\tau(\theta)=\mu$ +175,$d>2$ +176,$X\mid\theta$ +177,$n_i$ +178,$(X_n)_{n \geq 1}$ +179,$\kappa(s)=\log\mathsf{E}[e^{sX_1}]$ +180,$\{ \omega \mid \P(B\mid\G)(\omega) < 0 \}\in\G$ +181,$3000$ +182,"$P(A, ω) = 1_A(ω)$" +183,"$(0,t_1]$" +184,$f_0\in L^1(\mathbb R)$ +185,$E\setminus F\in R$ +186,"$L(e,t)\sim D(et, \phi)$" +187,$X_j$ +188,"$\psi=1_\Gamma(S,T)$" +189,$\forall A\in\F$ +190,$\tau(\theta)$ +191,$D_i^n$ +192,$1 \times 10^{24}$ +193,$V(\mu)=\mu^p$ +194,$p \in G$ +195,$L_X(s)/L_Y(s)\to\infty$ +196,$\BB(S)$ +197,$c^{p-2}\to\infty$ +198,$1+x^2$ +199,$\mu_x = -d\log({{}_tp_x})/dt = \lim_{t\downarrow 0} {}_tq_x/t$ +200,$V(m)\sim c_0m^p$ +201,$a(h_1+h_2)\le 2aw$ +202,$A_n$ +203,$j(x)\propto x^{-3/2}e^{-\beta x}$ +204,$B\in\B$ +205,$V(\mu)\propto \mu$ +206,$\P(A)=0$ +207,"$p:\Omega\times\F\to [0,1]$" +208,$\theta\in\Theta$ +209,$1/\sqrt{\alpha}\to 0$ +210,$I_x$ +211,"$\Omega=[0,1]$" +212,$\xi(\cdot\mid t)$ +213,$V_T(m)= m^3V_X(1/m)=m^2$ +214,$\iota\lambda$ +215,"$[P,2P)$" +216,$\rho E/(1-\tau) - rA$ +217,$\mathsf{E}[X\mid Y=y]$ +218,$A=\bigcup A_{ij}$ +219,$N_t = 0$ +220,$|\hat f|$ +221,$a^*=q(p^*)$ +222,"$\mathrm{Tw}_{3-p}(\mu, \sigma^2)$" +223,$X_1=aX$ +224,$0.45 \times 8.6 = 3.9$ +225,$\mathbb{P}(A) < \delta$ +226,$t\mapsto e^{-2\pi it}$ +227,"$9.81 \, \text{m/s}^2$" +228,$(x+t)\Gamma(x+t)=\Gamma(x+t+1)$ +229,$\hat \theta_s$ +230,$W'\subset\mathcal W$ +231,$m=m_1/a$ +232,$P=nb$ +233,$N_t$ +234,$T_*\propto |v|^2\propto GM/R$ +235,$\kappa(\theta)=\theta-\theta\log(-\theta)$ +236,"$1,2,\dots$" +237,$m\to 0$ +238,$\int c(y)dy=\infty$ +239,$\lim_{\mu\to 0} V(\mu)=0$ +240,$\mu^3\sigma^2$ +241,$kS=m+Ra$ +242,$\mathsf E[XY]\not=\mathsf E[X]\mathsf E[Y]$ +243,$\P\vert_\F = λ$ +244,"$(-\infty, 0)$" +245,$x_{\min{}}=9750$ +246,$\hat\theta_s>0.5$ +247,$S=T=$ +248,$\mathcal{C} = \{ A \in \Sigma : \mu(A) = \nu(A) \}$ +249,$N_t = 1$ +250,$\C$ +251,$0.03$ +252,$\BB(S)\otimes \A$ +253,$L^\infty$ +254,$\kappa$ +255,$\mathsf{CP}_1$ +256,$(1-{}_b\bar V)$ +257,$a\theta^2=c$ +258,$32$ +259,$1/\sqrt{\lambda}$ +260,"$p(t, B)=\P(S\in B\mid T):E\times \S\to [0,1]$" +261,"$\forall A\forall p[\forall x\in A\exists !y\phi(x, y, p)\rightarrow\exists Y\forall x\in A\exists y\in Y\phi(x, y,p)]$" +262,$a(w+h_2)$ +263,"$(2, 3)$" +264,"$, apply the advanced form of the Lagrange Inversion Formula\footnote{$" +265,$p=23$ +266,$\lambda = \P \times \Q$ +267,$\forall B\in\G$ +268,$\bar a_x$ +269,"$-l, -l+1, \dots, 0, \dots, l-1, l$" +270,$a=1/c$ +271,"$(\Omega, \F)$" +272,"$\min(\alpha_X, \alpha_Y) >2$" +273,"$\inf\,\Omega=0$" +274,$\frac{1}{2}$ +275,$2\lim_{\mu\to 0} V(\mu)/\mu^2=\lim_{\mu\to 0}V''(\mu)$ +276,$\lim_{\theta\uparrow 0} -\theta\log(-\theta) =0$ +277,$\mathsf{SD}(G')=\nu$ +278,$g\ge 0$ +279,$\P(A\mid\G) = \mathsf{E}[1_A\mid\G]$ +280,$l/P$ +281,$Y_t\to 0$ +282,$x/(1+x^2)$ +283,"$P:\F\times\Omega\to[0,1]$" +284,$Z=0$ +285,$P(\cdot | \mathcal{G})(\omega)$ +286,$\mu^3$ +287,"$p\neq 1,2$" +288,$\mathsf{Var}(s) =\mathsf{E}[s^2]$ +289,$|x_i-x_j|<1/k$ +290,$\kappa=0$ +291,$t=m/n$ +292,$X_t = X_0 e^{(\mu - \frac{1}{2} \sigma^2)t + \sigma W_t}$ +293,$10^2 - 10^4$ +294,$f\in L^1(\mathbb R)$ +295,$\omega\not=\omega'$ +296,$l(t)$ +297,$r$ +298,$\mathsf{E}[T]$ +299,$\mathsf{E}[D_t \mid \F_{t-1}] \geq 0$ +300,$10^{-6} - 10^{-3}$ +301,$\tau(\tau^{-1}(\mu))=\mu$ +302,$P = 53.565 = v EL + d \max(L) = 46.6 / 1.15 + (0.15 / 1.15) \times 100$ +303,$\omega\mapsto \P(X\in B\mid \G)(\omega)$ +304,"$\mathsf p=(p_0,\dots,p_{n-1})$" +305,$\exp(a\arcsin(z))=\sum_n \frac{p_n(a)}{n!}z^n$ +306,$\xi:\A\times M\to \mathbb{R}$ +307,$A_{x+b}$ +308,$\partial d/\partial\mu=-2(y-\mu)$ +309,$X_t - X_{t-1}$ +310,$1 <\alpha < 2$ +311,$1/n$ +312,$\omega^l=(\omega^n)^j=1$ +313,$8.617 \times 10^{27}$ +314,$A_x\cap A_y\not=\emptyset$ +315,$E\cap U_0=E$ +316,"$(\omega^{ij})_{i,j}$" +317,$N(t)$ +318,$Y_t=t-N_t$ +319,$00$ +330,$2^{402653211}$ +331,$\P(A\mid\G)_{\omega_0}\ge 0$ +332,"$P=\mathrm{EL} + r\,Q$" +333,$\Omega\times \Omega$ +334,$ Integration by parts shows $ +335,$h > 0$ +336,$X+Y$ +337,"$767 billion of capital, part of $" +338,$\mathsf{E}[X\mid\theta]$ +339,$\alpha_X > \alpha_Y + 2$ +340,$[0; -k]$ +341,$X \mapsto X + a$ +342,$8.75=10.5 / 1.2$ +343,$\epsilon_t$ +344,$\mathsf P(G)$ +345,$m_p$ +346,$g(X_n)=1$ +347,$1/b$ +348,$\rho(X) =\mathsf{TVaR}_p(X)$ +349,"$a\,\mathsf{E}\left[\dfrac{X_1}{X}\mid X\ge a \right]$" +350,$\mathbb{R}^2$ +351,$10^0$ +352,$l(y;\mu)=y\log(\mu) + (1-y)\log(1-\mu)$ +353,$1/2$ +354,$\mathsf{E}[X \mid X > q(p)]$ +355,$\bar a_{x:n\!\urcorner}$ +356,$p_s$ +357,$6.022\times 10^{23}$ +358,$y\in\Omega$ +359,$f_{X+Y}$ +360,$\phi(s)=\mathsf{E}[e^{isY}]$ +361,$A=E\cup P$ +362,$\mathsf{Pr}(\omega)$ +363,$\mathbb R$ +364,$(F((k-1/2)b)-F((k+1/2)b)) / b$ +365,$ which is an extreme stable with Lévy distribution $ +366,"$[0,1]$" +367,$10^{32}$ +368,$\\sigma=2.58$ +369,$\mathsf{TVaR}(X)=q(p)=100.0$ +370,$f(t)=\overline{f(-t)}$ +371,$Pa=kg /ms^2=N/m^2$ +372,$\delta > 0$ +373,$Y_s$ +374,$\alpha_1$ +375,"$g, g^2, \dots,g^{q-1}, g^q\equiv 1$" +376,$\log(2)$ +377,$(\sum_i \nu_i)(\kappa(t+\theta) - \kappa(\theta))$ +378,$E[Y_N]\le 0$ +379,"$\int_0^{100} g(S(x))\,dx$" +380,$Y(ω)=\mathsf{E}[X\mid\G](ω)$ +381,$h(X) = \prod_{i=1}^{n} \frac{1}{x_i!}$ +382,$ which is an $ +383,$\mathcal{G} \subseteq \mathcal{F}$ +384,$\mathsf{E}(G)=1$ +385,$E\cup F\in R$ +386,$\mathbf{v}$ +387,$A_{ij}=\{\omega\mid F_{r_j}(\omega) < F_{r_i}(\omega) \}$ +388,"$[0, \infty)$" +389,$A \subset \mathbb{R}$ +390,$p_s=\Phi((\hat\theta_s-0.5)/\hat\sigma_s)>0.5$ +391,$d=rv$ +392,$X\sim$ +393,$\mu = \nu$ +394,$602.6 billion and converted to net premium based on $ +395,$M_X(t)^n$ +396,$F^{\times}_{23}$ +397,$S(x) = x\cup \{x\}$ +398,$n\times 1$ +399,"$S:(L,\A)\to ??$" +400,$\frac{1}{2}kT$ +401,$1 \times 10^{20}$ +402,$\Q$ +403,$\forall x$ +404,$13$ +405,$\mathsf{E}[X \mid X > q(p^*)]=103.333$ +406,$A \in \mathcal{M}$ +407,$(T\lambda)\{\lambda_t\Omega=\infty\}=0$ +408,"$\int_0^1 j(x)\,dx<\infty$" +409,$2^9=512$ +410,"$1_{(0,1)}$" +411,$\mu < \frac{1}{2} \sigma^2$ +412,$\exists \mu$ +413,$p_k$ +414,$Y_1$ +415,$d^*$ +416,$\pi$ +417,$P^T_S(\cdot\mid\cdot):\A\times M\to\mathbb{R}$ +418,$m_s$ +419,$y^*-x^* \ge \epsilon$ +420,$00$ +459,$k=1.333$ +460,$2.35 \times 10^{-4}$ +461,$[\P]$ +462,$p<\infty$ +463,$m / s^2$ +464,$\{ a_n\}$ +465,$\mu=-1-\dfrac{1}{\theta}$ +466,"$\displaystyle\int_{T^{-1}(B)} \mathsf{E}[X\mid T]\,d\P$" +467,$1/P$ +468,$g(t)=h(t) / (1+l(t)) > 0$ +469,$\omega\in\Omega$ +470,$\nu\otimes P$ +471,"$\displaystyle\int_B \mathsf{E}[X\mid\G]\,d\P$" +472,$T^{-1}(A)$ +473,$(\mu)$ +474,"$\mathsf E[X] =\displaystyle\int_0^\infty xf(x)\,dx = \displaystyle\int_0^\infty S(x)\,dx$" +475,$\mathsf{Pr}(L'= l)$ +476,$\kappa'(\theta)>0$ +477,$\mathsf{E}[e^{sX_{m/n}}]=\mathsf{E}[e^{sX_{1}}]^{m/n}$ +478,$\ge \mathsf{VaR}$ +479,$y$ +480,$_1F_1$ +481,"$\mathrm{ED}^*(\theta, \lambda)/\lambda$" +482,$b=1/1.2=0.83$ +483,$\sum_x x q_x$ +484,$a\theta=1$ +485,"$F(\omega, \cdot)$" +486,"$a\in(0,2)$" +487,$\mathcal{A} \subseteq \mathcal{C}$ +488,"$P(A, \F)$" +489,$0\in\Theta$ +490,$|\hat F(f)|$ +491,"$\int_B \P(\bigcup_i A_i\mid\G)\,d\P$" +492,$d$ +493,$x_i+y_j$ +494,$\sigma^2=1 / \lambda$ +495,$v^b{}_bq_x(1-{}_b\bar V)$ +496,$x\le 1$ +497,$X\P=\P(X^{-1}(\cdot)))$ +498,"$\forall E\in\A,\ \forall F\in \B$" +499,$\lim_{\mu\to 0}V'(\mu)=\delta$ +500,$m=(K^{-1}Km)$ +501,$l(y;\mu)=y\theta(\mu)-\kappa(\theta(\mu))$ +502,$Z(\omega)$ +503,$B_s$ +504,$p > 1$ +505,${\lambda\alpha}/{\beta}$ +506,$\mathsf{E}[Y]=\mathsf{E}[N]\mathsf{E}[X]$ +507,$B=2\mathbb Z + \xi\mathbb Z$ +508,$p_\alpha\in B\cap F_\alpha$ +509,$X_S\ge \mathsf{E}[X_T\mid \F_S]$ +510,$\le x$ +511,$\sqrt{g d}$ +512,$K_\theta$ +513,$4\times$ +514,$B=M$ +515,$-stable distribution with Lévy density $ +516,"$r_1 = 1643984129.762957 \approx 1,643,984,129.8$" +517,$\sup_n E[|X_n|]<\infty$ +518,$q=11$ +519,$\sigma_T$ +520,$\sigma=0.5$ +521,$P^T(T\not=t\mid t)=0$ +522,$(Mg+\alpha a)w$ +523,$\partial l/\partial \beta_i=0$ +524,$\px=\P(B\cap A)$ +525,$3.2 \times 10^{15}$ +526,$f=1/b$ +527,$255$ +528,"$I=[0, 32]$" +529,"$\mathrm{Ga}(\mu, \sigma^2)$" +530,$\nu(dy)=(e^{2y}-1)dy$ +531,$g^a\equiv n\pmod{p}$ +532,"$(3,\infty)$" +533,"$F_\alpha - \bigcup_{\beta<\alpha} \{p_\beta, q_\beta\}$" +534,$X_t\ge \mathsf{E}[Y\mid \F_t]$ +535,$U_0$ +536,$\forall a\forall b\exists x[a\in x \wedge b\in x]$ +537,$-\gamma|\theta|^\alpha$ +538,$\mathsf{E}[X_\theta]=\infty$ +539,$10^3 - 10^{-1}$ +540,"$[0, 1]$" +541,$\alpha > 1$ +542,$\G=\sigma(Y)$ +543,$f_0$ +544,$X_1=0$ +545,$v_i$ +546,$K_{k+X}(t)=kt+K_X(t)$ +547,$\exists x[\forall z(z=\emptyset)\rightarrow z\in x \wedge \forall x\in x\forall z(z=S(x)\rightarrow z\in x)]$ +548,$\mathsf P$ +549,$d(y\mu) > 0$ +550,$\mathsf{E}[X \mid \G]$ +551,$\sum_ x x\mathrm{Po}(j(x)dx)$ +552,$\Gamma\in\G$ +553,$R=g^k\pmod p$ +554,$e[d_t d_s] = e[d_t]e[d_s]$ +555,$Z=\sum Z_i$ +556,$\Delta$ +557,"$e(f, y)$" +558,$n$ +559,"$\mathrm{St}(\alpha, \beta, 1, 0)=\mathrm{St}(\alpha, \beta)$" +560,$H(x)$ +561,$T_x=\inf\{ t > 0 \mid X_t\ge x\}$ +562,$F_n(x) := 1 - J_n(x)/J_n(0)$ +563,$-\kappa_T(y)$ +564,$r'=0$ +565,$1_G(\omega)$ +566,"$(\Omega, \mathcal{F}, \mathbb{P}, \{\mathcal{F}_t\}_{t \geq 0})$" +567,$\alpha/\beta^2$ +568,"${x}\times A\subset [0,1]^2$" +569,$\mathsf v = (\hat F(t_l))_l$ +570,$\theta=f(\mu)$ +571,$\kappa(\theta)=-\alpha\log(-\theta)$ +572,"$C=[0,\infty)$" +573,$\\alpha=0$ +574,$\frac{1}{2}mv^2$ +575,$u = 1-e-l-r^*$ +576,$10^{17}$ +577,$\Omega=C^\circ$ +578,"$g_n = \max \{f_1,\dots,f_n\}$" +579,"$\Lambda_{a,b}$" +580,"$\P^x f(x, Tx) = \mu^tP^x_t\,f(x,t)$" +581,$\forall X\exists P\forall z[z\subset X\rightarrow z\in P]$ +582,$x_{\min{}}$ +583,"$X_1, X_2, \ldots, X_n$" +584,$i^{-a}=e^{-a\log i}=e^{-ia\pi/2}$ +585,$\forall A\in\A$ +586,"$y_1,\dots, y_n$" +587,$4 \times 10^{-7} - 10^{-8}$ +588,$Ca_3Al_2(SiO_4)_3$ +589,$\kappa_X(\theta)=\log\mathsf{E}[e^{\theta X_t}]=\log\mathsf{E}[e^{\theta\sigma B_t + ct\theta}]=t(c\theta+ \sigma^2\theta^2/2)$ +590,$\sigma^2\mu^p$ +591,$Q$ +592,$n-1$ +593,$-1$ +594,$c^*(y):=e^{l(y;y)}=c(y)e^{y\tau^{-1}(y)-\kappa(\tau^{-1}(y))}$ +595,$(\omega^l)^n = (\omega^n)^{l} = 1$ +596,$J(x)/J(0)$ +597,$D∪ C^c$ +598,$CV(L(1))/\sqrt{\nu}$ +599,$X_1=99$ +600,$2^\lambda < \kappa$ +601,$1+1/3+1/5+\cdots$ +602,$dt$ +603,"$\int_\Omega f_0\,d\mu=\alpha$" +604,$tE[\text{jumps}]$ +605,"$(X,\A)$" +606,$\lambda\to (\alpha+1)/\alpha$ +607,$\nu(dx)$ +608,$0.5$ +609,$e^{-\beta s}s^{-1}\approx s^{-1}$ +610,"$b\in[-1,1]$" +611,$X=X^+-X^-$ +612,"$S=\{0,1,2,\dots\}$" +613,$x_i-x_j$ +614,$1.5\times 10^{37}$ +615,$\alpha(t) = t$ +616,$M[G]$ +617,$R^*$ +618,$\lambda^2\sigma=\lambda$ +619,$c(y)e^{y\theta - \kappa(\theta)}$ +620,"$a,b$" +621,$d(y;\mu)=|y-\mu|$ +622,"$f(s,t)=\lim_n f_n(s,t)$" +623,$D(E)$ +624,$0 \leq s < t$ +625,$P(Z\cap \Gamma) = \frac{1}{2}\mu(\Gamma) = P(Z)P(\Gamma)$ +626,$(-a)\Gamma(-a) = \Gamma(1-a)$ +627,$S=X$ +628,"$j(x)/J(1)1_{[1,\infty)}$" +629,"$(-2,-1)$" +630,"$(Y,\G,\Q)$" +631,$M(u) = c_1 / |u|^\alpha$ +632,$|X_t| \leq M$ +633,$K_\delta * f\to f$ +634,$w=\sum_i w_i$ +635,$|X_\alpha| > c$ +636,"$(\alpha,\beta)$" +637,$y\pm 2\sqrt{V(\mu)}$ +638,$\lambda > 0$ +639,$\rho=0.12$ +640,$g(T(X)\mid \lambda) = \lambda^{\sum_{i=1}^{n} x_i} e^{-n\lambda}$ +641,"$\int_0^{100} S(x)\,dx$" +642,$A=\mathbb Z[\xi]$ +643,$a=1$ +644,$\delta=\log(1+i)$ +645,$\text{VaR}_{\alpha}(A)$ +646,$\omega_c$ +647,"$f(x_i;\theta) = g(\theta, \sum_i x_i)h(x_i)$" +648,"$p\not\in\{1, 2\}$" +649,$8$ +650,$\nu>0$ +651,$m_X(s) \to \mathsf E[X]$ +652,$X_n(\omega)$ +653,$p\neq 2$ +654,$ achieves the left-shift and $ +655,$x\in\mathbb{R}$ +656,$\sup X_n=1\not=\sup X=0$ +657,$\dfrac{e^{\theta x}}{x^{\alpha+1}}$ +658,$(\alpha-1)(1-p)=1$ +659,$\sigma>0$ +660,$m_Y(s)\to\infty$ +661,"$t\mapsto p(t, A)$" +662,$<0.01$ +663,$\frac{1}{2-p}=\frac{\alpha-1}{\alpha}$ +664,$\P(A\mid\F)(\cdot)$ +665,$y_i$ +666,"$F(\cdot, F)=\P(T\le x\mid \G)(\omega)$" +667,$\P(A\mid\G)(\cdot)$ +668,$U$ +669,$a=P+Q=\max(L)=100$ +670,$10^{-1} - 1$ +671,$2(Mg+\alpha a)w$ +672,$8.617 \times 10^{4}$ +673,$4 \times 10^3 - 10^4$ +674,$\infty$ +675,$\theta_d=0.60$ +676,$\mathsf{E}[X]$ +677,"$[0,\theta]$" +678,$a_i + b_i\ \mathit{EL}$ +679,"$(\P_S,\sigma(T))$" +680,$V_\kappa$ +681,$B=g^b\pmod p$ +682,"$(-\infty,0]$" +683,$10000 \times (1-\alpha)$ +684,"$(\omega,\omega')\in\Delta$" +685,"$p\not\in (0,1)$" +686,$\mathsf{E}[D_t D_s] = \mathsf{E}[(X_t - X_{t-1})(X_s - X_{s-1})]$ +687,$λ$ +688,"$[-100, 1000)$" +689,$ is the total return on invested assets and $ +690,$\mathcal{W}$ +691,"$\mu_0 : \mathcal{A} \to [0, \infty]$" +692,"$\{\omega\mid P(\Gamma\cap Z, \omega) = \frac{1}{2}1_\Gamma(\omega) \ \forall\Gamma\in\mathscr I\})$" +693,$2^{17}=131072<150000<2^{18}=262144$ +694,$E=hc/\lambda = 10^{-6}/\lambda$ +695,$S_n=X_1+\cdots + X_n$ +696,$(\tau^{-1})'(\mu)=1/V(\mu)$ +697,"$\int_B X\,d\P$" +698,${}_b\bar V=1-\bar a_{x+b}/\bar a_x$ +699,$(E\cap U)$ +700,"$f(s, \cdot)$" +701,$p_1$ +702,$\hat \theta>\theta_d$ +703,$y^*-x^* < \epsilon$ +704,$M < \infty$ +705,$M_\odot \approx 1.989 \times 10^{30} \ \text{kg}$ +706,$\xi:\Omega\to \mathbb{R}$ +707,$1 \times 10^{19}$ +708,$10^{20}$ +709,$\approx 45\%$ +710,$\theta_s=\hat\theta_s+\hat\sigma_s Z_s$ +711,$f(x) \mapsto kje^{\theta x} f(x)$ +712,"$ are better. By a shaping argument, assume $" +713,$\nu$ +714,$ for different values of $ +715,$\kappa'$ +716,$p\approx 0.99$ +717,$p\neq 1$ +718,$\mu\to 0$ +719,$C=B+1$ +720,$T:\Omega\to Y$ +721,$\mathsf{E}(X\mid\G)$ +722,$h_2\le w$ +723,$t\sigma^2$ +724,$\mathsf{Var}(Z)=\sigma^2\mu^p$ +725,"$50 of the amount allowed on each claim in the classes under subsections 3, 4, 4-B, 5 and 6 must be deducted from the claim and included in the class under subsection 8. Claims may not be cumulated by assignment to avoid application on the $" +726,$dt=1/n$ +727,$\mathsf{E}[x_t x_s] = \mathsf{E}[x_t \mathsf{E}[x_s \mid \F_t]] = \mathsf{E}[x_t^2]$ +728,$X_t=\mathsf{E}[Y\mid\F_t]$ +729,$A = X_1 + \cdots + X_d$ +730,$S=X+Y$ +731,$\sigma(\mathcal{A})$ +732,$\G_1\subset \G_2$ +733,$\phi$ +734,"$h=0,1/2, 1$" +735,$(\F)$ +736,$0<\mathsf{E}[X_i]\le 1$ +737,$x\not= y$ +738,$\text{VaR}_{\alpha}$ +739,$\epsilon /2^{n+1}$ +740,$c_a\mapsto c_\lambda$ +741,$-\theta$ +742,$\lim_{\mu\to 0} V(\mu)/\mu^2=\infty$ +743,$a=1.75$ +744,"$\mathrm{DM}^*(\theta, \sum_i \nu_i)$" +745,$X_{t-1}$ +746,$\mathcal{F}_{k-1}$ +747,$\kappa_t(s) = t \kappa(s)$ +748,$R$ +749,$n - 1$ +750,$\nu_B(A)=\P(B\cap T^{-1}(A))$ +751,$F_2$ +752,$x\!\urcorner$ +753,$\sin{}$ +754,$\bold x$ +755,$Y_N$ +756,$\rightarrow$ +757,$\theta=\log(\mu/(n+\mu))$ +758,"$X_n,X$" +759,"$k=0,1,\dots,n-1$" +760,$E[X_0] = 0$ +761,$10^5 - 10^{12}$ +762,$p\ge r\ge 1$ +763,"$\forall A\in \A,\ f(A) \in \B$" +764,$s$ +765,$(k \approx 8.617 \times 10^{-5} \text{ eV/K}$ +766,$f'=\partial f/\partial\mu$ +767,$M^+_F$ +768,$B\in \B$ +769,$k(i)$ +770,$p=0.9$ +771,$\mathcal{M}$ +772,$10^7$ +773,"$[2P,3P)$" +774,$N$ +775,"$p, q \in G$" +776,$m(m+1)$ +777,"$\forall \gamma>0: \int_{|y|>\gamma} K_\delta(y)\,dy=1$" +778,$2.6 \times 10^{12}$ +779,$m^*(O \setminus A) < \epsilon$ +780,"$\mathsf{CP}(\lambda,X)$" +781,$10^{-6}$ +782,$t=0$ +783,$X\times Y$ +784,"$X\sim \mathrm{St}(\alpha, \beta=1, 0, 0; S1)$" +785,"$t\in[t, t+dt]$" +786,$2.2 \times 10^{24}$ +787,$-24$ +788,"$\psi(S,T)=1$" +789,$t \geq \tau_n$ +790,$\epsilon_2$ +791,$\\hat f(x)$ +792,$b\mu_x v^b$ +793,$j(x)=1/x^{\alpha + 1}$ +794,$f:\Lambda\to\Omega$ +795,$ where $ +796,"$. As usual, reduce to premium of 1 per unit time by adjusting the time period. $" +797,$l(y;y)$ +798,$\alpha_Y \le \alpha_X < \alpha_Y + 1$ +799,$V(\mu)\propto \mu^2$ +800,$2n^2$ +801,$N'/(Mg+\alpha a) - w\le w$ +802,$p=\infty$ +803,$q \in D$ +804,$F:\Omega\times\mathbb{R}\to\mathbb{R}$ +805,$\forall y\in Y$ +806,$3.1 - 100$ +807,$c > C$ +808,$\kappa'(\tau^{-1}(\mu))=\tau(\tau^{-1}(\mu))=\mu$ +809,$\mathsf E[g(X_n)]\to \mathsf E[g(x)]$ +810,$p = 0.5$ +811,$|\phi(-2\pi f)|$ +812,"$[0,0] \succeq [-k -k]$" +813,"$T:(\Omega,\F)\to(M,\B)$" +814,$\bar a_{n\!\urcorner}$ +815,$y\in C\subset\mathbf R$ +816,$\theta=c=\nu^2$ +817,$B'$ +818,$A$ +819,$\theta=-e^{-\mu}$ +820,$\mathsf{Var}(r(X))\ge 1/\mi(\mu)$ +821,$D ⊃ C N$ +822,"$|\mathcal W(g, W)|$" +823,$(x)$ +824,$10^6 - 10^9$ +825,$\pi_1(E_1)\times \pi_2(E_2) = \P_1(E_1)\P_2(E_2)$ +826,${}_1F_1$ +827,$\mu=1$ +828,$\beta_1+\beta_2-\beta_0$ +829,$\Omega_p$ +830,$10^{15} - 10^{19}$ +831,$1/x$ +832,"$(0, \infty)$" +833,$\mathsf{Pr}(\mathsf{CP}=n)=\sum_{k\ge n}\mathsf{Pr}(\mathsf{CP}=n\mid N=k)\mathsf{Pr}(N=k)$ +834,$\mu_*(M\cap E)=\mu_*(M'\cap E)=0$ +835,$A_1 \subseteq A_2 \subseteq \cdots$ +836,$Ann+V$ +837,$K > 0$ +838,$\mu-\sigma^2/2<0$ +839,$\alpha/\beta=\alpha\mu^{p-1}/(\alpha+1)\to 1$ +840,$2^{\aleph_0}$ +841,$Y=1-X$ +842,$1 < a < 2$ +843,$x_1 < x_2$ +844,$1-2^{n-1}$ +845,$0\ge \lambda \le 1$ +846,$\Lambda$ +847,$T_1-1$ +848,$x\downarrow 0$ +849,$Z_\nu$ +850,"$B(b)\approx -b\mu_xv^b \approx {-}_bq_xv^b = -A^{\, 1}_{x:b\!\urcorner}$" +851,$1-p=\frac{1}{\alpha-1}$ +852,$\lim J(x)\to\infty$ +853,$g^mA^R=g^{m+Ra}$ +854,"$X=[0,1]^2$" +855,"$\int |X_n(\omega) - X(\omega)| \,\mathsf{Pr}(d\omega)\to 0$" +856,$\sigma^2\mu^p=\lambda\alpha(\alpha+1)/\beta^2$ +857,$M_\oplus \approx 5.972 \times 10^{24} \ \text{kg}$ +858,"$\int e^{\theta y}c(y)\,dy< \infty$" +859,$\mathsf E[\log(X_1 / X_0)]$ +860,$J_n(0) < \infty$ +861,$N\mid G$ +862,$p^*=0.752$ +863,$\omega\in B_0$ +864,$R\to\infty$ +865,$\exists x\ [\forall z\ (z=\emptyset)\rightarrow z\in x \wedge \forall x\in x\forall z\ (z=S(x)\rightarrow z\in x)]$ +866,$2^2\rightarrow 3^3-1=2\times 3^2 + 2\times 3 + 2 = 26$ +867,$3\mu(U)/2$ +868,$\mathsf{E}[X_2\mid X \ge a]$ +869,$a=0$ +870,$N=0$ +871,$r_i 0$ +926,$P_X$ +927,$1 — 1_B$ +928,$e^{st}-1\approx st + O(s^2)$ +929,$\int rf =\mathsf{E}[r]=\mu$ +930,$P_{x+b}-P_x > 0$ +931,$X_2=0$ +932,$A_1 \supseteq A_2 \supseteq \cdots$ +933,$\P(B\mid\G)$ +934,$V(\mu)=\mathsf{Var}(\mathsf{CP}_2)=\lambda(\mu/\lambda)^2x_2=\mu^2(x_2/\lambda)$ +935,$b\!\urcorner$ +936,$l = jn$ +937,$\bar\mu$ +938,$x_0\to\infty$ +939,$y_{0}=1$ +940,$r=0.045$ +941,$a$ +942,"$\phantom{P}= v\,\mathrm{EL} + d\,\max(\mathrm{loss})$" +943,"$\mathsf{CP}(\lambda, X)$" +944,$500g=4900N$ +945,$\omega$ +946,$N=40$ +947,$26 \rightarrow 2\times 4^2 + 2\times 4 + 1=41 \rightarrow 60 \rightarrow 83 \rightarrow 109\rightarrow\dots$ +948,$t=0.5$ +949,$\mathbb{R}^n$ +950,$. Thus $ +951,$X_i\sim L(1)$ +952,"$[0,x]$" +953,$a\theta=1-s$ +954,$\phi(t)\approx \phi(0)$ +955,"$\phi:(E, \mathsf{E}E)\to(\mathbb{R}, \BB(\mathbb{R}))$" +956,"$(M, \B)$" +957,"$Y\sim\mathrm{Ga}(\mu,\alpha)$" +958,$10^{23}$ +959,$1-r$ +960,$\mu+\mu\sqrt{\mu+2-2\sqrt{1+\mu}}$ +961,$\alpha>2$ +962,$Q:\Omega\times \mathsf{E}E\to\mathbb{R}$ +963,$m_X(·)$ +964,$\mathcal X\times M$ +965,$\alpha_X = \alpha_Y + 1$ +966,"$k=0,\dots, n$" +967,$V_1$ +968,$D_t = X_t - X_{t-1} = \epsilon_t$ +969,"$(0, 1)$" +970,$N=n$ +971,$D_s$ +972,$N_2$ +973,$\R$ +974,$x_1$ +975,$r \in G$ +976,$\P(A)$ +977,$13.8$ +978,$\phi\circ X$ +979,$P^T(\cdot\mid t)$ +980,$m(A) = m^*(A)$ +981,$\alpha_i(x)$ +982,"$\kappa_\nu(\theta)=\nu\,\kappa(\theta)$" +983,$\mathsf{Var}(s)=1/\sigma^2$ +984,$s < t$ +985,$L(\nu)$ +986,$\sum_i x_i$ +987,$X=1_A$ +988,$\P(A)>0$ +989,"$T:(\Omega, \F) \to (M,\B)$" +990,$\G=\sigma(T)$ +991,$c=1.124$ +992,$\frac{1}{1-p}+1=\frac{2-p}{1-p}$ +993,$J(1)<\infty$ +994,$D_t = (X_{t-1} + \epsilon_t)^2 - X_{t-1}^2 - 1 = 2X_{t-1}\epsilon_t + \epsilon_t^2 - 1$ +995,"$p:\Omega\times \F \to [0,1]$" +996,"$\mu f=\int f\,d\mu=\int f(x)\mu(dx)$" +997,$A \in \mathcal{C}$ +998,$p-1=22$ +999,$Q-0.4$ +1000,$A_n \uparrow A$ +1001,$ is the cumulant generating function for a CP with expected frequency $ +1002,$\mathsf{Var}(G)=a\theta^2$ +1003,"$\P(A)=\P(A\cap\Omega)=\int_\Omega 1_B\,d\P=\P(B)$" +1004,$g(s)=d + sv$ +1005,$a:=\lim_{m\downarrow 0} V(m)/m$ +1006,$j(x)\to\infty$ +1007,$r=0.15$ +1008,"$A_t = \min(t, T)$" +1009,$\alpha + 1$ +1010,$\mathsf{E}[D_t \mid \F_{t-1}] \leq 0$ +1011,$0\in\Theta_p$ +1012,$1/2+1/4+1/6+\cdots$ +1013,"$g(s)=\min(1, s / (1-p))$" +1014,"$F(\omega, x) = \P(X\le x\mid\G)(\omega)$" +1015,$A\in\tF$ +1016,$\bar A_{x+b}$ +1017,$\forall t\in E$ +1018,$p=1.0005$ +1019,$A_\omega$ +1020,$\BB(S)\otimes\A$ +1021,$L^*$ +1022,"$\mathsf{CP}(\lambda, X) = X_1+\cdots +X_N$" +1023,$5 \times 10^9$ +1024,$. Insurance interpretation: $ +1025,$F_X$ +1026,$10^9$ +1027,$\mathcal{C}$ +1028,$x_n\downarrow 0$ +1029,"$\mathsf{CP}(\lambda, \hat X)$" +1030,$\F=\G$ +1031,$(b-a)U_N$ +1032,$\mu$ +1033,$ is not differentiable at $ +1034,$0\le x\le 200$ +1035,$x=0$ +1036,$\mathsf{NA}(X_i) = \mathsf{CoTVaR}_p(X_i)$ +1037,$\hat\theta$ +1038,$J(x)\to\infty$ +1039,$X_1=X+b$ +1040,$\omega\in Z$ +1041,$\sigma^2$ +1042,$\omega<1/n$ +1043,$y\neq\mu$ +1044,$i\in I$ +1045,$\lambda=1$ +1046,$\alpha e^{-\beta x}/x$ +1047,$a=\mathsf{TVaR}(p^*)$ +1048,$\F_t$ +1049,$u''' > 0$ +1050,$\mathsf{Pr}(0)>0$ +1051,$\mu(E\cap U)\ge \alpha\mu(U)$ +1052,"$n_{\text{water, 20°C}} = \frac{2338 \times 1}{8.314 \times 293.15}$" +1053,$10^{-8}$ +1054,"$(x_1,x_2]$" +1055,$1/\sqrt{\alpha}$ +1056,$m(A)=0$ +1057,$\theta \neq 0$ +1058,$\epsilon>0$ +1059,$p_B$ +1060,"$P(A, \cdot) = \P(A\mid \F_1)(\cdot)$" +1061,$P_{x+b}-P_x$ +1062,"$[a, b] \subset \mathbb{R}$" +1063,${\lambda\alpha(\alpha+1)}/{\beta^2}$ +1064,"$\rho = 1000 \, \text{kg/m}^3$" +1065,$\mu-\sigma^2/2$ +1066,$\{X_\alpha\}$ +1067,"$\mu^*(E)=\inf\left\{ \sum_{n\ge 1} \bar\mu(E_n) \mid E_n\in S(R),\ E\subset\bigcup_n E_n \right\}$" +1068,"$\xi(\phi^{-1}(t)\mid t) = 1,\ \forall t\in M$" +1069,$\P(X\le r\mid\G)=E[1_{X\le r}\mid\G]$ +1070,"$a\,\dfrac{\mathsf{E}[X_1\mid X \ge a]}{\mathsf{E}[X\mid X \ge a]}$" +1071,$0 < a-1 < 1$ +1072,$t=t_0$ +1073,$\alpha<\omega_c$ +1074,$\nu\ll P$ +1075,"$[\epsilon_1, \epsilon_2] \succeq [0, \epsilon_1+\epsilon_2]$" +1076,"$L(e,t)$" +1077,$\forall X\ \exists U\ [\forall Y\ \forall x\ (x\in Y \wedge Y \in X)\rightarrow x\in U]$ +1078,"$ x = 0, 1, 2, \ldots $" +1079,$\mathsf{VaR}$ +1080,$X_n=1$ +1081,$\exists !x\phi(x)\leftrightarrow \exists x\phi(x)\wedge \forall x\forall y(\phi(x)\wedge \phi(y)\rightarrow x=y)$ +1082,$\bar P_{x+b}$ +1083,"$X_j=(x_{1j}, x_{2j},\dots,x_{Mj})^t$" +1084,$m_X(s) \to \infty$ +1085,"$\omega\in [k2^{-m}, (k+1)2^{-m}]$" +1086,$\alpha(2)=0$ +1087,"$(0,t]$" +1088,$p_\alpha$ +1089,$r-\mu$ +1090,$V(\mu)$ +1091,$10^3$ +1092,$\bar a_x = (1-\bar A_x)/\delta$ +1093,$\theta\in\tilde\Theta$ +1094,$K=B^a=g^{ak}$ +1095,"$p\not\in\{0,1,2\}$" +1096,$p=0.271$ +1097,$w(Z)/\mathsf{E}[w(Z)]$ +1098,$C_v$ +1099,$x_{\min{}}=0$ +1100,$\mu_f$ +1101,"$p\not=0,1,2,3$" +1102,$\le$ +1103,$\P(A\mid \F_1)$ +1104,$\mathsf{E}[x_t x_{s-1}] = \mathsf{E}[x_t \mathsf{E}[x_{s-1} \mid \F_t]] = \mathsf{E}[x_t x_{t-1}]$ +1105,$\hat\sigma_s$ +1106,$X_t = X_{t-1} + \epsilon_t$ +1107,$x_{i2}$ +1108,$+l$ +1109,$(\alpha-1)/\beta$ +1110,$\lambda < \kappa$ +1111,$\forall A\in\mathsf{E}E$ +1112,$x_\mathrm{range}= x_{\max{}}-x_{\min{}}$ +1113,"$50) of the amount allowed on each claim in the classes under subsections (3) to (7), inclusive, of this section, shall be deducted from the claim and included in the class under subsection (9) of this section. Claims may not be cumulated by assignment to avoid application of the fifty dollars ($" +1114,$\xi\mathbb Z$ +1115,$21$ +1116,$\delta_\mu$ +1117,$\A\otimes\B$ +1118,"$\px=\sum_i\int_B \mathsf{E}[X_i\mid \G]\,d\P$" +1119,$\G(\omega)=\{\omega\}$ +1120,$X_c$ +1121,$A \subseteq O$ +1122,$BM^2$ +1123,$w(z)$ +1124,$x+t$ +1125,$\forall x[\exists y(y\in x)\rightarrow \exists y(y\in x \wedge \neg\exists z(z\in x \wedge z\in y))]$ +1126,$\nu_B\ll T\P$ +1127,"$\Delta m_{32}^2 \approx 2.44 \times 10^{-3} \, \text{eV}^2$" +1128,$\mathrm{EL}$ +1129,$S(R)$ +1130,"$(\Omega, \F,\P)$" +1131,$1.65 - 3.1$ +1132,$J(0)<\infty$ +1133,$X_s \le \mathsf{E}[X_t \mid \F_s]$ +1134,$\tau=0.5$ +1135,$\lambda$ +1136,$1+\epsilon$ +1137,$1-1/n$ +1138,$\gamma = \frac{1}{\sqrt{1 - (v/c)^2}}$ +1139,$\P(B\mid\A)$ +1140,$\nu=\theta m$ +1141,$d=1/(1+r)$ +1142,"$\mu(E) = \inf\{ \mu(U) \mid E\subset U, U\text{\ open} \}$" +1143,"$p(\cdot, A)$" +1144,$10^{-11} - 10^{-15}$ +1145,"$\int_0^1 x^2 j(x)\,dx$" +1146,"$697.6 billion in 2016, $" +1147,$\aleph_0$ +1148,$\P(A\mid Y=y)=\mathsf{E}[1_A\mid Y=y]$ +1149,$\mathsf E(G')=1-f$ +1150,$\beta<\alpha$ +1151,$y<0$ +1152,$(y-\mu)^2$ +1153,$\lambda=1/m$ +1154,$\eta(\theta)$ +1155,$\kappa_{T_x}$ +1156,$d^2$ +1157,$l(y;\mu)=\log(c(y))+y\tau^{-1}(\mu)-\kappa(\tau^{-1}(\mu))$ +1158,$1_A$ +1159,"$Y\sim N(\mu, \sigma^2)$" +1160,$aggfft*aggfft = aggfft$ +1161,$10^{-3} - 7 \times 10^{-7}$ +1162,$\mathcal W$ +1163,$\mu_x = A+Bc^x$ +1164,$n=100$ +1165,$(c)$ +1166,$h(y) = -\log y$ +1167,$N(u) = -c_2/u^\alpha$ +1168,$1\le p\le 2$ +1169,"$\mathsf{E}[\psi(S,T)] = \displaystyle\int_M \P_T(dt)\int_L\psi(s,t)P^T_S(ds\mid t)$" +1170,$2\nu$ +1171,$B_t$ +1172,$\{X\le x\}$ +1173,$\ge 2$ +1174,"$(-1,0)$" +1175,"$\int_{\mathbb{R}} K_\delta(y)\,dy=1\ \ \forall \delta>0$" +1176,"$\mu^*(E)=\inf\left\{ \sum_{n\ge 1} \bar\mu(E_n) \mid E_n\in \bar S,\ E\subset\bigcup_n E_n \right\}$" +1177,$Z_1$ +1178,$GM/R$ +1179,$PV=N'$ +1180,$0<\alpha<1$ +1181,$2^{32}=4$ +1182,$\{T=t\}=\{t\}$ +1183,$\forall x\ \forall y\ \forall z\ (z \in x \leftrightarrow z \in y)\rightarrow x=y$ +1184,$\bar P_{40}=6908.82$ +1185,$D_i-N_i > 0$ +1186,$0.3$ +1187,$X_\alpha$ +1188,$A=0.00022$ +1189,$\mu=\tau(\theta)=\tan{\theta}$ +1190,$\{X_t\}$ +1191,"$P:\F\times M\to [0,1]$" +1192,$\frac{p}{(1-p)^2}=\frac{p}{1-p}(1+\frac{p}{1-p})$ +1193,$\mu(A) = \nu(f(A))$ +1194,$g-br-v$ +1209,$\mathsf{E}[Y]=\mu$ +1210,"$-b, b$" +1211,$\mu^0=1$ +1212,$= n \times 6.022 \times 10^{23} = 2.38\times 10^{24}$ +1213,$\sqrt{2Np}=19$ +1214,$\alpha=1/2$ +1215,$P^T_S(\cdot\mid\cdot)$ +1216,$g'$ +1217,$L_X(s)/L_Y(s) = k$ +1218,"$P(\omega, B)=\P(B\mid\G)(\omega)$" +1219,$\mathbb{Q}$ +1220,$\{x\}=x-\lfloor x\rfloor$ +1221,$k/n$ +1222,$e^{-\lambda\nu}$ +1223,"$f_0, f_{1/2}, f_1$" +1224,"$\iota: x\mapsto (x, Tx)$" +1225,$F = m v^2 / r$ +1226,"$j=1,2, \dots, d$" +1227,$m_X(s)\to\infty$ +1228,$g_n$ +1229,"$\mathit{Tw}_p(\mu, \sigma^2)$" +1230,$\omega = \dfrac{d\theta}{dt}$ +1231,"$\psi(S,t)=1_{\{t\}}$" +1232,$10^{10}$ +1233,$t = 1$ +1234,$-1.805$ +1235,$\propto$ +1236,"$n=2,3$" +1237,$P=(kN/V)T\propto$ +1238,"$\mathsf{CP}(\lambda, \mathrm{gamma}(\alpha, \beta))$" +1239,$M=E_0+B$ +1240,$\le 1$ +1241,$V(\mu)=\kappa''(\tau^{-1}(\mu))$ +1242,"$200 of losses otherwise payable to any claimant under this subsection. All claims under life insurance policies and annuity contracts, whether for death proceeds, annuity proceeds or investment values, must be treated as loss claims. Claims may not be cumulated by assignment to avoid application of the $" +1243,"$ makes the left tail thinner, the right tail thicker, and increases the mean. The effect on the right tail is manageable because it is thinner than a normal, @Zolotarev1986, @Carr2003a. As $" +1244,$\bar X$ +1245,$\mathsf{E}[e^{sX_1}]=\mathsf{E}[e^{sX_{1/n}}]^n$ +1246,$X_1 + X_2 \sim 2^{1/\alpha}X$ +1247,"$x\in\Omega,L, t\in M$" +1248,$f\in L^1(\mathbb R)$ +1249,$H$ +1250,$\alpha=\alpha(p)$ +1251,$A \in \mathcal{F}$ +1252,$m_X(s)$ +1253,$\bfx$ +1254,$\phi(t)$ +1255,$V(m) = m^3V^*(1/m)$ +1256,$t < s$ +1257,$C_p$ +1258,$CV=\nu=\sqrt{a}\theta$ +1259,$f(L)\in \BB$ +1260,$\beta_i$ +1261,$\tau^{-1}(\mu)=-1/(2\mu^2)$ +1262,$\psi$ +1263,$F((k+1/2)b)-F(k-1/2)b)$ +1264,$3.2 \times 10^{18}$ +1265,$10^{-2}$ +1266,$v$ +1267,$n>0$ +1268,$\backslash$ +1269,"$(X, \Sigma)$" +1270,$X_t = e^{B_t - t^2/2}$ +1271,$\mathsf{Pr}(X ≥ x_0) = 1-p$ +1272,$1-p$ +1273,$t \neq s$ +1274,$\int c(y)dy = 1$ +1275,$P=53.565$ +1276,$\ge 0.98$ +1277,"$\mathsf{cov}(m_X(S), m_Y(S))\ge 0$" +1278,$\lambda_x\uparrow \infty$ +1279,$\Theta=\{0\}$ +1280,$\P_X$ +1281,$p\ge 1$ +1282,$\Delta G$ +1283,$\hat F(x)=\sum_{i:x_i\le x} \lambda_i/\lambda$ +1284,$X_1=aX+b$ +1285,$x_t$ +1286,"$\kappa_{T_x}=x\,\kappa_{T}$" +1287,$\mathsf{Pr}(X_1)$ +1288,$b(\theta)=e^{-\kappa(\theta)}$ +1289,$\P HX$ +1290,$\cap$ +1291,$\tau_n$ +1292,$\alpha=-1/2$ +1293,"$200 of losses otherwise payable to any claimant under this subsection other than the federal government. All claims under life insurance and annuity policies, whether for death proceeds, annuity proceeds or investment values, shall be treated as loss claims. Claims may not be cumulated by assignment to avoid application of the $" +1294,$G=f+G'$ +1295,$8.617 \times 10^{10}$ +1296,$A\in\S$ +1297,$ is an interior point of $ +1298,$m$ +1299,$f$ +1300,$\forall\omega\in\Omega$ +1301,$\mathbb{R}$ +1302,"$L(e,t)=eR_t$" +1303,$100$ +1304,$10^{-3} - 10^{-1}$ +1305,$\mathsf{E}[s]=0$ +1306,"$(M,\B)$" +1307,$> \mathsf{VaR}$ +1308,$M_t = t$ +1309,$x\in D(E)$ +1310,$X_\nu/\nu$ +1311,$Z_c(3900)$ +1312,$P_t\{T\not=t\}=0$ +1313,$M=\Omega$ +1314,$p^*$ +1315,$k/T=\beta$ +1316,$1/r$ +1317,$a=0.75$ +1318,"$T(y)=(y, y^2)$" +1319,$\kappa_T(y)=-\sqrt{-2y}$ +1320,$X = (x_{ij})$ +1321,$\tau \leq T$ +1322,"$ power is convolution, giving the hitting probability to the level $" +1323,$\A$ +1324,$y\mapsto P_y(E)$ +1325,$m(F)=0$ +1326,"$\mathsf{CP}_2(\lambda, (\mu/\lambda)X)$" +1327,$V(\mu)=\mu^2$ +1328,$x^20$ +1337,$\int X_n\to 0$ +1338,$\mathsf P(A\mid \mathscr G)$ +1339,"$\omega = [0, 1]$" +1340,$\mathsf E[X] = \mathsf{TVaR}_0(X)$ +1341,"$\omega\mapsto Q(\omega, B)$" +1342,"$Z \sim \mathrm{DM}^*(\theta, \nu)$" +1343,$X_k - X_{k-1}$ +1344,$=\mathsf{E}[X]/(1-p^*)$ +1345,$k(0)=\log(1)=0$ +1346,$X_n=X$ +1347,$B'=\bigcup_i G_i'$ +1348,$X_s \ge \mathsf{E}[X_t \mid \F_s]$ +1349,$x\neq 0$ +1350,$g=f+\epsilon 1_B>f$ +1351,$m(A)$ +1352,$\sigma(T)$ +1353,$\mathsf{E}[X_1] / \mathsf{E}[X]$ +1354,"$\mathsf{CP}(\lambda_i, x_i)$" +1355,$u=0.271>0$ +1356,$\mathsf{E}E$ +1357,$\mathrm{NEF}(c)$ +1358,"$\mathrm{inf}\,S=0$" +1359,$S\P=\P_S$ +1360,$\mathcal{A}$ +1361,$p_t(A)$ +1362,$\{y\mid c(y)\neq 0\}$ +1363,$b=0.53$ +1364,$d^*(0)=0$ +1365,$L'$ +1366,$\log(c(y))$ +1367,$Y\sim$ +1368,"$j=1,\dots,d$" +1369,$\lambda=\sum_i\lambda_i$ +1370,$P_Y(y)=\P(Y\le y)$ +1371,$\mathsf{Pr}(Z=0)$ +1372,$\mathsf{E}[Z]=\mu$ +1373,$\mu A=\mu 1_A$ +1374,$x_n(\mathrm{Po}(\lambda_n) - \lambda_n)$ +1375,$ for $ +1376,$\P H\xi$ +1377,$g\in \mathcal{W}$ +1378,$\mu(U)>1/k$ +1379,$\mu\sqrt{1+2\mu}$ +1380,$\bar\theta_s$ +1381,$M\times d$ +1382,$1 \times 10^{15}$ +1383,$ be the waiting time until accumulated surplus equals $ +1384,"$(-1,-1/2)$" +1385,$1/\nu$ +1386,$+$ +1387,$b > 0$ +1388,$K(x)=(\pi(x^2+1))^{-1}$ +1389,$2 \times 10^{14}$ +1390,$\mu = \dfrac{m_1 m_2}{m_1 + m_2}$ +1391,$E\cap F=(E\cup F) \setminus (E\triangle F)$ +1392,$\log(1-\Phi(x))$ +1393,$\frac{1}{2}mv^2 = \frac{3}{2}kT$ +1394,$(1 + \mu^2)^{3/2}$ +1395,$\P(T\in A\mid\G)(\cdot)$ +1396,$\omega=\exp(-2\pi i / n)$ +1397,$1.25 \times 10^{14}$ +1398,$\pm\infty$ +1399,$< \cdots <$ +1400,$\text{VaR}_\alpha$ +1401,$\mathit{PFL}$ +1402,"$ is average invested assets, equal to $" +1403,$V(m)=m$ +1404,$V(\mu)=1$ +1405,"$U_n(a, b)$" +1406,$(x^{-1}-x^{-3})\phi(x)$ +1407,$\omega\mapsto \P(A\mid G)(\omega)$ +1408,$f=1_A$ +1409,"$=\displaystyle\int_B^{\phantom{X}} \mathsf{E}[X\mid T=t] \,\P_T(dt)\quad$" +1410,"$\psi(s,t)=1_B(t)\mathsf{E}[X\mid S=s,T=t]$" +1411,$\mathsf{E}[Y_\theta]=y$ +1412,$\theta=0$ +1413,$l(t)=1$ +1414,"$\displaystyle\int H\xi\,d\P$" +1415,$0.87$ +1416,$\mathsf{E}[X_1\mid X \ge a]$ +1417,$83.3=100/1.2$ +1418,$f\to\infty$ +1419,$G:=T_1-1$ +1420,$\P(C) = 1$ +1421,$\mathsf{CP}_2$ +1422,$\frac{an^2}{V^2}$ +1423,$r=1$ +1424,$\G$ +1425,$6\times 10^{14}$ +1426,${{}_tp_x} \mu_{x+t}$ +1427,"$(n,p)$" +1428,$\BB$ +1429,$\sec\theta$ +1430,$p=3$ +1431,$X_t = \mathbb{E}[X \mid \mathcal{F}_t]$ +1432,"$=\displaystyle\int_B^{\phantom{X}} \mathsf{var}phi \,d\P_T\quad$" +1433,"$P(A,t)=P_t(A)$" +1434,$X_n(\omega)=n$ +1435,"$(\Omega,\F)$" +1436,$d^*(x)\neq 0$ +1437,$\{\lambda_t\}$ +1438,"$Z=\mathsf{CP}(\lambda,\text{gamma}(\alpha,\beta))$" +1439,"$0,1,2,\dots$" +1440,$N=1000$ +1441,$\prod_{n\ge N}(1-\frac{1}{n})=0$ +1442,$17$ +1443,$\mathsf{TVaR}$ +1444,$\P_S=\P_T P^T_S$ +1445,$c=-\Gamma(-a)(c_1 + c_2)\cos (a\pi/2)$ +1446,$b_i$ +1447,$V_j$ +1448,$C_p / C_v$ +1449,$\beta_i(x) / \alpha_i(x) > 1 > S(x) / g(S(x))$ +1450,$X=X_1+X_2$ +1451,$E[X_\tau] = 1$ +1452,$\\cdots$ +1453,$\hat F$ +1454,$5 \times 10^{14}$ +1455,$\mathsf{E}(G)=a\theta$ +1456,"$S,T$" +1457,$f^{-1}(\F)$ +1458,"$\Gamma(S,T)\in\A\otimes\B$" +1459,$2$ +1460,$\dfrac{1}{1-p^*}\displaystyle\int_{1-p^*}^1 q(s)ds = q(p) = 100.0$ +1461,$ρ(X)$ +1462,"$t=0,1,2,\dots$" +1463,$1500 = 250 \times (1+11)/2$ +1464,$z^n$ +1465,$\theta=\log(\mu)$ +1466,"$(1,2)$" +1467,$(\delta(t-1) + \delta(t+1))/2$ +1468,$y-\mu$ +1469,$\kappa(\theta)=-\log(-\mu)$ +1470,$\theta\mapsto -\kappa(\theta)$ +1471,$\tilde \Theta$ +1472,$\phi\mu \ll \nu$ +1473,$V(\mu)=\mu$ +1474,$t=2$ +1475,$\px=\sum_i \P(B\cap A_i)$ +1476,"$g(s) = \min(1, s / (1-p))$" +1477,$\kappa(\theta)=n\log(1+e^\theta)$ +1478,$x^{-\alpha}$ +1479,"$\mathsf{CP}_n:=\mathsf{CP}(J_n(0), X_n)$" +1480,$j_n(x)=j(x)\wedge n$ +1481,$\mu_*$ +1482,"$(B, Km)$" +1483,$\theta=(1-f)/a$ +1484,$\kappa''(\theta)=\tau'(\tau^{-1}(\mu))=1/(\tau^{-1})'(\mu))=V(\mu)$ +1485,$x_i+y_{k(i)}$ +1486,$7$ +1487,$)$ +1488,$d^*(x)=x^2$ +1489,$\mu=\kappa'(\theta)$ +1490,$P_c(4380)^+$ +1491,$\mathsf{E}[X\mid T]$ +1492,$\beta_0+\beta_1$ +1493,$\log(1-1/n)<-1/n$ +1494,$X_t = \mathsf{E}[X \mid \F_t]$ +1495,$\P(\bigcup_i B_i \mid\G)_{\omega_0} = \sum\P(B_i\mid\G)_{\omega_0}$ +1496,$\P(\cdot\mid\G)$ +1497,$x_0+x_1+x_2$ +1498,$\A_y$ +1499,$1\le p<\infty$ +1500,$\alpha\to\infty$ +1501,"$\alpha\in (0, 2]$" +1502,$z$ +1503,$t\pi/2$ +1504,"$\Gamma(\alpha,\beta)$" +1505,$\mathsf{E}[Y]=\mu=np$ +1506,$\P(\cdot\mid \G)(ω)$ +1507,$Z_s$ +1508,$\mathsf{E}[x_s \mid \F_t] = x_t$ +1509,$\{X=x\}$ +1510,$f = J/2^N$ +1511,"$\P(B\cap A)=\int_B \P(A\mid\G)\,d\P$" +1512,$\alpha < 1$ +1513,$V(m)=mW(m)$ +1514,$>0$ +1515,$\test$ +1516,$-br-v=0.258$ +1517,$x_1\leftrightarrow y_1$ +1518,$n\ge 3$ +1519,$10^{16}$ +1520,$p=0.74$ +1521,$-\frac{1}{2}$ +1522,$e^t$ +1523,$10^2$ +1524,$nb=P$ +1525,$\P H\xi = \P HX$ +1526,$\{X \ge a\}$ +1527,$\xi(\omega)=\xi(X \mid \G)(\omega)$ +1528,$\mathsf{E}[e^{tY}]=\sum \mathsf{E}[(tY)^n/n!]$ +1529,"$B_1,B_2,\dots$" +1530,$10^6$ +1531,$q \leq p$ +1532,"$\omega\mapsto P(\omega, B)$" +1533,$e^{-k(x/\alpha)^{\alpha/(\alpha-1)}}0$ +1540,"$B\mapsto p(t, B)$" +1541,$X>a$ +1542,$\alpha + Mg/a$ +1543,$X(t)$ +1544,$x^{-\alpha-1}e^{\theta x}$ +1545,$n^3$ +1546,$x>0$ +1547,$1 - T$ +1548,$v = \omega r$ +1549,$\alpha_X$ +1550,$-7$ +1551,$g \leftrightarrow$ +1552,$x\ge 0$ +1553,$\frac{\partial f}{\partial \mu}=\frac{\partial f}{\partial \mu}\frac{f}{f} = \frac{\partial l}{\partial \mu}f$ +1554,$t/|t|$ +1555,$x - y$ +1556,$F(x)$ +1557,$0<\alpha\le 1$ +1558,$\alpha=\infty$ +1559,$Y_n=n-S_n$ +1560,$\mu^*(M\cap E)=\mu(E)$ +1561,$m^*$ +1562,$\nu\uparrow\infty$ +1563,$J(0)=\infty$ +1564,$g'(s) \ge 0$ +1565,$e^{ct}$ +1566,$B \in \F$ +1567,$\lambda_t \Omega$ +1568,$\int c(y)dy\neq 1$ +1569,$\approx 10^{-40}$ +1570,$\Sigma$ +1571,"$\P(A\cap G)= \int_G \P(A)\,d\P$" +1572,$\tau=0.156$ +1573,$t > T$ +1574,$\nu(B)=\P 1_B\xi$ +1575,$E[X \mid \mathcal{F}_t] - E[X \mid \mathcal{F}_{t-1}]$ +1576,$\mathsf{CTE}_p <= \mathsf{CTE}^+_p <= \mathsf{TVaR}_p$ +1577,$X=1_B$ +1578,"$\Omega:=\tau(\mathrm{int}\,\Theta)$" +1579,$S=\bigcup_j D^n_j$ +1580,"$(-\infty,0)$" +1581,$\mathsf{E}[X_T]=\mathsf{E}[X_0]$ +1582,"$e',s', r', Q$" +1583,$F_1$ +1584,$d \bar S/da$ +1585,$10^{-2} - 10$ +1586,$V(\mu)=\mu^2/\alpha$ +1587,$\rho(X) = (1-r)\mathsf E[X] + r\mathsf{TVaR}_p(X)$ +1588,"$u_A, u_E$" +1589,$\Delta S$ +1590,$\P(\cdot\mid\A)$ +1591,$\Omega= \mathrm{int}\ \mathrm{conv}(S)$ +1592,$\P_T(B) = \P(T^{-1}(B)) = \P(T\in B)$ +1593,$\bar P_x:=\bar A_x / \bar a_x$ +1594,$2^8-1=255$ +1595,$105$ +1596,$6\times 16= 96$ +1597,$P(A | \mathcal{G})(\omega)$ +1598,$\le 0$ +1599,$10^{-12}$ +1600,$\bar S$ +1601,$P^T(F\mid t)$ +1602,$g(s) = vs + d$ +1603,$N=kg m/s^2$ +1604,$2.38\times 10^{24}$ +1605,$\sum_n x_n^2\lambda_n$ +1606,$0$ +1622,$P^T(A\mid t)$ +1623,$\mathsf{E}[X\mid Y]$ +1624,$\lambda=\mathsf{E}[X]<1$ +1625,"$\int_B \xi\,d\P$" +1626,$E=\Omega$ +1627,$V'$ +1628,$e'=0.24$ +1629,$\lambda=\lambda_1 + \lambda_2$ +1630,$\nu_0$ +1631,"$k=0,\dots,n-1$" +1632,$< 10^{-22}$ +1633,$\mathring\Theta$ +1634,"$\partial\eta/\partial\beta_1=0,1,2$" +1635,$A_x\setminus A_y$ +1636,$j(x)=x^{-5/2}$ +1637,$y_{2}=3$ +1638,$b - a$ +1639,$p \in P$ +1640,$\theta=s\theta_1+(1-s)\theta_2$ +1641,$\delta$ +1642,$X_n = 1$ +1643,$A=f^{-1}(B)$ +1644,$Z = Y\lambda$ +1645,$\mu^2$ +1646,$\delta^{18}O$ +1647,$X_n = 2^n \cdot I(A_n)$ +1648,$\{ P_t\}_t$ +1649,$\beta=0.57$ +1650,$L=L(\nu)$ +1651,$X_{T_x}=x$ +1652,"$\sim 696{,}000 \ \text{km}$" +1653,$\eta(\theta)=\theta$ +1654,$E[X_{n+1} \mid \mathcal{F}_n] \leq X_n$ +1655,$10^{12} - 10^{15}$ +1656,$t0$ +1686,$\{X_n\}$ +1687,$X_n$ +1688,$dx$ +1689,$=14\times 16+2=226$ +1690,"$\int f\,d\mu$" +1691,$10^4 - 10^6$ +1692,"$x\in(-\frac{1}{2}\mu(U), \frac{1}{2}\mu(U))$" +1693,$f(kb)$ +1694,$\sigma_s$ +1695,$k$ +1696,$-stable distribution with Lévy density $ +1697,$J$ +1698,$\sum_x dx = a$ +1699,$\theta_1$ +1700,$t=b$ +1701,"$(2,\infty)$" +1702,"$1, =$" +1703,$p_1+p_2=1$ +1704,"$X:(\Omega,\F)\to(\mathbb{R},\BB(\mathbb{R}))$" +1705,$X_i$ +1706,"$\mathsf{Var}(X_x)=\mathsf{Var}(\mathsf{CP}(j_n(x)\delta, x))=x^2j_n(x)\delta$" +1707,$\exp(\sigma \sqrt tZ)=\exp(\sigma^2t/2)$ +1708,$\kappa_X$ +1709,$. Let $ +1710,$B=\{p_\alpha\}$ +1711,$2.0-3.0$ +1712,"$\alpha_X\in(\alpha_Y+1/3, \alpha_Y+1/2)$" +1713,$-8$ +1714,"$(L,\A)=(\Omega,\F)$" +1715,$\P(B)=0$ +1716,$\P(A\cap B)=\P(A)\P(B)=\int_B P(A)\P_T(dt)$ +1717,$m=\kappa'(\theta)$ +1718,"$(\Omega,\F,\mu)$" +1719,"$A_0,\dots,A_r\in\F$" +1720,$\omega\in J$ +1721,"$T:(\Omega, \F)\to(E,\mathsf{E}E)$" +1722,"$\{ \omega\mid p(\omega, A)=1_A(\omega),\ \forall A\in\G \}$" +1723,"$1_{[0,1]}$" +1724,$\sigma^2/n_i$ +1725,$Y=X_1+\cdots +X_N$ +1726,$p=0.99$ +1727,$e^{\theta x}j(x)$ +1728,$\bar a_x = \bar a_{x:b\!\urcorner} + v^b{}_bp_x\bar a_{x+b}$ +1729,$0\le t\le T_x$ +1730,$X \mapsto kX$ +1731,$r(X)$ +1732,$f(x)$ +1733,$r_1$ +1734,$q(g)$ +1735,$t-1$ +1736,$ct$ +1737,$P=vL + da$ +1738,$f(x)=\cos(2\pi\omega x)$ +1739,"$X,Y$" +1740,$\alpha<0$ +1741,$t-s$ +1742,"$r=0.05, 0.15, 0.25$" +1743,$f(x)=x^{-\alpha}L(x)$ +1744,"$X,Y,T$" +1745,$G\in\mathscr G$ +1746,$s\downarrow 0$ +1747,$\tilde y$ +1748,$N' := kNT$ +1749,$N=365$ +1750,$l(y;\theta)=\log(c(y)) +y\theta-\kappa(\theta)$ +1751,$g^a=g^{\log_g(n)}=n$ +1752,"$(\mu,\sigma^2)$" +1753,$m^2W(m)$ +1754,$m(m-1)$ +1755,$-2$ +1756,$\bar P_x$ +1757,$Y=y$ +1758,$0\not\in\Omega_p$ +1759,$N'$ +1760,$R>0$ +1761,$A=g^a \pmod p$ +1762,$\{\tau_n\}$ +1763,$q \in G$ +1764,$2T+U=0$ +1765,$.} with $ +1766,$2.38 \times 10^{24}$ +1767,$Y$ +1768,$X\mapsto aX+b$ +1769,$A'$ +1770,$P_y$ +1771,$W(\cdot)$ +1772,$X = 1$ +1773,$\bar\alpha=-\alpha$ +1774,$x^{-\alpha-1}e^{\theta x}/x$ +1775,$u_i$ +1776,$2.592 \times 10^{16}$ +1777,$j(x)$ +1778,$\P(B)=1$ +1779,$M_T(y)=w(e^{y})$ +1780,$\mathsf{CV}(G) = \mathsf{SD}(G') = \nu$ +1781,$A_n \in \mathcal{C}$ +1782,"$n=1,\dots,6$" +1783,$c(y)=\binom{n}{y}$ +1784,$\kappa_1$ +1785,$N':=kN$ +1786,$L-L^*$ +1787,$t \geq T(\omega)$ +1788,"$(Y,\B)$" +1789,$\mathscr I$ +1790,$2\pi$ +1791,$\mathsf{E}[|X_t|] < \infty$ +1792,$2K + U = 0$ +1793,$\mathsf{E}[X_{t+1} \mid \F_t] = X_t$ +1794,$6.3\times 10^{11}$ +1795,$\mu^*$ +1796,"$(T,\mu)$" +1797,"$Ca(Mg,Fe)Si_2O_6$" +1798,$x^*$ +1799,$10^4$ +1800,$Z(ω)$ +1801,"$Q(\omega, \cdot)$" +1802,"$L=L(e,t)$" +1803,$10$ +1804,$y=0$ +1805,"$30,000 per accident up to $" +1806,$M_B(t)=(1-p)+pe^t$ +1807,$\sigma(X)$ +1808,$p_n(a)$ +1809,"$\theta\in\Theta\setminus\mathrm{int}\,\Theta$" +1810,$1.7 \times 10^{-3}$ +1811,"$(\Omega\times M, \F\otimes\B)$" +1812,$\tau(\theta)=\kappa'(\theta)$ +1813,$|X_\alpha|$ +1814,$m_1$ +1815,$m_X$ +1816,$\psi=1_\Gamma$ +1817,$\\infty$ +1818,$\mathsf{E}[e^{sX_1}]^{1/n}=\mathsf{E}[e^{sX_{1/n}}]$ +1819,$j(x)=x^{-3/2}$ +1820,$\B\subset \F$ +1821,$Nk$ +1822,$10^{-5}$ +1823,$U_N$ +1824,$\{X_{t \wedge \tau}\}$ +1825,$p_k=F((k+1/2)b)-F(k-1/2)b)$ +1826,$1/8.6$ +1827,$E[X_t \mid \mathcal{F}_{t-1}] = X_{t-1}$ +1828,$-15$ +1829,$dx/x$ +1830,"$50 of the amount allowed on each claim in the classes under paragraphs II, V, and VI except claims of the guaranty associations as defined in RSA 404-B, 404-H, 404-D, and 408-B shall be deducted from the claim. Claims may not be cumulated by assignment to avoid application of the $" +1831,$u=-0.295$ +1832,$\forall\omega$ +1833,"$100 billion; and costs associated with providing regulated insurance paper, such as underwriting, product management, regulatory and compliance costs, taxes licenses and fees, billing, policy maintenance and policy issuance, of nearly 10% of premium, or $" +1834,$p=1.005$ +1835,$r=0.038$ +1836,$X=X_s + X_l$ +1837,$\mathsf{Pr}(X_n=1)=1/n$ +1838,"$(-\epsilon, \epsilon)$" +1839,"$A=\mathbb Q\cap [0,1]$" +1840,$L^1(\mathbb R)\to L^1(\mathbb R)$ +1841,$a\to 2$ +1842,$MgAl_2O_4$ +1843,$6 \times 10^{}$ +1844,$-\alpha$ +1845,$\hat f(t)$ +1846,$\bar y - \kappa(\theta)=0$ +1847,"$\partial d/\partial\mu=-2\,\partial l/\partial \mu$" +1848,$\lambda=\beta/\alpha$ +1849,${}_b\bar V$ +1850,$-2< -\alpha<-1$ +1851,$\to$ +1852,$G$ +1853,$\int_0^\epsilon (e^{sx}-1)j_n(x)dx \approx \int_0^\epsilon sxj_n(x)dx$ +1854,$\theta=-\alpha/\mu$ +1855,$A\in\G$ +1856,"$(\mu,\alpha)$" +1857,$\mathsf{E}[z^T]$ +1858,$x_{\max{}}$ +1859,$0$ +1860,$\mathsf{Var}(aX)=a^2\mathsf{Var}(X)$ +1861,"$\mathsf E_g|\mathcal W(g, W)|$" +1862,$\Gamma(\alpha):=\int_0^\infty x^{\alpha-1}e^{-x}dx$ +1863,$V(\mu)=1/(\tau^{-1})'(\mu)=\mu^2$ +1864,"$Y=[0,1]$" +1865,$3$ +1866,$0\le t<\beta$ +1867,$\G\subseteq \F$ +1868,"$\{(\A_y, P_y)_{y\in Y} \}$" +1869,$σ$ +1870,$10^{13}$ +1871,$g^{kS}=R^S=g^{m+Ra}=g^mA^R$ +1872,$2^{-(i+j)}$ +1873,$\gamma = 2/\sqrt(a) = 2\nu$ +1874,$\sigma=1$ +1875,$\alpha<2$ +1876,$0\le \tau\le 1$ +1877,$y-\kappa'(\theta)=0$ +1878,$F_\alpha$ +1879,$C_p = \frac{7}{2}R$ +1880,$N\times 1$ +1881,$g$ +1882,$\mathsf{Pr}(X_n>\epsilon)\to 0$ +1883,$agg$ +1884,"$\F,\A,\B$" +1885,$E'=X\setminus E$ +1886,$\mu g=0$ +1887,$\sigma$ +1888,$1/V$ +1889,$\mathsf{Var}(Y)=np/(1-p)^2$ +1890,$\exp(\mu t)$ +1891,$\{a_n\}$ +1892,$1-p_s>0.5$ +1893,$1.38 \times 10^{-23}$ +1894,$\mathsf{E}[\mathsf{E}[X\mid \G_2]\mid \G_1]=\mathsf{E}[X\mid \G_1]$ +1895,$x \!\!\urcorner$ +1896,$10^5$ +1897,$n=2^m+k$ +1898,$\mathrm{NE}(\mu)$ +1899,$\{A_i\}$ +1900,$D_i$ +1901,$P(\omega)$ +1902,$\kappa'(\theta)=\mu$ +1903,$P_i+Q_i$ +1904,$10^{15}$ +1905,$X_l$ +1906,$G_i$ +1907,$d=2$ +1908,$X_{t \wedge \tau_n}$ +1909,$10^{-1} - 10^{-3}$ +1910,"$195 million or three points of losses on natural catastrophes including Hurricane Ida, the floods in Europe, and Winter Storm Uri. This compares to a combined ratio of 98 for 2020 which included $" +1911,$|x|$ +1912,$p>2$ +1913,"$\{\emptyset, Ω\}$" +1914,$c_0(y)=c(y)e^{l(y;y)}$ +1915,"$(0,t_1+t_2]$" +1916,$Km\pmod p$ +1917,$x_i=n_i + i\xi$ +1918,$|x|^2$ +1919,"$\beta\in[-1,1]$" +1920,$P^T_S(A\mid t)$ +1921,$\max X_i$ +1922,"$\forall \omega\not=\omega',\ \exists G\in\G:\ 1_G(\omega)\not=1_{G'}(\omega')$" +1923,$K_\delta(x) = K(x/\delta) / \delta$ +1924,$N=(1-\alpha)M$ +1925,$X_t = \exp(B_t - t^2/2)$ +1926,$\alpha\ge 1$ +1927,$Y_t=\exp(B_t - t/2)$ +1928,$A \in \mathcal{A}$ +1929,$\mathsf{E}[x_{t-1} x_s] = \mathsf{E}[\mathsf{E}[x_{t-1} x_s \mid \F_{t-1}]] = \mathsf{E}[x_{t-1} \mathsf{E}[x_s \mid \F_{t-1}]] = \mathsf{E}[x_{t-1} x_{t-1}] = \mathsf{E}[x_{t-1}^2]$ +1930,$\Omega=N\cup (N+1)$ +1931,$\alpha\le 1$ +1932,$X=0$ +1933,$\sigma_n^2 = x_n^2\lambda_n$ +1934,$r = r_1 + r_2$ +1935,$T(X)$ +1936,$\mathsf{Pr}(N=n)=e^{-\lambda}\lambda^n/n!$ +1937,$ is allowable (oscillating) but tilts with mean $ +1938,$5 \times 10^{10}$ +1939,$m_2$ +1940,$\sigma^2=1/\lambda$ +1941,$\kappa(\theta)=\log \sum_n e^{\theta n}/n!$ +1942,$697.6 billion underlying @tbl-equity-what-if this implies $ +1943,$8.617 \times 10^{22}$ +1944,$c(y)$ +1945,$(\alpha-1)(p-1)=-1$ +1946,$\cos{}$ +1947,"$Mg,Fe)SiO_3$" +1948,$n\times n$ +1949,$Y_\nu$ +1950,$(\beta)$ +1951,$A\in\A$ +1952,"$S:(\Omega, \F) \to (L,\A)$" +1953,$10 - 10^2$ +1954,$0<\alpha< 1$ +1955,$E\subset U_0$ +1956,$0.01 \text{ kWh/GB}$ +1957,$\bar A^{1}_{x:n\!\urcorner}$ +1958,$N=\sum_{i\in I} N_i + N_a$ +1959,$l = L/P$ +1960,$2/\sqrt{a}= 2\nu/(1-f)$ +1961,$m_i(s)\to\mathsf E[X_i]$ +1962,$<1$ +1963,$\F\subset\mathscr P(\Omega)$ +1964,$b=1/f$ +1965,$y=-x$ +1966,$b>0$ +1967,$\mathsf{Pr}(T=n)$ +1968,"$n=1,2,3,\dots$" +1969,$C_v = \frac{5}{2}R$ +1970,$PV = nRT$ +1971,$10^{9}$ +1972,$A=g^a\pmod p$ +1973,"$\int_K^\infty \mathbb{P}(|X_i| > t) \, dt$" +1974,$n\ge 1$ +1975,$-4$ +1976,$\mathsf{E}[D_t \mid \F_{t-1}] = \mathsf{E}[X_t - X_{t-1} \mid \F_{t-1}] = 0$ +1977,"$P(\{ω\}, ω) = 1$" +1978,$\kappa(\theta)$ +1979,$b=-1$ +1980,$w$ +1981,$\lambda(e^\theta-1)$ +1982,$e^{\theta x - t\kappa_X(\theta)}f_t(x)$ +1983,$-br-v=0.341$ +1984,$V(0)=1$ +1985,$\mathbb{R}\to\mathbb{R}$ +1986,"$\psi(S(\omega), t)=1_{L(t)}(S(\omega))$" +1987,"$\forall E\in\A,\ \exists N\in\B:\ QN=0$" +1988,$\Phi$ +1989,"$X = \{X_t, t \geq 0\}$" +1990,$D_0 = \mathsf{E}[X]$ +1991,"$\phi:(L, \A) \to (M, \B)$" +1992,$\mathscr G$ +1993,$F^{\times}_{359}$ +1994,$m_1=m+b$ +1995,$\exp(h(y))$ +1996,$\theta=\mu$ +1997,"$R = 8.314 \, \text{J/(mol·K)}$" +1998,$nb$ +1999,$E\setminus E\in R$ +2000,$V(\cdot)$ +2001,$X_s$ +2002,"$P(N, ω) = 1$" +2003,$ is a gamma process with density $ +2004,$N'/\alpha a\le w$ +2005,$1 \le l \le n-1$ +2006,$kP$ +2007,$ and rate $ +2008,$:=a$ +2009,"$\mathcal{W}(g, W)$" +2010,$f_X$ +2011,$1 \times 10^{16}$ +2012,${}_tq_x=1-{{}_tp_x}$ +2013,$\mathsf{E}[X_T] = \mathsf{E}[X_0]$ +2014,$y=kx$ +2015,$\epsilon$ +2016,"$[\theta_r, \theta_d]$" +2017,$Y_n$ +2018,$p=(2+\bar\alpha)/(1+\bar\alpha)$ +2019,$\alpha/\beta$ +2020,$Y=e^{-X}$ +2021,$\mu h<\infty$ +2022,$10^{-18} - 10^{-22}$ +2023,$\px=\P(\bigcup_i (B\cap A_i)$ +2024,"$, $" +2025,$i$ +2026,$s_3$ +2027,$g'(S(X))$ +2028,$g(t)=h(t)\{ l(t) < 1\}$ +2029,$a<1$ +2030,$v_X = \text{Var}_{0.99}(X)$ +2031,"$(L, \A)$" +2032,$\mathsf{E}[1_A\mid\G]$ +2033,$=kA$ +2034,$11$ +2035,$h_2$ +2036,$\log_2\ge 1$ +2037,$\bar L_i$ +2038,"$\A=\{A_1,A_2,\dots\}$" +2039,"$i=A,E$" +2040,$\mathsf{E}[N]$ +2041,$g=\mathsf{E}(G^3)=\nu^3 \mathsf{skew}(G')+3c+1$ +2042,$C$ +2043,$\mu=-\sigma^2/2$ +2044,$X_\infty=\mathsf{E}[Y\mid\F_{\infty^-}]$ +2045,$\mathsf{Pr}(X_n=0)=1-1/n$ +2046,$\mu(A) = \nu(A)$ +2047,$(T\lambda)g=\mu g$ +2048,$g(s)=vs +d > s$ +2049,$\mathsf{E}[X]=1$ +2050,$F = G m_1 m_2 / r^2$ +2051,$\mathsf P(A)$ +2052,$\mu\sqrt{\mu^2+4\mu}$ +2053,$\int_A Xd\P$ +2054,$g(\mu)=\eta$ +2055,$10^1$ +2056,$\lambda(F(x_2)-F(x_1))$ +2057,$D_t = X_t - X_{t-1}$ +2058,$0<\epsilon<1$ +2059,$A\in\G_Y$ +2060,$2 \times 10^{19}$ +2061,$j$ +2062,$\beta>0$ +2063,$\mathsf{E}[X_1\mid X > a^*]$ +2064,"$Y=\{y_1,\dots,y_n\}$" +2065,$4l + 2$ +2066,$y_{1}=2$ +2067,$T_x$ +2068,$\bar A$ +2069,"$1, 2, 10$" +2070,$j(x)=1/x^{a+1}$ +2071,"$\nu(B)=\int_B 1_A\,d\P$" +2072,$nG$ +2073,$V_\nu$ +2074,$c^*$ +2075,$V(\mu)=e^\mu$ +2076,"$A, B$" +2077,$\theta=\log(p)<0$ +2078,$X_2$ +2079,$\lambda\downarrow 0$ +2080,$g(s)$ +2081,$y\in Y$ +2082,$B\in\mathsf{E}E$ +2083,$\omega'\in\Delta_\omega$ +2084,"$(\Omega, \B)$" +2085,$c_1=c_2=0$ +2086,$n=2$ +2087,$\Omega\times\Omega$ +2088,$\mathsf{E}[\cdot\mid\G]$ +2089,$\mathsf{E}[X_\nu]=\nu\mathsf{E}[X_1]$ +2090,"$\Gamma = \{(\omega,\omega)\}$" +2091,$\hat F(t)=\phi(-2\pi t)$ +2092,$W$ +2093,$n=3$ +2094,$\kappa_1'(-\kappa(\theta))\kappa'(\theta)=1$ +2095,$\approx 0.9999999999$ +2096,$\max_{s\le t} X_s - X_t$ +2097,$m^2$ +2098,$2^{-1}$ +2099,$\mathcal F$ +2100,$g=3$ +2101,$T\P$ +2102,$10^{19}$ +2103,$b = B/P = 1/1.2 = 0.83$ +2104,$\int_{-f_{\max{}}}^{f_{\max{}}}$ +2105,$J(x)$ +2106,$X\equiv 1$ +2107,$E\in\A_y\ \forall y\in Y\setminus N$ +2108,"$\{E\subset\Omega\mid E\in\sigma(\C'), \C'\subset\sigma(\C),\text{ countable}\}$" +2109,$g(\mu)=\log\mu$ +2110,$\mathsf{E}(G^3)=g$ +2111,$X_t=t-G_t$ +2112,$ is constant. This NEF is regular because $ +2113,$(X_N-a)^-$ +2114,"$(X,\F,\P)$" +2115,$T^2$ +2116,"$\Phi(a, b)=(\phi(f(a)), \phi(b))$" +2117,"$(\P,\sigma(T))$" +2118,$X = X_1 + X_2$ +2119,$(x+b)$ +2120,$c(y)=\dfrac{1}{\pi}\dfrac{1}{1+y^2}$ +2121,$S(\omega)=\omega$ +2122,"$\G=\{ b(a_i,2^{-j}) \}_{i,j}$" +2123,$B(b)$ +2124,$\mu=0$ +2125,$ corresponding to an extreme stable distribution with $ +2126,$0.999999$ +2127,$g(s)=d + vs$ +2128,$1 \times 10^{23}$ +2129,$M(t)=\mathsf E[\exp(tx)]=\phi(-it)$ +2130,$e^x-1$ +2131,$\to 0$ +2132,$Pa$ +2133,"$Y=\mathrm{ED}(\mu, \sigma^2)$" +2134,$A\in \A$ +2135,$\tan{}$ +2136,$\mathcal{F}_n$ +2137,$\Q=\pi_Y(\R)$ +2138,${{}_tp_x}=\exp(-\int_0^t \mu_{x+s}ds)$ +2139,$k(\theta)=\left(\int c(y)e^{\theta y}dy\right)^{-1}$ +2140,$K_T(\theta)=-\log(1-\theta/\lambda)$ +2141,$\F_1 \subset \F$ +2142,$V(\mu)=\mathsf{Var}(\mathsf{CP}_1)=\mu x_2$ +2143,$(E\cap U) + x$ +2144,$\leftrightarrow$ +2145,$e+l+r^*$ +2146,$V_X(m)=m$ +2147,$B\subset Y$ +2148,$b^2 \mu_x /2$ +2149,$\lambda \uparrow 1$ +2150,$B$ +2151,$-br-v$ +2152,$-9$ +2153,$\theta\to 0$ +2154,"$c_1, c_2\ge 0$" +2155,$-\theta - \lambda(e^{-\theta} - 1)) = y$ +2156,$B_i$ +2157,$= 1 - T_\mathrm{sink} / T_\mathrm{source}$ +2158,$\alpha_Y$ +2159,$5.97 \times 10^{24}$ +2160,$V(\mu)=1/(\tau^{-1})'(\mu)=\mu^3$ +2161,$t\ge 0$ +2162,$\alpha \mu(U_n) \le \mu(E\cap U_n)$ +2163,$X-100$ +2164,"$B_1,B_2\in\F$" +2165,$\mathsf{Var}(G)=c$ +2166,$\nu<1$ +2167,$\tF$ +2168,$c=0$ +2169,$x \!\urcorner$ +2170,$K_\delta$ +2171,$-6$ +2172,$\mathsf{Pr}(\mathsf{CP}(\lambda)=0)=e^{-\lambda}$ +2173,$\delta(t)$ +2174,$nu$ +2175,$B\mapsto\P(B\mid\G)(\omega)$ +2176,$\mathsf{Pr}(X\le x)$ +2177,$\mathcal F = \{F_\alpha\mid \alpha<\omega_c\}$ +2178,"$n_{\text{water, initial}}$" +2179,$e^z/z$ +2180,$H(x)\not=H(y)$ +2181,$\B$ +2182,$p\to\infty$ +2183,$T\P(A)=\P(T^{-1}(A))>0$ +2184,$B=2.7\times 10^{-6}$ +2185,$\mu_1 = \mu_2$ +2186,$\mu^p$ +2187,"$P = 1 \, \text{GPa} = 1 \times 10^9 \, \text{Pascals}$" +2188,$\omega\in B_1\cup\dots\cup B_r$ +2189,$10^{-43}$ +2190,$\mathsf{Pr}(L=l)$ +2191,$P_t f$ +2192,$(\bar a_x - \bar a_{b\!\urcorner})/\bar a_x$ +2193,$\ge 1$ +2194,$T_x=x$ +2195,$\mu_1$ +2196,$\sech$ +2197,$\mathbf{B}$ +2198,$\theta$ +2199,"$5/8, 1/4, 1/8$" +2200,$\P(B\mid\A)=\P(B)$ +2201,$F = U - TS$ +2202,$L^r$ +2203,$\mu\{ l>1\}=0$ +2204,$i=\sqrt{-1}$ +2205,$2aw=0.0035$ +2206,$X=a$ +2207,$D-N = \sum_{i\in I} (D_i-N_i) - N_a$ +2208,$W(t)$ +2209,$(1-\alpha)M$ +2210,$s>0$ +2211,$\kappa_T(y)= -\theta=\log(w(y))$ +2212,$n\log_2(n)$ +2213,$A=F\triangle Q$ +2214,"$(a,b)$" +2215,"$\mathrm{ED}(\mu, \sigma^2 / n_i)$" +2216,$X_n(\omega)=1$ +2217,"$\{\omega\mid p(\omega, A(\omega))=1 \}$" +2218,$X$ +2219,$r \leq p$ +2220,$\theta=\nu\lambda$ +2221,$\mathsf{Pr}(B=1)=\mathsf{Pr}(X>x)=p$ +2222,$x^{-3}$ +2223,$(\bar P_{x+b} - \bar P_x)\bar a_{x+b}=\bar A_{x+b}-\bar P_x \bar a_{x+b}=: {}_b\bar V$ +2224,$\hat\theta_s$ +2225,$20$ +2226,$1.805$ +2227,$\alpha=1$ +2228,$\mu=\lambda/ \psi$ +2229,$G\in \sigma(\A\otimes \B)$ +2230,"$p(B, y)$" +2231,$\Gamma$ +2232,$z=1$ +2233,$m_X(s)\to\mathsf E[X] + k$ +2234,$Y_\epsilon$ +2235,$15$ +2236,"$\int_0^a \alpha_2(x)F(x)\, dx$" +2237,$\alpha=0.99$ +2238,"$\int_B \P(A\mid\G)(\omega)\,\P(d\omega)$" +2239,$-\kappa_1(-\kappa(\theta))=\theta$ +2240,$f_0\in\mathcal K$ +2241,$\mu=21.315$ +2242,$\P(A_0)=0$ +2243,$h_1$ +2244,"$g:[0,1]\to [0,1]$" +2245,$g(s)=s^{0.72}$ +2246,"$\F\times\Omega\to [0,1]$" +2247,"$380,000$" +2248,"$\mathcal{W}(g, W)\subset \mathcal{W}$" +2249,$\frac{1}{2}\sech(\pi y/2)$ +2250,$t^a$ +2251,$\bar y$ +2252,$p=\frac{\alpha-2}{\alpha-1}$ +2253,"$\forall A\ \forall p\ [\forall x\in A\ \exists !y\ \phi(x, y, p)\rightarrow\exists Y\ \forall x\in A\ \exists y\in Y\phi(x, y,p)]$" +2254,$0\in\Omega_p$ +2255,"$l=0,1,\dots,n/2+1$" +2256,"$(1, 2, 3, \dots)$" +2257,$m=0$ +2258,$10^{-1}$ +2259,$ is accumulated profit from an inflow of premium 1 per unit time and a cumulative claims process $ +2260,$V(m)=m^3$ +2261,$T\lambda$ +2262,"$\G=\sigma(A_1,\dots,A_r)$" +2263,$aw$ +2264,$\mi(\mu):=\mathsf{Var}(s)$ +2265,"$(i,j)$" +2266,$x_2\leftrightarrow y_1$ +2267,$e$ +2268,$\frac{1}{4}(1 + 8\mu-\sqrt{1+8\mu})$ +2269,"$e(fT, y) = f(y)$" +2270,"$I=[0,P)$" +2271,$(y-t)/V(t)$ +2272,$p=1.995$ +2273,$B\in\G$ +2274,$\mathsf{Pr}(L'\ge l)$ +2275,$q(1-s)$ +2276,$A(\omega)$ +2277,$X_\nu$ +2278,$\mathsf{Pr}(|X_n(\omega)-X(\omega)|>\epsilon)\to 0$ +2279,$x\!\!\urcorner$ +2280,$\H$ +2281,$1 million punitive award is grossly excessive and unconstitutional in a case where Dr. Mann had only $ +2282,$\mathsf{E}[X\mid T] = \phi(T)$ +2283,$X_i=x_i$ +2284,$T_1 = 20°C = 293.15$ +2285,"$(-\infty,0)\subset\Theta$" +2286,$x_{ij}$ +2287,$\delta\to 0$ +2288,"$\psi(S,T)$" +2289,$\sigma(G)$ +2290,$\tau(\theta):=\kappa'(\theta)=\mu$ +2291,$X_2=100$ +2292,$\alpha=0$ +2293,$S\P(A) = \P(S\in A)$ +2294,$n\to \infty$ +2295,$\theta_s$ +2296,$T_0=T-1$ +2297,$\F_1$ +2298,$q(p^*)=0.0$ +2299,"$X_{i}=\mathsf{CP}(j_n({x_i})\delta, {x_i})-j_n({x_i})\delta {x_i}$" +2300,$f^{-1}(\BB(\mathbb{R})) = \sigma(A_n)$ +2301,$ on $ +2302,$P$ +2303,$2^{256}\approx 10^{77}$ +2304,$T\P(A)=\P(T^{-1}(A))$ +2305,$T = 2L / \sqrt{g d}$ +2306,$x\subset X\leftrightarrow \forall z(z\in x\rightarrow z\in X)$ +2307,$g_n(x)=f_j(x)$ +2308,$\approx 1$ +2309,$p=0$ +2310,$J(x)<\infty$ +2311,"$\displaystyle\int_B \P(A\mid\G)\,d\P$" +2312,$d\bar S_i/da$ +2313,$V_1 / V_2 = 12:1$ +2314,$\phi(x)/x$ +2315,$\forall X[\forall x\in X(x\not=\emptyset) \wedge \forall x\in X\forall y\in X(x=y\vee x\cap y=\emptyset)]\rightarrow\exists S\forall x\in X\exists !z(z\in S\wedge z\in x)$ +2316,$\mathsf{VaR}_p(X)-f(\mathsf{VaR}_p(X))$ +2317,"$(L,\A)$" +2318,$x^2\mathsf{Var}[N]$ +2319,$m_1=am$ +2320,$10^{11}$ +2321,$\mathsf{Pr}(B=0)=1-p$ +2322,$\\alpha=1$ +2323,$\mu+\mu^3/2+(\mu^2/2)\sqrt{2+\mu^2}$ +2324,$\omega=\exp(2\pi i / n)$ +2325,"$|\mathcal{W}(g,W)|$" +2326,$J(x) - J(x+dx) \approx j(x)dx$ +2327,$x_{\max{}} = x_{\min{}} + n$ +2328,$\theta>0$ +2329,$(m+1)$ +2330,"$\lim_{\mu\to 0} V(\mu)/\mu=\delta:=\inf\,\{S\setminus \{0 \}\}$" +2331,$\approx 10^{-5}$ +2332,$X(ω)$ +2333,$6 \times 10^7$ +2334,$b$ +2335,$T^2 \propto r^3$ +2336,$\alpha < 2$ +2337,"$A\subset[0,1]$" +2338,$A^c$ +2339,$\mathsf{Pr}(N=n)=p_n$ +2340,"$\{\lambda_t\}, with $" +2341,$PV = kNT$ +2342,"$\mathsf{CP}(1,X)$" +2343,$\phi:M\to\mathbb{R}$ +2344,$b=12$ +2345,$\mathsf{E}[X_x] = 0$ +2346,$\eta$ +2347,$A\in \F$ +2348,$10^{27}$ +2349,$\Omega$ +2350,$j(x)=x^{-\alpha-1}$ +2351,$f_X\sim cf_Y$ +2352,$C= \mathrm{conv}(S)^-$ +2353,$A_i=\{X=x_i\}$ +2354,$\P(B\cap A)$ +2355,$K=B^a=A^b=g^{ab}\pmod p$ +2356,$E[\tau]$ +2357,"$\int_0^1 x^2j(x)\,dx<\infty$" +2358,$\nu\ge 0$ +2359,$\scriptstyle #1$ +2360,$\theta\mapsto -\kappa_1(\theta)$ +2361,$\sigma_s=0$ +2362,$x_2\leftrightarrow y_2$ +2363,$\sup_{\theta\in\Theta} y\theta-\kappa(\theta)$ +2364,$\mathsf{Pr}(T_x<\infty)=1$ +2365,$\mathsf{Var}(Y)=\mu^2/\alpha$ +2366,$nt$ +2367,$y_{3}=7$ +2368,$\P(B\mid \G)$ +2369,$\forall x\forall y[\forall z(z \in x \leftrightarrow z \in y)\rightarrow x=y]$ +2370,$\mathcal{F}_t = \sigma(X_s : s \leq t)$ +2371,$F_{\bar X}(x)=p_1F_1(x) + p_2F_2(x)$ +2372,$1 \times 10^{10}$ +2373,$f_t$ +2374,$F\subset M$ +2375,$\bar\theta_s<0.5$ +2376,$J(0)$ +2377,"$\alpha\in [0,1)$" +2378,$\mathsf{Pr}(X_n=Y)=\mathsf{Pr}(X=Y)=0$ +2379,"$. It falls into the compensated IACP, case 3 group, discussed in [Part III](./2020-10-20-Probability-Models-for-Insurance-Losses/), and takes any real value, positive or negative, despite only having negative jumps. It has a thick left tail and thin right tail. Its mean is zero, but the variance does not exist. A tilt with $" +2380,"$[a, b]$" +2381,$X_t = ct+\sigma B_t$ +2382,"$1000 \, \text{kg/m}^3$" +2383,$2.725$ +2384,$p\uparrow 2$ +2385,$\mathsf E(G)= f + \mathsf E(G') = 1$ +2386,"$c_1,c_2\ge 0$" +2387,$\mu=\tau(\theta)=ne^\theta/(1+e^\theta)=np$ +2388,$\approx 27\%$ +2389,$v=1/1.15 = 0.87$ +2390,$X_n(\omega)\to 0$ +2391,$\tau$ +2392,$\log$ +2393,$\theta=-\mu^{-1}$ +2394,$L$ +2395,$x\mapsto x^c$ +2396,$\int_A m(Y)d\P=\int_A m(Y(\omega))\P(d\omega)$ +2397,"$(\Omega, \F, \P, \{\F_t\}_{t \geq 0})$" +2398,$kX$ +2399,$\int (r-\mu)f=0$ +2400,$N_t - A_t = 1 - T$ +2401,"$P(·, ω)$" +2402,$\{S\le T\}$ +2403,$X^{\tau_n}$ +2404,$1$ +2405,$d\mu = \kappa''(\theta)d\theta$ +2406,$+\frac{1}{2}$ +2407,$9$ +2408,"$B\mapsto p(\omega, B)$" +2409,$\mathsf{E}(G)=M_G'(0)=1$ +2410,$(x-a)^+\wedge b$ +2411,$2.44 > 2 \times 1.2$ +2412,$E=U+T=-T$ +2413,$e^{-\lambda}>0$ +2414,"$r_2 = 3852972862.741849 \approx 3,852,972,862.7$" +2415,$x^n=e^{n\log(x)}$ +2416,$\kappa(\theta(\mu))=n\log(n/(n+\mu))$ +2417,$E[X_\tau]$ +2418,$\mathbb R^\times$ +2419,$2/\sqrt{\alpha}$ +2420,"$\phantom{P}= \displaystyle\frac{1}{1+r}\,\mathrm{EL} + \displaystyle\frac{r}{1+r}\,a$" +2421,$A_n \downarrow A$ +2422,$G=G_1-1$ +2423,$\{X_t\}_{t \geq 0}$ +2424,$\times T=\rho T$ +2425,$1 \times 10^{17}$ +2426,$k\in\mathbb Z$ +2427,$I_k$ +2428,"$\alpha=n/2,\beta=1/2$" +2429,$r\in\mathbb{R}$ +2430,$\P(\Omega\mid\G)_{\omega_0}=1$ +2431,$5 \times 10^5$ +2432,$p\leftrightarrow p_1 = 3-p$ +2433,$\mu/V(\mu)$ +2434,$0 \le \alpha \le 1$ +2435,$r^*$ +2436,$6$ +2437,$p=2$ +2438,$q_1$ +2439,$0.4$ +2440,$g=W$ +2441,$\Omega\times\F$ +2442,$\mu(F)=0$ +2443,$\Delta = g^{-1}(\Delta_\mathbb{R})$ +2444,$\bigcup_{i=1}^{\infty} A_i \in \mathcal{M}$ +2445,$10^{-3}$ +2446,$T_\nu$ +2447,$n\ge 2$ +2448,$x=60$ +2449,$\sqrt{2Np}$ +2450,"$=\displaystyle\int_{T^{-1}(B)}^{\phantom{X}} \mathsf{var}phi(T) \,d\P\quad$" +2451,$b=1$ +2452,$D_t$ +2453,$T\lambda \ll \mu$ +2454,$y_1 < y_2$ +2455,"$\int |X_n(\omega) - X(\omega)|^p\, \mathsf{Pr}(d\omega)\to 0$" +2456,$p=e^\theta$ +2457,$v=1-d$ +2458,$m(y)$ +2459,$\alpha_X > 2$ +2460,"$[0,1] \times \{y\} \subset X$" +2461,$t_l = l f_{\max{}} / n$ +2462,$l(t)<\infty$ +2463,$x\to 0$ +2464,$K(t)=\log M(t)$ +2465,"$a_1,a_2\in A$" +2466,$16$ +2467,$\mu=T\lambda$ +2468,"$Y_l:=\mathsf{CP}(J(1), X_l)$" +2469,$n\ge N$ +2470,"$1 million in punitive damages from Steyn, $" +2471,$\G=\sigma(\A)$ +2472,$\sup_\Omega |X_n - X| \to 0$ +2473,$\mathsf{Var}(X)=\mathsf{E}[X^2]-\mathsf{E}[X]^2$ +2474,"$\mathsf{E}[e^{\theta X_t}] = e^{t\,\kappa_X(\theta)}$" +2475,"$(x_i, y_{k(i)})$" +2476,"$T:(\Omega,\F)\to (E,\mathsf{E}E)$" +2477,$\int X=0$ +2478,$\omega_0$ +2479,$\Delta\in\F\otimes\F$ +2480,$\mu(a+b\mu)$ +2481,$n=2^{\log_2}$ +2482,$\lambda < 1$ +2483,$\beta_0+\beta_1+\beta_2$ +2484,$\log_g(n)=a$ +2485,$F\subset E\in \H\implies F\in \H$ +2486,$c(k)=\binom{n+k-1}{k}$ +2487,$M_t$ +2488,$x_{\min{}} = m_0b$ +2489,$(\tau^{-1})'(\mu)=1/V(\mu)=\mu^{-p}$ +2490,$\theta < 0$ +2491,$P^T(T=t\mid t)=\P(\{t\})=0$ +2492,$\cos(iz)=\cosh(z)$ +2493,$e^{\theta X_t -t\kappa_X(\theta)}$ +2494,"$0, 1, \dots, n - 1$" +2495,"$J=\{\omega\mid P(\Gamma\cap Z, \omega) = \frac{1}{2}1_\Gamma(\omega) \ \forall\Gamma\in\G\}$" +2496,$\lambda/r<1$ +2497,$0\leq f \leq 1$ +2498,$\mathsf v$ +2499,$10^{-15}$ +2500,$X\ge a$ +2501,$\mathcal{M} = \sigma(\mathcal{A})$ +2502,$x_{i1}$ +2503,$|b|$ +2504,"$b,-2b,b$" +2505,$2^1+1\rightarrow 3^1+1-1=3^1 \rightarrow 4^1-1=3 \rightarrow 2 \rightarrow 1 \rightarrow 0$ +2506,"$[-\infty, 2]$" +2507,$X_\infty(\omega)$ +2508,"$\mathcal W(g, W)$" +2509,$p$ +2510,$\kappa_1'(-\kappa(\theta))=1/m$ +2511,$t \to \infty$ +2512,$\lambda_t$ +2513,"$[-10, 10)$" +2514,$(T\lambda)g = \mu^t g(t)l(t) \le \mu h < \infty$ +2515,$E[\tau] = 2$ +2516,$ if $ +2517,"$(E, \mathsf{E}E)$" +2518,$\H(R)$ +2519,$\mathsf{E}[X\mid\G]=\mathsf{E}[X]$ +2520,"$\psi:(L\times M, \A\otimes \B)\to\mathbb{R}$" +2521,$\sum_{m\in\mathbb Z} F((mn + k +1/2)b)-F(mn + k - 1/2)b)$ +2522,$\partial l/\partial \mu$ +2523,$n\mathsf{Var}(X)$ +2524,"$\alpha_X\in(\alpha_Y+1/2, \alpha_Y+1)$" +2525,$\F$ +2526,$X_t$ +2527,"$\px=\int_B \sum_i\mathsf{E}[X_i\mid \G]\,d\P$" +2528,$m = n \times \text{molar mass} = 114.6g$ +2529,$\hat F / |\hat F|$ +2530,$f_Y$ +2531,$m=et$ +2532,$\omega = 2\pi / T$ +2533,$1 \times 10^{14}$ +2534,$J/K$ +2535,$f(\mu)= \mu$ +2536,"$f(X,Y)$" +2537,$> 10^{19}$ +2538,$F = M v^2 / r$ +2539,$ which is an extreme stable with Lévy distribution $ +2540,$\mathsf{Pr}(X_n\in A)=1$ +2541,$h_f$ +2542,$T(x)$ +2543,$\sup_{\alpha \in A} \mathsf{E}[|X_\alpha|^p] < \infty$ +2544,$K=A^k\pmod p$ +2545,$3.2 \times 10^9$ +2546,$T''$ +2547,$\Delta H$ +2548,$N'/w-\alpha a\le Mg$ +2549,$\hat X$ +2550,"$p(\omega, \cdot)$" +2551,"$F_n,F$" +2552,"$M = \{M_t, t \geq 0\}$" +2553,$x_2$ +2554,"$(\P_X, \sigma(T))$" +2555,$\P_S$ +2556,$l(y;\mu)=\log(a(y) + y\tau^{-1}(g^{-1}(\bfx\beta))-\kappa[\tau^{-1}(g^{-1}(\bfx\beta))]$ +2557,$p=1.6$ +2558,$b=0$ +2559,$\mathrm{conv}(S)$ +2560,$\kappa(\theta)=-n\log(1-e^\theta)$ +2561,$\mathsf{E}[X\mid\sigma(T)]$ +2562,$\P(D ∪ C^c) = 1$ +2563,$\lambda = 2$ +2564,$x_i$ +2565,$t < T$ +2566,$Y_\nu=Z_\nu/\nu$ +2567,"$\lambda,\mu$" +2568,$5 trillion business. Property casualty insurers write $ +2569,$10^{-4} \times$ +2570,$t = T$ +2571,$X=\sum_i X_i$ +2572,$dgS$ +2573,$10^9 - 10^{12}$ +2574,$\tau^{-1}$ +2575,$k(t)l(t)<\infty$ +2576,"$\mathit{ED}(\mu, \sigma^2)$" +2577,$f \in C(X)$ +2578,$\theta=\tau^{-1}(\mu)$ +2579,$\theta=\log\left(\frac{p}{1-p}\right)$ +2580,$G = H - TS$ +2581,$\mu_2$ +2582,$1-0.271 = 0.729$ +2583,"$\int_0^1 xj(x)\,dx=\infty$" +2584,$<$ +2585,"$(\Omega, \F, λ)$" +2586,$25M limit above \$ +2587,$[(k-1/2)b)-F((k+1/2)b))$ +2588,$K''(0) = \kappa''(\theta) = \tau'(\theta)$ +2589,$1/x^2$ +2590,$\sigma^2>0$ +2591,$\approx 10^{-1}$ +2592,$V(m)=m^2$ +2593,"$[0, \epsilon_2]$" +2594,$P(P(\omega))$ +2595,$\sigma(\A\otimes \B)$ +2596,$Y = \operatorname{frac}(2\omega)=\{2x\}=x -\lfloor x\rfloor$ +2597,$N_i$ +2598,$G_i'$ +2599,$\sup(X)$ +2600,$\Gamma(z + 1) = z\Gamma(z)$ +2601,$A \subseteq \mathbb{R}$ +2602,$j_n(x)=j(x)$ +2603,$(\hat f(l/P))_l$ +2604,$l(t)=\lambda_t \Omega$ +2605,$X=Y$ +2606,$N \mid G$ +2607,$\kappa'(\theta)$ +2608,$\beta=\alpha/\mu$ +2609,$\P(A\cap B)=0$ +2610,$\mathrm{Re}(a)>0$ +2611,$A_t = T$ +2612,$(\alpha+1)/\alpha\to 1$ +2613,"$P^T_S(\cdot\mid\cdot):\A\times M\to [0,1]$" +2614,$r_1 = 44594.593127378765$ +2615,$\px=\P(B\cap \bigcup A_i)$ +2616,$\tau = \inf\{t \geq 1 : M_t = t \}$ +2617,"$i=1,\dots, M$" +2618,"$X:(\Omega,\F)\to(S,\S)$" +2619,$s(x;\mu) = \partial l/\partial \mu$ +2620,$1 < \alpha < 2$ +2621,"$\mathcal{W}(\mathrm{fuzzy}, W)$" +2622,$1-\alpha$ +2623,$n=4$ +2624,$96\le x \le 103$ +2625,"$F:\Omega\times \mathbb{R}\to[0,1]$" +2626,$X_1+\cdots +X_n$ +2627,$t \wedge \tau_n$ +2628,$L^2$ +2629,$q(p)$ +2630,$G_\delta$ +2631,$\alpha_i(x):=\mathsf{E}\left[\dfrac{X_i}{X}\mid X > x\right]$ +2632,$E$ +2633,"$x_1,\dots,x_{k+1}$" +2634,$X_0=1$ +2635,"$[9750, 10550]$" +2636,$S$ +2637,"$1,2,\dots, 11$" +2638,$w(z)=zg(w(z))$ +2639,$P(z)=\sum_n p_nz^n$ +2640,$\mathcal{F}_t$ +2641,$p=1$ +2642,"$\forall A\in\F,\ \omega\to P(A, \omega)$" +2643,"$P(·, \omega)$" +2644,$\hat f$ +2645,$x_0=a/2\pi$ +2646,$0x$ +2660,$x^2$ +2661,$q_\beta$ +2662,$10^6A_{40}=121059.21$ +2663,$f:S\times T\to \mathbb{R}$ +2664,$b=0.3$ +2665,$q(s)$ +2666,$5.184 \times 10^{19}$ +2667,$\\alpha=2$ +2668,$g(x)=e^{2\pi i x\theta}$ +2669,$\epsilon\mu(A\cap B)\le \nu_0(A\cap B)$ +2670,$A \in\F$ +2671,$A=A_1\cup\cdots\cup A_n$ +2672,${}_{dt}q_{x+t}\approx dt\mu_{x+t}$ +2673,$(6.022 \times 10^{23} \text{ molecules/mol})$ +2674,"$\psi(s,t)=\mathsf{E}[\psi(S,T) \mid S=s,T=t]$" +2675,$p=29$ +2676,$f_{\max{}}=1/b$ +2677,$\sum_x x p_x$ +2678,$g(x)=0$ +2679,"$\{0,1,2,\dots\}$" +2680,$\sigma^2=1/\nu$ +2681,$-\theta - \lambda(e^{-\theta}-1)=y$ +2682,"$\Delta_\omega=\{w \mid (\omega,w)\in\Delta\}$" +2683,$x^{-1/2}$ +2684,$q$ +2685,$0.99999$ +2686,$C=2\mathbb Z + 1 + \xi\mathbb Z$ +2687,$1<\alpha<\omega_c$ +2688,$d(d-1)\approx d^2$ +2689,$D(F)$ +2690,$P_S$ +2691,$SiO_4^{4-}$ +2692,$\cos\arctan(\mu)=1/\sqrt{1+\mu^2}$ +2693,$\mathsf{E}[X\mid T]=\mathsf{var}phi(T)$ +2694,$=p$ +2695,$\P H\xi=\P HX$ +2696,"$P(\omega, \cdot)$" +2697,$g_1$ +2698,$e^{\theta_1 y}$ +2699,$=$ +2700,"$E, F$" +2701,$2.09 \times 10^{14}$ +2702,$c$ +2703,$L=\sum_s l_s B_s$ +2704,$s_0$ +2705,$ because it has a cusp). In the case $ +2706,$f:L\to M$ +2707,$R_t$ +2708,$q_\alpha\in B'\cap F_\alpha$ +2709,$\frac{1}{2}1_\Gamma(\omega)$ +2710,$F_\sigma$ +2711,$1 \times 10^{18}$ +2712,$I$ +2713,$B=\bigcap_i G_i$ +2714,$10^{18}$ +2715,$ and gamma severity. Distributions with $ +2716,"$(\Omega, \mathcal{F})$" +2717,$F(x_{\max{}})-F(x_{\min{}})$ +2718,$\kappa(\theta(\mu))$ +2719,$\mathcal{F}$ +2720,$A\cap B\subset B$ +2721,$\lim_s\inf m_X(s)\ge \mathsf E[X]$ +2722,$C_p = \frac{5}{2}R$ +2723,$c\ge 0$ +2724,$|X_t|$ +2725,$1_{T^{-1}(B)}$ +2726,"$G(t, 1/\nu)$" +2727,$4\times 10^{19}$ +2728,$R^S=g^mA^R$ +2729,"$ which is the cumulant function for the normal distribution. As expected, $" +2730,$\nu\{0\}=0$ +2731,$(y-\mu)/V(\mu)$ +2732,$G_\delta\subset\R$ +2733,"$(0,\infty)$" +2734,$\tilde x$ +2735,$\alpha_Y + 2 > \alpha_X > \alpha_Y+1$ +2736,$(\lambda)$ +2737,$10^{-18}$ +2738,$a = \omega^2 r = v^2 / r$ +2739,$n=16$ +2740,$(1-p)(1-\alpha)=(p-1)(\alpha-1)=-1$ +2741,$X_n(\omega)\to X(\omega)$ +2742,$\mu_x$ +2743,$y^*$ +2744,$B\in\F$ +2745,"$D = \{ω : P(N, ω) = 1\}$" +2746,$0\le \hat \theta\le 1$ +2747,$u$ +2748,$A_t = t$ +2749,$2\times 10^{30}$ +2750,$N_t - A_t = -t$ +2751,$\P(A\cap G)=\P(A)\P(G)$ +2752,$|e^{isy}|=1$ +2753,$\hat\theta_s<0.5$ +2754,$\Omega\times \F$ +2755,$P^T(F\mid t)=\P(F)$ +2756,$(X)$ +2757,"$x_i, y_i$" +2758,$c(z; \nu)$ +2759,$f(\mathsf{VaR}_p(X))$ +2760,$\{\mathcal{F}_t\}$ +2761,$A=1$ +2762,$p\to 2$ +2763,"$\alpha_X\in(\alpha_Y, \alpha_Y+1/3)$" +2764,$T = 20°C + 273.15 = 293.15 K$ +2765,"$i=1,2,\dots,10000$" +2766,$\forall x\ [\exists y\ (y\in x)\rightarrow \exists y\ (y\in x \wedge \neg\exists z(z\in x \wedge z\in y))]$ +2767,$\alpha + F_1/a$ +2768,"$(y_1, y_2)$" +2769,$t\ge T$ +2770,$\tau'(\theta)=\kappa''(\theta)=\mathsf{Var}(Y_\theta)>0$ +2771,$A = \{ω\}$ +2772,$\alpha_i(x)S(x)$ +2773,$\int_1^\infty j(x)dx<\infty$ +2774,$\lambda_2/\lambda$ +2775,$T(X) = \sum_{i=1}^{n} X_i$ +2776,$X = \sum_i X_i$ +2777,$-1/2$ +2778,$(X_n)$ +2779,$\P(A \cap B)$ +2780,$h_1\le w$ +2781,$\hat p_s$ +2782,"$10,280$" +2783,$D ∪ C^c ⊃ N$ +2784,$\P[\cdot\mid Y]$ +2785,$R_x:=T_x-x$ +2786,$K(t; \theta)$ +2787,$\leq$ +2788,$X(x)=1/x$ +2789,"$r \approx 384{,}400 \ \text{km}$" +2790,$Z\cap J\not=Z\cap (J\setminus\{\omega\})$ +2791,$\P(A\mid\G)=P(A)$ +2792,"$\mathsf{CP}(\lambda, B)$" +2793,$d(y;\mu)>0$ +2794,$\mathsf Q$ +2795,$10: \left\Vert K_\delta\right\Vert_{L^1}\le c\ \forall \delta>0$ +2800,$\mathsf{E}[X\mid X \ge x]\ge x$ +2801,$0.01 \text{ kWh}$ +2802,$\bar a_{75}=9.81$ +2803,$\sum_n 1/n$ +2804,$x_1+y_1 \le x_1+y_2\le x_2+y_2$ +2805,$\mathsf{E}[XY\mid\G]=X\mathsf{E}[Y\mid \G]$ +2806,$(T\lambda)k = \mu kl < \infty$ +2807,$\P_T$ +2808,$g'(s)$ +2809,$X_1 = aX + b$ +2810,$10^{-15} - 10^{-18}$ +2811,$m^2(1+m)$ +2812,$n=2^5=32$ +2813,$ and also that $ +2814,$m/s^2$ +2815,$k(\theta)>0$ +2816,$N\sim\mathrm{Po}(\lambda)$ +2817,$. The distribution is very thick tailed and does not have a mean ($ +2818,$x/c$ +2819,$\mathsf E[X_i \mid X]$ +2820,$10^{21}$ +2821,$R=L^*+R^*$ +2822,$kT$ +2823,$(y-\mu)/\sqrt{V(\mu)}$ +2824,$T$ +2825,$D_s = X_s - X_{s-1}$ +2826,$\mathsf E[|X_1|]<\infty$ +2827,$\ln(10)=2.302585$ +2828,$m_l$ +2829,$-k$ +2830,$F\subset A$ +2831,$L \propto M^{3.5}$ +2832,"$, resulting in a distribution with support on $" +2833,$K$ +2834,$\mathsf{Pr}(N>32)=7.37\times 10^{-9}$ +2835,$S\mid \B$ +2836,$\mu=\tau(\theta)=\dfrac{ne^\theta}{1-e^\theta}$ +2837,$D_t = Y_t - Y_{t-1}$ +2838,"$(-\mu(U)/2, \mu(U)/2)$" +2839,$\lambda_1/\lambda$ +2840,$(1+\rho)\mathsf{E}[C]$ +2841,$0\le t\le 1$ +2842,$f=0$ +2843,$\mathsf{E}[X]=\infty$ +2844,$\mathsf{E}[X]=\mathsf{Var}(X)=1$ +2845,$Y_\nu=X_\nu/\nu$ +2846,$f(\cdot;\theta)$ +2847,$u=0$ +2848,$\{T=t\}$ +2849,$\times$ +2850,$\sigma^2=1$ +2851,$\alpha_Y = \alpha_X=\alpha$ +2852,$A\leftrightarrow 1_A$ +2853,$\theta^2/2$ +2854,$\mu=T\P$ +2855,"$\px=\sum_i\int_B X_i\,d\P$" +2856,$N'/2w - Mg - \alpha a> 0$ +2857,$\mathsf{Var}(Y)=npq=\mu(1-p)=\mu(1-\mu/n)$ +2858,$\beta\to (\alpha+1)/\mu$ +2859,$x \urcorner$ +2860,$D_t = (X_t^2 - t) - (X_{t-1}^2 - (t-1)) = X_t^2 - X_{t-1}^2 - 1$ +2861,"$\mathsf{E}[X\mid T=t] = \displaystyle\int_L \mathsf{E}[X\mid S=s,T=t]P^T_S(ds$" +2862,"$(S,d)$" +2863,$10^{-35}$ +2864,"$X = (X_1, X_2, \ldots, X_n)$" +2865,$ for integer $ +2866,$K_\theta(t)=\kappa(\theta+t)-\kappa(\theta)$ +2867,$g(s) = \Phi(\Phi^{-1}(s) +\lambda)$ +2868,$\A=\{\Omega\}$ +2869,$10^{-8} - 10^{-11}$ +2870,$T\lambda=\mu$ +2871,$p=1/2$ +2872,$s\to\infty$ +2873,$\sum_i x_i^2$ +2874,$\nu\ll\P$ +2875,$\int_{-\infty}^\infty$ +2876,$y_s$ +2877,$aggfft$ +2878,$\mu_0$ +2879,$f:\Omega\to\mathbb{R}$ +2880,$\theta_1=\theta_2$ +2881,$B\subset\R$ +2882,$> 0.98$ +2883,$f_{\max{}} = 1 / b$ +2884,$g(t)=0$ +2885,$\mathsf{E}[h(T)X]=\mathsf{E}[h(t)\mathsf{E}[X\mid T]]$ +2886,$X_{t\wedge T}$ +2887,$\mathbb{E}[|X_i|]$ +2888,$a \ge 1$ +2889,$x_2:=\mathsf{E}[X^2]$ +2890,$\int_0^1 \frac{dx}{x}=\infty$ +2891,"$I=[x_{\min{}}, x_{\max{}}]\subset\mathbb R$" +2892,$\pm 3\sigma$ +2893,$\mathbb Q$ +2894,$C_p - C_v = R$ +2895,"$p_1,p_2$" +2896,$λ^*(N)=1$ +2897,$p_2$ +2898,$f(y;\theta)=c(y)e^{y\theta-\kappa(\theta)}$ +2899,$\nu>1$ +2900,"$H:(\Omega,\G)\to(\mathbb{R},\BB(\mathbb{R}))$" +2901,$\P_S(A\mid T)$ +2902,$x^2 - 2 = 0$ +2903,$\Gamma=\Phi^{-1}(D)$ +2904,$Y=Y_l+Y_s$ +2905,$\{\F_t\}_{t \geq 0}$ +2906,$\hat \theta<\theta_r$ +2907,$E[M_0]=0$ +2908,$\approx 0.999999999$ +2909,$J(x)=x^{-\alpha}$ +2910,$d=r/(1+r)$ +2911,$I_2$ +2912,$-\alpha >0)$ +2913,$-N_a<0$ +2914,$V_1(m)=m^3V(m)=\sigma^2m^3$ +2915,$V(m)=1-m$ +2916,$250$ +2917,$(-t)^a=|t|^a$ +2918,"$\omega\mapsto (\omega, T\omega)\in \Omega\times M$" +2919,"$Z\sim\mathrm{ED}^*(\theta,\lambda)$" +2920,$\bar A_x$ +2921,$g(s) = 1 - (1 -s)^\beta$ +2922,"$b\in (-1,1)$" +2923,$\alpha/\beta=\mu\alpha/(\alpha+1)$ +2924,$0.26$ +2925,$X=(x_{ij})$ +2926,"$\Delta m_{21}^2 \approx 7.53 \times 10^{-5} \, \text{eV}^2$" +2927,$1-X$ +2928,$\mathsf{E}[Y^n]$ +2929,$\approx 10^{-3}$ +2930,$\mathsf{Pr}(L\ge l)$ +2931,$h>0$ +2932,$1_G(ω)$ +2933,"$\{1,2,\dots,10000\}$" +2934,$(Z)$ +2935,$\xi$ +2936,"$\mathsf{CP}(\lambda, B) \sim \mathrm{Poisson}(\lambda p)$" +2937,$P_T$ +2938,$K_0(t)=\kappa(t)$ +2939,$G_1$ +2940,$J_n$ +2941,$1/CV^2$ +2942,$t\in\Omega$ +2943,"$\forall B\in\BB(\mathbb{R}),\ \int_B \mathsf{E}[X\mid T=t]\P_T(dt)=\int_{T^{-1}}(B) X\,d\P$" +2944,$V(0)=0$ +2945,$f(\mu) = 2 / \mu$ +2946,"$p:E\times \S\to [0,1]$" +2947,"$\px=\int_B \sum_i X_i\,d\P$" +2948,$T_x\ge x$ +2949,"$T:(M,\B)\to ??$" +2950,$l(y;\mu)=-y/\mu - \log\mu$ +2951,"$(x)^- = \max(-x, 0)$" +2952,"$\forall x\ \forall p\ \exists y[\forall u(u\in y\leftrightarrow(u\in x\wedge \phi(u,p)))]$" +2953,$s0$ +2993,$X(\omega)$ +2994,$\P(\cdot\mid X)$ +2995,"$X^{\tau_n} = \{X_{t \wedge \tau_n}, t \geq 0\}$" +2996,$A = \bigcap_{n=1}^{\infty} A_n$ +2997,$A_j$ +2998,$P_t$ +2999,$T(y)=y$ +3000,$ROL = EL + \lambda (\mathit{EL} (1 - \mathit{EL})/w)^{1/2}$ +3001,"$\Gamma(T)=\{ (\omega, T\omega) \mid \omega\in\Omega \}$" +3002,$\log(\sqrt{2\pi})=0.399090$ +3003,$\lambda = 1 / \sigma$ +3004,"$n_{\text{water, initial}} = \frac{P_{\text{water}}V}{RT} = \frac{4240 \times 1}{8.314 \times 303.15}$" +3005,$\{Y=y\}$ +3006,$f(x)=(x-d)^+1_{\{x \le m \}}$ +3007,$e^{\theta_2 y}$ +3008,$\beta_2$ +3009,"$\displaystyle\int HX \,d\P$" +3010,$\bar a_{b\!\urcorner}=(1-v^b)/\delta$ +3011,$\kappa'(\tau^{-1}(\mu))=\mu$ +3012,$L^1$ +3013,$x+\tau$ +3014,"$(\Omega, \F, \P)$" +3015,$S(x)$ +3016,$\alpha > 0$ +3017,$n\to\infty$ +3018,$\mathsf{Pr}(L'\ge 270)=65.1\%$ +3019,"$50 of the amount allowed on each claim in the classes under subs. (3) to (6), except for claims of the federal government under subs. (3) and (3c), shall be deducted from the claim and included in the class under sub. (8). Claims may not be cumulated by assignment to avoid application of the $" +3020,$f \mapsto e^{\theta X} f$ +3021,$\forall X\ \exists P\ \forall z\ [z\subset X\rightarrow z\in P]$ +3022,"$(V,\Theta)$" +3023,"$X_0, X_1, ..., X_t$" +3024,$\rho=\min_x\{x + \mathcal V(X-s\}$ +3025,$E\triangle F=(E\setminus F)\cup (F\setminus E)$ +3026,$10^{-13}$ +3027,$\alpha(\alpha+1)/\beta^2$ +3028,$\mathbb{Q}(\sqrt{2})$ +3029,"$P(\Gamma\cap Z, \omega)$" +3030,$\G\subset\F$ +3031,$\mu_*(E_0)=0$ +3032,$2aw$ +3033,$n^2$ +3034,"$[0,1]\times \{y\}$" +3035,"$\displaystyle\int_0^a \alpha_i(x)\,dx$" +3036,$t-\delta t$ +3037,$\{X=a\}$ +3038,$\theta_s>0.5$ +3039,$\mathsf{E}[X_t]=\lambda t$ +3040,$K_{X+Y} =\log M_{X+Y}=\log (M_{X}M_Y)=\log (M_{X}) + \log(M_Y)=K_X+K_Y$ +3041,"$\forall x\forall p\exists y[\forall u(u\in y\leftrightarrow(u\in x\wedge \phi(u,p)))]$" +3042,$p \leq q$ +3043,"$x_0, x_1, x_2$" +3044,$\kappa(\theta)=\theta^2/2$ +3045,$\alpha_X > \alpha_Y + 1$ +3046,$1_B$ +3047,$1 \times 10^9$ +3048,$f_0(x) = \sup_n f_n(x)$ +3049,$|X_n - X_{n-1}| \le K$ +3050,$A_x=\bigcap \{A\in\F\mid x\in A\}\in\F$ +3051,$iR$ +3052,$\bar A_{x+b} - \bar P_{x+b}\bar a_{x+b}=0$ +3053,$\mathrm{CV}(Z_1)/\sqrt{\nu}$ +3054,$(1/2x_0)\mathbb Z$ +3055,"$g(\omega_1,\omega_2)=(f(\omega_1), f(\omega_2))$" +3056,$\mathsf{E}[XM]$ +3057,$\xi(A\mid \G)(\omega)$ +3058,$\mathsf{E}[X_t \mid \F_{t-1}] = X_{t-1}$ +3059,$s_1$ +3060,$N=2^{256}=10^{77}$ +3061,$P(1-e)$ +3062,$p_0$ +3063,$\mathbb{R}^\infty$ +3064,"$[0,1]^\infty$" +3065,$\mathsf{E}[X]=\lambdax)$ +3067,$G\in\G$ +3068,$\mathsf{E}[\cdot\mid Y]$ +3069,"$\G=\{\emptyset, \Omega\}$" +3070,$\P(A\mid\G)$ +3071,$\mathsf{Pr}(X)$ +3072,$e^x=\sum_{n\ge 0} x^n/n!$ +3073,$\theta_s<0.5$ +3074,$a=1/2$ +3075,$2X_{t-1}\epsilon_t$ +3076,$W(0)\neq 0$ +3077,"$(L,\A, \mu)$" +3078,"$[-k, \epsilon] \succeq [0, \epsilon - k]$" +3079,$0.9$ +3080,$v+d=1$ +3081,$\eta=0.49$ +3082,"$(M,\B, \nu)$" +3083,$\ell(I_k)$ +3084,$\theta=0\in\Theta_p$ +3085,$g(0)=0$ +3086,$\alpha_Y \le \alpha_X$ +3087,"$\forall\,\omega,t\in M$" +3088,$\alpha\to 0$ +3089,$\A_y\supset \A$ +3090,$ and so $ +3091,"$(0,t_2]$" +3092,$Mg+F_2>Mg$ +3093,$\P(A\mid\G) = 1_A$ +3094,$x_0$ +3095,$S\P$ +3096,$\mathsf E[X_i \mid X](\omega) = \mathsf E[X_i \mid X=X(ω)]$ +3097,$\theta-\log(1+\theta))=y$ +3098,$O$ +3099,$d=iv=1-v$ +3100,"$I_x=[x, x+dx]$" +3101,$x=1$ +3102,$y_{i}$ +3103,"$642 billion of gross written premium in 2017, and, after reinsurance, net earned premium was $" +3104,$Y_t$ +3105,$\mathcal X$ +3106,"$\Q(F)=\R(X\times F),\ \forall F\in\B$" +3107,$M_F$ +3108,$V_1(m)=m^3V(1/m)$ +3109,$q_2$ +3110,"$f:(L,\A)\to(M, \B)$" +3111,$\mathsf{E}[1_B\mid \G]$ +3112,"$p(\cdot, y)$" +3113,$\mathcal{F}_k$ +3114,$\mathsf{Pr}(L\ge 270)=77.1\%$ +3115,$\tilde\Theta$ +3116,"$(-\infty,\infty)$" +3117,$1-1/2 + 1/3-1/4+\cdots$ +3118,$\forall t\in M$ +3119,$\emptyset = \forall x(x!=x)$ +3120,"$P(\omega, B)$" +3121,$p_1=3-p$ +3122,"$\mathrm{ED}^*(\theta, \lambda)$" +3123,"$\alpha,\beta$" +3124,$-br$ +3125,$\Delta G = \Delta H - T\Delta S$ +3126,"$\mathrm N(0, \sigma_n^2)$" +3127,$\P(A\mid\sigma(T))$ +3128,$g(S(x)) > S(x)$ +3129,$\kappa_p(\theta)$ +3130,$T(x)=\{2x\}$ +3131,$T'$ +3132,$\{\F_t\}$ +3133,$p-1$ +3134,$T_n = \sum_{i=1}^{n} T(X_i)$ +3135,"$ Finally, since we are interested in the left-shift $" +3136,$f_{\max{}}$ +3137,$E_0+A$ +3138,$D(F)\cap A$ +3139,$\sqrt{x_2/\lambda}$ +3140,$p=0.7$ +3141,$r \leq q$ +3142,$f_0 * f= f$ +3143,"$\ddot{a}_{x, n\!\urcorner}$" +3144,"$p\not=1,2$" +3145,$3^{-n}$ +3146,$\{A_n\}$ +3147,$E[X_\tau] = E[X_0]$ +3148,$I_2=t^ae^{-ia\pi/2}\Gamma(-a) \ge 0$ +3149,$\mathsf{Pr}(N\in I + kP)$ +3150,$V(\mu)\propto \mu^p$ +3151,$\mathbb R^+$ +3152,$d(y;y)=0$ +3153,$\kappa(\theta)=-\log(-\theta)$ +3154,$s(X)=s(X;\mu)$ +3155,"$\displaystyle\int_B X \,d\P$" +3156,$f_P$ +3157,$9 = 2^3 + 1 = 2^{(2^1 + 1)} + 1$ +3158,$ROL = a + b\ \mathit{EL} + c \ C(t)$ +3159,$A=g^a$ +3160,$X_t \neq \mathbb{E}[0 \mid \mathcal{F}_t]$ +3161,$1<\alpha<2$ +3162,$1< \alpha<2$ +3163,$\partial L/\partial \mu$ +3164,$X_t=ct + \sigma B_t$ +3165,$\mathcal{G}$ +3166,$b \urcorner$ +3167,$T=t$ +3168,$a=\mathsf{E}[X\mid X\ge a^*] =\mathsf{E}[X_1\mid X\ge a^*] + \mathsf{E}[X_2\mid X\ge a^*]$ +3169,$a=b^c$ +3170,$N-A$ +3171,$b + v_i / r$ +3172,$X:\Omega\to\mathbb{R}$ +3173,$X_n=1/n$ +3174,"$\omega\mapsto p(\omega, A)$" +3175,$A^c \in \mathcal{M}$ +3176,$R^2$ +3177,$B_t\le x$ +3178,$\bar\theta_s>0.5$ +3179,$λ_*(N)=0$ +3180,$\mathsf{Var}(X)=1$ +3181,$\sup_t \mathbb{E}[|X_t|^p] < \infty$ +3182,$\sigma(\C)$ +3183,$g:\Omega\times\Omega\to\mathbb{R}^2$ +3184,$x_{\min{}}\not=0$ +3185,$c=\mathsf{Var}(G)=\nu^2$ +3186,$\mathsf{E}[X\mid \G](\omega)$ +3187,$V(\mu)=\kappa''(\tau^{-1}(\mu))=1/(\tau^{-1})'(\mu)$ +3188,$\text{Maximum number of electrons in a shell} = 2n^2$ +3189,$X^T$ +3190,$A\in\F$ +3191,"$[0, P)$" +3192,$\mu=\lim_{\alpha\to-\infty} (2-\alpha)/(-\theta)$ +3193,$(T\lambda)\{\lambda_t\Omega=0\}=0$ +3194,"$(\Omega, \mathscr F, \mathsf P)$" +3195,"$(X_1, \dots, X_d)$" +3196,$V(\phi)$ +3197,$a=(1-f)^2/\nu^2=(1-f)^2/c$ +3198,$-10$ +3199,$N'/aw$ +3200,$B_n$ +3201,$\Gamma=\mu\otimes\Lambda$ +3202,$\mathsf{E}(G^r)=\theta^r\Gamma(a+r)/\Gamma(a)$ +3203,$\rho\ge 0$ +3204,$E_0+a_2$ +3205,$\mathit{LGD}$ +3206,$V_2$ +3207,$T_n=T\wedge n$ +3208,$\delta\downarrow 0$ +3209,$G'$ +3210,$f(A)=B\cap f(L)$ +3211,$\times T^4 \propto 4\pi R^2T^4$ +3212,$\mu\{ l<1\}=0$ +3213,$t=1$ +3214,$h$ +3215,$t\to\infty$ +3216,$2^1\rightarrow 3^1-1=2 \rightarrow 1 \rightarrow 0$ +3217,"$f(\cdot, t)$" +3218,"$A\mapsto p(t, A)$" +3219,"$L=\min(\max(X-a, 0), y)$" +3220,$-1<-\alpha<0$ +3221,$T_{x/n}$ +3222,$-t^2/2$ +3223,$C_v = \frac{3}{2}R$ +3224,$p=(\alpha-2)/(\alpha-1)$ +3225,$\mathsf E[X_i \mid X](\omega) = \mathsf E[X)i \mid X=X(ω)]$ +3226,$M_G(\zeta) = (1-\theta\zeta)^{-a}$ +3227,$52.6 \times 10^6$ +3228,$\F_s$ +3229,"$\mathrm{rcdf}(\P,\G)$" +3230,$\lim_{\mu\to 0} V'(\mu)=\delta$ +3231,"$k=6,7,8,9$" +3232,$\Theta_p$ +3233,"$j=1,\dots, d$" +3234,$e^{\theta y}$ +3235,$M_1$ +3236,$7.35 \times 10^{22}$ +3237,$ω$ +3238,$\lambda_i$ +3239,$\mathsf{CP}_n$ +3240,$-br-v_i<0$ +3241,$X0$ +3250,"$[1,\infty)$" +3251,$B(b)<0$ +3252,"$\int_0^1 j(x)\,dx=\infty$" +3253,$d(y;\mu)$ +3254,$P^T_S(\cdot\mid t)$ +3255,"$\P,\mu,P^T_S(\cdot\mid\cdot)$" +3256,$\beta_0$ +3257,"$([0, 1], \tF,\P)$" +3258,$\lambda=\mu^{2-p}(\alpha+1)/\alpha\to \mu(\alpha+1)/\alpha$ +3259,$g(x)=1$ +3260,"$p:\Omega\times\mathsf{E}E\to [0,1]$" +3261,"$(\B,Y\setminus N)$" +3262,$t\in M$ +3263,$\tau^{-1}(\mu)=-\mu^{-1}$ +3264,"$S, T$" +3265,$h(0.05) = 1-g(1-0.95) = 0.0203$ +3266,$\mathsf{skew}(G)=\mathsf{skew}(G')$ +3267,$\alpha \uparrow 2$ +3268,$n=5$ +3269,$P_t\{T=t\}=1$ +3270,$q_\alpha$ +3271,$\lambda_{\text{max}}$ +3272,$X \mapsto bX$ +3273,"$p\not=1,2,\infty$" +3274,$f=f_0$ +3275,$10^{-10}$ +3276,$ rather than 1. Since $ +3277,$E[\tau] < \infty$ +3278,"$\int_1^\infty xj(x)\,dx$" +3279,$\nu\ll T\P$ +3280,$1 + \mu^2 + \sqrt{1+\mu^2}$ +3281,$0.8\times 1.2 = 24/25$ +3282,$f(kb)b$ +3283,$\theta_r=0.40$ +3284,$\mathsf{E}[M_t | \mathcal{F}_s] = M_s$ +3285,$5.186592 \times 10^{19}$ +3286,$1 \le p \le 2$ +3287,$\lambda F$ +3288,$\bar\alpha+1=1/(p-1)$ +3289,"$\P(A\mid \G):\Omega\to [0,1]$" +3290,"$\nu(B)=\int_B X\,d\P=\mathsf{E}[X1_B]$" +3291,"$(-\\infty, \\infty)$" +3292,$\bar\theta=0.5$ +3293,$^{128}$ +3294,$\exp(4t^2\sigma^2/2)$ +3295,$T_1$ +3296,$n=2^4=16$ +3297,"$g = 9.81 \, \text{m/s}^2$" +3298,$A=G\triangle P$ +3299,$\P(A\cap B)=\P(A)$ +3300,"$\mathrm{ED}(\mu, \sigma^2)$" +3301,$8.617 \times 10^{5}$ +3302,$\mu=1/c$ +3303,$\mathsf{E}[X_1]\ge 0$ +3304,$ah_1$ +3305,$\mathsf{Pr}(X_i=0)>0$ +3306,"$b, -b$" +3307,$r\times$ +3308,$x_{i0}$ +3309,$G_t$ +3310,$A\subset B$ +3311,$T_2$ +3312,$A=A(J)$ +3313,$U_n$ +3314,$\G_Y=\sigma(Y)$ +3315,$K'(0)=\kappa'(\theta)=\mu$ +3316,"$Mg,Fe)_2SiO_4$" +3317,$P^T_S$ +3318,$L(t)=S(T^{-1}(t))=\{ S(\omega)\mid T(\omega)=t \}$ +3319,$Y=-X$ +3320,$a = v^2 / r$ +3321,"$\mathsf{E}[X\mid\G](\omega)=\int X(\nu)\,p(\omega, d\nu)$" +3322,$\mathcal{F}_{t-1}$ +3323,$X_T = \mathsf{E}[X_\infty\mid\F]$ +3324,$(1-p + pe^t)^n$ +3325,$(-2N\log(1-p))^{1/2}=22.49$ +3326,$3.8\times 10^{26}$ +3327,$\mathbb R/A$ +3328,"$\nu(B)=\int_B P(A\mid\G)\,d\P$" +3329,$\int f(-x)dx=-\int f(y)dy=-F(y)=-F(-x)$ +3330,$2(2l+1)$ +3331,"$S:(\Omega,\F)\to(L,\A)$" +3332,$r=0.05$ +3333,$\ge$ +3334,$\P(D) = 1$ +3335,$Y_{s+t}-Y_s$ +3336,$ the net effect of this transformation is to shift left by $ +3337,$c(y;\nu)$ +3338,$3.40 per $ +3339,$-18$ +3340,$\ge x$ +3341,$P_t B$ +3342,"$(\Omega, \G)$" +3343,"$C = \{ω:P(\{ω\}, ω) = 1\}$" +3344,$V_{G_\nu}$ +3345,$(\theta)$ +3346,$\{\omega\}\in\G\ \forall\omega\in\Omega$ +3347,$chi^2$ +3348,$+1$ +3349,$k+1$ +3350,$E(G-E(G))^3 = 2a\theta^3$ +3351,$b\!\!\urcorner$ +3352,$ It is the reciprocal of $ diff --git a/greater_tables/data/book.csv b/greater_tables/data/book.csv new file mode 100644 index 0000000..436e08c --- /dev/null +++ b/greater_tables/data/book.csv @@ -0,0 +1,5849 @@ +,expr +0,$\mathbf {s_3}$ +1,$\bar M$ +2,$(1+r)\lambda \mathsf{E}[X]$ +3,$m(1)=m_3=0$ +4,$X_2=2$ +5,$a=1$ +6,$e^{-kX}/\mathsf{E}[e^{-kX}]$ +7,$U < s$ +8,$n \le pN < (n+1)$ +9,"$\mathsf{TI,\ MON}$" +10,$\log(g')$ +11,$\mathsf{E}_{\mathsf Q}[X\mid \mathcal F]=\mathsf{E}[XZ\mid \mathcal F]/\mathsf{E}[Z\mid \mathcal F]$ +12,$(.*?)\$ +13,$\rho(X)=\infty$ +14,$F(x-) = \lim_{t\uparrow x} F(t)$ +15,"$\mathsf{MON,\ TI,\ PH}$" +16,$Y\succeq Z$ +17,$|S|$ +18,$\mathsf{CONVEX}$ +19,$s^{1/2}$ +20,$1000e^{\mu}$ +21,$p^* =0.7501$ +22,$X=\sum_j X_j$ +23,$\beta_{2}$ +24,$\sigma=0.50$ +25,$Z(s)=\Phi^{-1}(s)$ +26,$\hat p=1-g^{-1}(1-p)$ +27,$\kappa_i(X)=\mathsf{E}[X_i\mid X]$ +28,$\mathsf{E}[X_i\mid X](\omega)$ +29,$\sigma^2 t$ +30,$\uparrow\uparrow$ +31,$F(x)=1-e^{-x/\mu}$ +32,$g(S(X))$ +33,$0<\rho\le 1$ +34,$P = \mathsf{E}[X] + \pi\mathsf{E}[X]$ +35,$\bar Q_{0}=a_{0}-\bar P_{0}$ +36,$s\downarrow 0$ +37,$X=\frac{1}{n}\sum_i X_i$ +38,$>(s_0/2^{n+1})2^n\bar q(s_0)=s_0\bar q(s_0)/2$ +39,$\rho(X)>\max(X) g(0+)=\infty$ +40,$\mathsf{E}[Z]\le 1$ +41,$\lambda\to\infty$ +42,$\mathsf{j}(a)=6$ +43,"$g(s)=w+(1-w)s, s>0$" +44,$\mathsf{TVaR}_{0.65}$ +45,$c(S)=g(\mathsf{Pr}(S))$ +46,$c(S\cup\{i\})=c(S)+c(i)$ +47,$\mu(\{p_j\})$ +48,$\mathsf{Pr}(E')+\mathsf{Pr}(E)=\mathsf{Pr}(\Omega)=1$ +49,$q(Y)$ +50,"$(\Omega, \mathcal F, \mathsf{Pr})$" +51,$Z_A$ +52,$\mathcal D(X)\ge 0$ +53,$p=\text{Pr}[L^* > A]$ +54,"$\beta_H:=\mathsf{cov}(r_H, r_M)/\mathsf{var}(r_M)$" +55,$X_{t+dt}=X_t + \mu dt + \sigma dW_{dt}$ +56,$\rho(X)\ge \mathsf{E}[X]$ +57,$u(x)=-v(-x)$ +58,$g(x)=1$ +59,$F_{\mathbf{v}}(x)=s$ +60,${n}-X_2$ +61,$U_X > p$ +62,$b_i$ +63,$\rho(\nu Z) \le \nu\rho(Z)$ +64,$\Phi(x):=\int_{-\infty}^x \phi(t)dt$ +65,$\mathsf{E}[X]=27.5$ +66,$U = A$ +67,$X\le l$ +68,$U_X < p$ +69,$g'(1-p) \frac{q\wedge \alpha}{q}$ +70,$rpq$ +71,$c>0$ +72,$Y=0$ +73,$\mathbf \Omega$ +74,$\rho(X)=\max_k \mathsf{E}_{\mathsf Q_k}[X]$ +75,$1-p_0$ +76,$L(X)=k(X-\mathsf{E} X)$ +77,$P = \mathsf{E}[Xe^{\pi X}]/\mathsf{E}[e^{\pi X}]$ +78,"$(p, 1-g^{-1}(1-p))=(p,\hat p)$" +79,$\mathit{MV}(a)$ +80,$Z_4$ +81,"$\kappa_i(\mathbf{v}, x)$" +82,"$x=A,L,S$" +83,$c(S)=\rho(\sum_{i\in S} X_i)$ +84,$S_X(a)$ +85,"$a,b=\pm 1/n$" +86,$\mathbf {X_{2}(a)}$ +87,"$x_{1,i}, x_{2,i}$" +88,$1_{X>a}$ +89,"$\int_0^\infty -z(x)\,dF(x)=-1$" +90,$k\mapsto k\rho(X)$ +91,$\rho_g(X)=\mu+\lambda\sigma$ +92,$\hat q$ +93,$F_X^{-1}(V)=q_X(V)$ +94,$0\le\beta<1$ +95,$p>S(x^*)$ +96,$a\le X\le b$ +97,$P(x)=A(1_{X>x})=g(S(x))$ +98,$g(S)\Delta X'$ +99,$1<\lambda=k+f$ +100,$\rho(X)=\mathsf{E}[X] + c\mathsf{Var}(X)$ +101,$1./16=0.0625$ +102,"$\alpha>1,0\le\beta\le 1$" +103,$\mathsf{Pr}(A)=1-p$ +104,$g''(s)\le 0$ +105,$S(x_{max})=0$ +106,$\{X=x\}$ +107,$\rho_g(X\wedge a)$ +108,$Z=(1-p)^{-1}1_{\tilde X>q_{\tilde X}(p)}$ +109,$Z_1$ +110,"$X_{t-1,1}$" +111,$X_2(10)$ +112,"$X_{t,3}$" +113,$X\le x$ +114,$r = (g(s)-s)/(1-g(s))$ +115,$1_A/\mathsf{Pr}(A)$ +116,$\mathsf{TVaR}_1(X)$ +117,$\rho(Y)=\rho(X)g(p)=g(q)g(p).$ +118,$M(x)=g(S(x))-S(x)$ +119,$Y_{1}$ +120,$\mathsf{Pr}(X0$ +124,$X_n(\omega)\to X(\omega)$ +125,$^{***}$ +126,$\bar S(a)$ +127,$\mathsf{E}[X_i g'(S(X))]$ +128,$\sum (X\wedge a)p$ +129,"$\{1,2,\dots, N\}$" +130,$D\rho_{X_g}(X_c)$ +131,$(g(s)-s)/(1-g(s))=\iota$ +132,"$P_X(a,b] = F(b)-F(a)$" +133,$k > 0$ +134,$X_n\downarrow X$ +135,$x\to \infty$ +136,$\Phi(Z(s))=s$ +137,$q^-(p) = \inf\ \{ x\mid F(x) \ge p\}$ +138,$Y(\omega_1)\le Y(\omega_2)$ +139,$v(A)\le v(B)$ +140,$\mathbf {1_{X>x}}$ +141,$\alpha_i(a) S(a)$ +142,$\mathsf{E}[X]=\mathsf{TVaR}_0(X)$ +143,$\mathbf {Z_2}$ +144,$\hat{\tilde p}=1-g^{-1}(1-[1-g(1-p)])=p$ +145,$\pi(X)=\log(m_X(\alpha)) / \alpha$ +146,$\log(\mathsf{E}[e^{\pi X}])/\pi$ +147,$E[s|W=t]$ +148,$S(x)\gg 0$ +149,$1-\beta_i(x)g(S(x))$ +150,$S_X(x)=\Phi(-(x-\mu)/\sigma)$ +151,$\pi(X) = \rho(X\wedge \alpha(X))$ +152,$a(\mathbf{v}) =\mathsf{VaR}_p(X(\mathbf{v}))= q_{\mathbf{v}}(p)$ +153,$\mathsf Q \in \mathcal Q$ +154,$a=D+S$ +155,"$\bar P_{t,0}$" +156,"$0, 8, 10$" +157,$Q(x)/(1-S(x))$ +158,$p=1/6$ +159,"$\rho_2(X)=\mathsf{E}[X] + \mathsf{cov}(X,Z)$" +160,$\mathbf {g(S)}$ +161,$\rho=\mathsf{TVaR}_{0.95}$ +162,$f(S_t)=\log(S_t)$ +163,$\int_0^\infty xdF(x) =\int_0^\infty xf(x)dx$ +164,$u_j(x)$ +165,$f_{xx}=-1/S_t^2$ +166,$\mathbf {M_{2}\Delta X}$ +167,$\mathsf{E}[X\mid \mathcal F_t]$ +168,$X$ +169,$t+2$ +170,$n\ge m$ +171,$\mathbf {Z_4}$ +172,$|f|$ +173,$b$ +174,$g'(S(x))$ +175,$\mathsf{var}(Y_{d})=\sum_{s>d} \sigma_s^2$ +176,$r_l$ +177,$\mathbf {Z_8}$ +178,"$\rho(Y_{2,0})$" +179,$1+\iota^*=(1+\iota)(1+\tau)$ +180,$r_f/(1+r_f)$ +181,$L^r$ +182,"$\mathsf{E}[(X_i-\mathsf{E} X_i)(X-\mathsf{E} X)]/\mathsf{SD}(X)=\mathsf{cov}(X_i,X)/\mathsf{SD}(X)$" +183,$u(0)=0$ +184,$(ng)$ +185,$\tilde Z = \mathsf{E}[Z\mid X]$ +186,$E[X|X>qp]$ +187,$\rho(X) + c = \rho(X+c)\ge \rho(X) + \mathsf{E}[cZ]$ +188,$1-g(S)$ +189,$a_{0}$ +190,$\bar M_t = \bar P_t - \mathsf{E}[Y_{t}]$ +191,$\rho_g(X \wedge a)$ +192,$\rho(0)=\rho(0 \times X)=0\times \rho(X)=0$ +193,$\rho_g(X)$ +194,$\mathbf {\mu}$ +195,$\displaystyle\int_\Omega X(\omega)p(\omega)\mathsf{Pr}(d\omega)$ +196,"$n={{n}}, p=1/{{p}}={{pf}}$" +197,$\Delta Q_{gc}(a) = a_{gc}-P(X_{0}(a_{gc}))-a$ +198,"$\bar S_i = \sum_{j} X_{i,j}p_j$" +199,$\mathcal G\subset\mathcal F$ +200,$\tilde X_2 = X_2 - \mathsf{E}[X_2]$ +201,$10^{-12}$ +202,$\rho(X)=\mathsf{E}[XZ]$ +203,"$x\in[0,\infty)$" +204,$\mathsf{Pr}(S_t > a)=\mathsf{Pr}(X_t > a/S_0)=1-\Phi\left([\log(a/S_0)-(r-\sigma^2/2)t]/\sigma\sqrt{t} \right)=\Phi(d^*-\sigma\sqrt{t})$ +205,$F_0 = \bar P_{act}-\bar P = R-\bar M$ +206,$\mathsf{E}_\mathsf{Q}[X+c]=\mathsf{E}_\mathsf{Q}[X]+c$ +207,$X_{-3}$ +208,$\bar\delta$ +209,$t>0$ +210,"$(\Omega, \mathcal F, \mathsf{P})$" +211,$\mathit{LGD}$ +212,$\mathsf{E}[L\wedge A]$ +213,$\mu_c$ +214,$p<0.5$ +215,$a_h=2-a_l<2-b_l=b_h$ +216,"$F(p)=\mu([0,p])$" +217,$\mathsf{E}_\mathsf{Q}$ +218,$\lambda dt\to 0$ +219,$0 < p_0 < p_1 < 1$ +220,$\mathsf{E}[X] + d(\max(X)-\mathsf{E}[X])$ +221,$p\mapsto g'(1-p)$ +222,$\omega=0.\omega_1\omega_2\dots$ +223,"$\mathbf {X\,\Delta S}$" +224,$BCD$ +225,$\beta_i(x)<\alpha_i(x)$ +226,$\nu=\nu(p)$ +227,$a_1 = a(Y_{1})$ +228,$\mathit{NPV}_{\infty}=2\times 2.5=5$ +229,$dG/dF$ +230,$\mathbf {X(a)}$ +231,$M = P - \mu_U= 0.505$ +232,$H_k(X)=H_k(Y)$ +233,$l(p)$ +234,$\bar Q$ +235,$\mathsf{E}[N]=2.0$ +236,$L_0^{l_1} + L_{l_1}^{l_1+l_2} = L_0^{l_1+l_2}$ +237,$\mathsf{E}[X_d]$ +238,$X''$ +239,$\mathsf{VaR}_{0.7}(X)=2.439 > 2 \times 1.204=2.408$ +240,$\mathsf{CTE}^+$ +241,$0 < p < 1$ +242,$\displaystyle\int_0^\infty xg'(S_X(x))dF_X(x)$ +243,$\pi=0$ +244,$h(p)=1-g(1-p)=1-(1-p)^{1/3}$ +245,$\alpha(\mathsf Q)=\infty$ +246,$\gamma$ +247,$x\in A$ +248,$p_j=\mathsf{P}(X=x_j)$ +249,"$F_n,F$" +250,$\mathsf{Pr}(\cup_i E_i)=\sum_i \mathsf{Pr}(E_i)$ +251,$\rho(\lambda X)=\lambda\rho(X)$ +252,$\nu^{-1}\mathsf{E}[\nu(X)]$ +253,$A(1_{X>x})$ +254,$g(s)=(\iota+s)/(\iota+1)$ +255,"$\max(x, 0)$" +256,$x\mapsto x^{n}$ +257,$\mathsf{E}_{\mathsf Q}[X_i\mid X\le a](1-g(S(a))) + a\mathsf{E}_{\mathsf Q}[X_i/X\mid X >a]g(S(a))$ +258,$E[G]=1$ +259,$\Lambda = \dfrac{E( r_{U} ) - r_{f}}{\sigma_{r_{U}}}$ +260,"$\{90,\dots,99\}$" +261,$P = 3.103$ +262,$g(s) \ge s$ +263,$\mathsf{MONETARY}$ +264,$\mathsf{TVaR}_{0.95}(X)=\mathsf{E}[XZ]$ +265,$p(\omega)=0$ +266,$a(X_i;X) = \lim_{t\to 0} (\rho(X+tX_i)-\rho(X))/t$ +267,$\sigma_{U} = \sqrt{1 - 2p - p^{2}} = 0.973$ +268,$\sigma_A$ +269,$\mathsf{E}[X_1Z]$ +270,$\beta$ +271,$\mathbf {x}$ +272,$\mathit{NPV}_1 = \bar Q - \bar Q = 0$ +273,"$X_4, X_5$" +274,"$g:[0,1]\to[0,1]$" +275,$X+Y$ +276,$\sup_\mathsf{Q} \mathsf{E}_\mathsf{Q}[X]$ +277,$Y=1-X$ +278,$A\subset\Omega$ +279,$g'(s)\ge 1$ +280,$K_h(t):=k(h+t)-k(t)$ +281,$\mathscr{E}_i$ +282,$\rho_2$ +283,$y_c$ +284,$\mathsf{E}[X\mid t]$ +285,$1-F(q(p));\alpha)$ +286,$w(X)=1_{X>X_p}$ +287,$\delta=0$ +288,$q(0)$ +289,$|x|$ +290,$Y_n$ +291,$X_1+({n}-X_2)$ +292,$w=0.06405$ +293,$\sum_j Y_j = 0$ +294,"$P_X(a,b]=\mathsf P(X\in (a,b])=F(b)-F(a)$" +295,$e^{kx}S(x)\to\infty$ +296,"$f(\cdot, \omega)$" +297,$N_i$ +298,$\lambda S(x)$ +299,$\mathbf {M=g(S)-S}$ +300,$t=2$ +301,$\mathsf{E}[X_2(a)\mid X_1(a)=x] \le a-x$ +302,$0\le s\le 1$ +303,$\rho(X) \le 0$ +304,$x_{i-1}$ +305,$Y_{0}$ +306,$\infty-\infty$ +307,$\mathsf{j}(a) = \max\{j:X_j < a \}$ +308,$s \ne s^\ast$ +309,$\mathbf {d=1}$ +310,$\sigma_d^2$ +311,$P=L + \iota Q = \nu L + \delta a=L(1+\rho)$ +312,$\rho(X)=x_p$ +313,"$\mu=7.4, \sigma=1.9$" +314,$\mathsf{E}[X]+kR(X)$ +315,$\bar q(s/2)\le 2\bar q(s)$ +316,$Q_1=0.125$ +317,$\mathsf{E}[Z_j\mid X]$ +318,"$D_n, D_n^*$" +319,$\rho(X)=\mathsf{E}_{\mathsf{Q}}[X]=\mathsf{E}_{\mathsf{Q}}[\sum_i X_i]=\sum_i \mathsf{E}_{\mathsf{Q}}[X_i]$ +320,$a>b_h$ +321,$\sum_t Q_t$ +322,$0\le \lambda < 1$ +323,$\mathbf {t+2}$ +324,$-u''(w)/u'(w)$ +325,$q(p)=-\log(1-p)\mu$ +326,$\mathsf{E}_Q[X_i\mid X]=\mathsf{E}[X_i\mid X]$ +327,$1=v+d$ +328,$n=2$ +329,$\mathsf{P}(1_{U < s}=1)=\mathsf{P}(U < s)=s$ +330,$X=U$ +331,$X(\omega') = \sum_\omega X(\omega)1_\omega(\omega')$ +332,$a'$ +333,$U_i$ +334,"$\bar P_{0,1}$" +335,$g_i=u_i^{1/b} < u_i$ +336,$\rho(X\wedge a)=\bar P(a)$ +337,$E(X\wedge a)=\bar S(a)$ +338,$1-g(0^+)$ +339,$\alpha\not\equiv 0$ +340,"$[0,1]\times [0,1]$" +341,"$X_{i,j}\Delta g(S_j)$" +342,$c_i=\displaystyle\sum_{i\not\in S\subset\Omega}\dfrac{|S|!(N-|S|-1)!}{N!}\times$ +343,$\mathbf {\sigma}$ +344,"$\mathit{MV}(X, a) = a - \rho(X\wedge a)$" +345,$u'(0)=1$ +346,$S(x)=0.1$ +347,$s=0.01$ +348,$\int_a^{a+y} g(S(x))dx$ +349,$\sum X_i(a)p$ +350,$\beta(x)\le \alpha(x)$ +351,$X_1=18$ +352,$g(s)$ +353,$Z'(s)=1/(\Phi'(Z(s)))=\sqrt{2\pi}\exp(Z(s)^2/2)$ +354,$D/L$ +355,"$S\,\Delta X$" +356,$a=11$ +357,$\rho(X+tY)\ge \mathsf{E}_{\mathsf Q_X}[X+tY]$ +358,$\mathsf{E}[X_1]=4.75$ +359,$\log(1-1/n)<-1/n$ +360,"$, which he describes as the standard way to obtain the $" +361,$\phi(p) = g'(1-p)$ +362,$\mathsf{VaR}_p(X_1+X_2)\le \mathsf{VaR}_p(X_1)+\mathsf{VaR}_p(X_2)$ +363,$P(X_i(a_{gc}))$ +364,$n$ +365,$t > 1/3$ +366,$\mathsf{E}[u(P-X)]=0$ +367,$\mathsf{Var}(\pi)$ +368,$g'(S(x))f(x)$ +369,"$(lee.west |- lee.north)+(0,-2.5)$" +370,"$D^n\rho_X(X_{i,\cdot})$" +371,$-x^2$ +372,$X_n\to X$ +373,$r_f/(1+ r_f) = 0.0196$ +374,$\mathbf {g_4(s)=s^{0.9}}$ +375,$D\rho_{X_n}(X_c)$ +376,$f_{opt} =(pb - q)/b$ +377,$\{n\mid X(n)\not =0\}$ +378,$\mathsf{TVaR}_0(\cdot)=\mathsf{E}[\cdot]$ +379,$\mathbf {\iota}$ +380,$\rho(X_0+Y) \ge \rho(X_0) + \mathsf{E}[YZ]$ +381,$\ge 1$ +382,$n-3$ +383,$Q = C + lg$ +384,"$(1-p, 1]$" +385,$\tilde X-X$ +386,$\Delta Q_{ro}(a)$ +387,$\mathsf{E}[Z_1]=\mathsf{E}[Y]$ +388,$\lim_{x\to\infty}F(x)=1$ +389,$\mathsf{E}[X_i]=14$ +390,$g^{-1}$ +391,$p=0.9973$ +392,$M=P-s$ +393,$f(x_i)$ +394,$\mathcal F'_0\subset\mathcal F_0$ +395,$M/EL$ +396,$\mathit{EER}$ +397,$a(c_1;X) = c_1$ +398,"$\delta = 34/39, \nu=5/39$" +399,$\mathsf{P}(\{\omega\})$ +400,$A(X)-B(X)$ +401,$\rho(X\wedge a) = \sum\rho(X_i(a))$ +402,$q(0)=0$ +403,$k=c/(e^c-1)$ +404,$\Lambda = \dfrac{M - K r_f}{\sigma_U}$ +405,$\nu < 1$ +406,$\rho_g(X) = \infty$ +407,$U''(x)<0$ +408,$M = P \mu_U = 0.3$ +409,$\bar S_i(a)$ +410,$y=$ +411,$g'(S(x))=v$ +412,$\mathsf{Pr}(\{\omega_1\})=1/3$ +413,$\bar Q(a)$ +414,$\mathsf{j}(a)=4$ +415,$\mathsf{TVaR}_{0.8}(X)$ +416,$L/P$ +417,$\bar P(a+da)-\bar P(a)$ +418,$t+d$ +419,$g(0+)M$ +420,$Z(\omega)\mathsf{P}(\omega)$ +421,$\mathsf{E}[X_0]=80$ +422,$\mathbf {X_{1}(a)}$ +423,$t > 0$ +424,$g'(S(x))f(x)dx$ +425,"$k\mathsf{E}[(X_i-\mathsf{E} X_i)(X-\mathsf{E} X)]=k\mathsf{cov}(X_i,X)$" +426,$v_f(\mathsf{E}_\mathsf{Q}[X_i] - \dfrac{\mathsf{E}_\mathsf{Q}[X_i]}{\mathsf{E}_\mathsf{Q}[X]}\mathsf{E}_\mathsf{Q}[(X-a)^+])$ +427,"$(\x*0.65, 3.75*2)$" +428,$\rho$ +429,$\mathsf{E}_\mathsf{Q}[X_i]$ +430,$\hat p = F(x) = 1-g^{-1}(1-p)$ +431,"$\min(x_1,x_2)$" +432,${\mathsf{Q}}$ +433,$0=\rho(0)=\rho(X-X)\le \rho(X) + \rho(-X)$ +434,$c(\mathsf{var}nothing)=0$ +435,$f'_-(x)\le f'_-(y)\le f'_+(y)$ +436,$v_f\mathsf{E}_Q[X_i]$ +437,"$(x_{1,1}, x_{1,2})$" +438,$\sum_n 1/n$ +439,"$\displaystyle\int_0^a \alpha_i(x)S(x)\,dx$" +440,"$\beta(X,M)=\mathsf{cov}(X,M)\sigma_M^2$" +441,$X_{-1}$ +442,$\mathcal Q=\{\mathsf Q\mid \alpha(\mathsf Q)=0 \}$ +443,$A_i$ +444,"$a(X,p)$" +445,$r\lambda\mathsf{E}[X]$ +446,"$(s,\iota)$" +447,$a-L_0^a(X)$ +448,$\tilde Z=\mathsf{E}[Z\mid X]$ +449,$S(a+x)=d/dx(\mathsf{E}[X \wedge (a+x)-X \wedge a)$ +450,"$[p_{-},p_{+}]$" +451,$y=x$ +452,$\inf_x \{ x + \alpha\mathsf{E}[(X-x)^+] + \beta\mathsf{E}[(X-x)^-] \}$ +453,$af$ +454,$M$ +455,$\mathsf{Pr}(\mathsf{var}nothing) =0$ +456,$\mathsf{TVaR}_{p^\ast}$ +457,$\mu=0.107$ +458,$E(X_{-1}(a))$ +459,$g'(S_X)$ +460,$j > 0$ +461,$a=\sum_i a\alpha_i(a) = \sum_i\kappa_i(a)$ +462,$\mu=0$ +463,$x>1$ +464,$F(p)=p$ +465,$X_i$ +466,$q_{\tilde X}$ +467,$a\le \dfrac{P-S}{\iota} + P\approx \dfrac{P-\mathsf{E}[X]}{\iota} + P$ +468,$\omega\in \Omega$ +469,$Y_c=(Y\mid Y > y_c)$ +470,$(m_1-m_0)/s_1$ +471,$q_B(p)=\sup B$ +472,$\mathsf{E}[X]+k\mathsf{var}(X)$ +473,$M_1\Delta X$ +474,"$(a,b]$" +475,$\rho(m)=\rho(0)-m$ +476,$\mathbf v$ +477,"$\omega=(1,0,0,1,0,0,\dots)$" +478,$g(S(x))=1$ +479,$0 < s < 1/4$ +480,$r_h$ +481,$X\ge a$ +482,$Q$ +483,$p\delta_p$ +484,$y^{\ast}$ +485,$\nu=1/(1+\iota)$ +486,$\mu=0.1$ +487,$s_1=0$ +488,$p=0.4$ +489,$g(S_{X}(x))$ +490,$\mathsf{Q}(B_k)=\mathsf{P}(B_k)/\mathsf{P}(B_k)=1$ +491,$m(t^\star)=3m/4$ +492,$n_s(1-g(s))$ +493,"$g,h:[0,1]\to [0,1]$" +494,$x_{(j)}-x_{(j-1)}$ +495,$\mathsf{SRM}$ +496,$v\in V_X$ +497,$a(X_i)$ +498,"$\mathsf{var}(W)=\sum_{d\ge 0} \mathsf{var}(Y_{-d,d})$" +499,$\mathsf{E}_{\mathsf{Q}}[X]$ +500,$A/L$ +501,$a_{2}$ +502,$\rho_g(X)=\bar P$ +503,$\arg \min_{q \in \mathbb{Q}} E_q[U(a)]$ +504,$X=X_1+X_2$ +505,"$n=(0.702, 1.163)$" +506,$\sum_i$ +507,$\phi'(p)$ +508,"$(X_{1,j},\dots,X_{m,j})$" +509,$E(X\wedge a)$ +510,$1/6$ +511,$\mathsf{Pr}(\{\omega_2\})=2/3$ +512,"$\Omega=\{\omega_1,\omega_2,\omega_3,\omega_4\}$" +513,$a(X_i;X)\ge \mathsf{E}[X_i]$ +514,$\nu = 1/\lambda$ +515,$\alpha \le 1$ +516,$n\times m$ +517,$\mathsf{Q}$ +518,${6 \choose 2}=15$ +519,$\mathsf{E}[X \mid U]$ +520,$\sup(\lambda X)=\lambda \sup(X)$ +521,$P+Q=a$ +522,$k=2$ +523,$f(x) \to 0$ +524,$X=1$ +525,$v_1X_1(1)$ +526,$\pi=\Pi/p\nu(p)$ +527,$\mathcal{N}_X(X_i(a))$ +528,$\mathcal B_p$ +529,$S(x)\le s^*$ +530,$q_A \le q_B$ +531,"$A_2=[\epsilon, \epsilon]$" +532,$X=\sum_i X_i$ +533,$K = A - P$ +534,"$(1-g(s), 1-s)$" +535,"$r=1,2,3,4$" +536,$0=x_0 1-p) \}$ +549,$X=g(Z)$ +550,$P = \mathsf{E}[X] + \pi\mathsf{Var}^+(X)$ +551,$Y\mid Y > y_c$ +552,$a_1' = a_0-X_1$ +553,"$X_{t-1,3}$" +554,$\mathbf{B}(t)$ +555,$\mathsf Q\in\mathcal Q(X)$ +556,$g''<0$ +557,$g(w s_1 + (1-w)s_2) \le w g(s_1) + (1-w) g(s_2)$ +558,"$k=1,\dots,m$" +559,$S_t=S_0 X_t$ +560,"$G=\mathrm{cl}\{\, (\mathsf{E}_\mathsf{Q}[X_i], \mathsf{E}_\mathsf{Q}[X]) \mid \mathsf Q\in\mathcal Q \, \}$" +561,$\rho(-X)$ +562,$\mathsf{E}[X]\le \mathsf{E}[Y]$ +563,"$[s_1,1]$" +564,"$[0, 1-p]$" +565,$X(\omega)=1-\omega$ +566,$1-g(S(x))$ +567,$T = \min\{ t:U(t)\le 0 \}$ +568,$x_0=q^-(p_0)$ +569,"$\beta_i(t\mathbf{v}, x)$" +570,$\lambda=g(\lambda_{obj})$ +571,"$[-2\pi, 2\pi]$" +572,"$\mathsf{E}[X_i\,\mathsf{E}[Z\mid X]]$" +573,$X(\lambda\mathbf{v})$ +574,"$\bar P_{t,0} = D\rho_{W_t}(Y_{t,0})$" +575,$a>1$ +576,$a=R+Q$ +577,$k-L_0^k$ +578,$p\ge 0$ +579,$\mathsf{E}[\iota Q] = \mathsf{E}[\iota]\mathsf{E}[Q]$ +580,$\int g(S)$ +581,$\mathcal E(X)=\mathsf{E}[(p X^+ + (1-p)X^-)/(1-p)]$ +582,$0\le f<1$ +583,"$I(q,p)=0$" +584,$1_{X < q(1-s)}$ +585,$g - s$ +586,$x_i=1$ +587,$x\ge q(1-s^*)=:x^*$ +588,$\mathsf{TVaR}_0(X)=\mathsf{E}[X]$ +589,$X\succeq Z$ +590,$0\le w\le 1$ +591,$\mathsf{CTE}$ +592,$\iota = \dfrac{\delta}{1-\delta}$ +593,$X=x$ +594,$g^{-1}(s)$ +595,$U(0)=2$ +596,$\alpha = 0.642.$ +597,$s>1-p$ +598,$M_i := \beta_ig-\alpha_iS$ +599,${}^2$ +600,$C_c$ +601,$ROL = a + b\ \mathit{EL} + c \ C(t)$ +602,$X_2=0$ +603,$M=\delta a'$ +604,$\alpha(x) S(x)>\beta(x) g(S(x))$ +605,$P(X_{-1}(a_{gc}))$ +606,$L = \text{E}[L^*\wedge A]$ +607,$c(S)$ +608,$A\cap B\subset B$ +609,$g(s) = 1 - (1 - s)/(1 + r_f + Ck(s))$ +610,$X-b\le 0$ +611,$a=\mathsf{E}_\mathsf{Q}[X]$ +612,$f(x)=(\sqrt{2\pi}x)^{-1}\exp(-(\log(x)-\mu)^2/2\sigma^2)$ +613,$r_f=0$ +614,$\mathsf{VaR}_p(X)-f(\mathsf{VaR}_p(X))$ +615,$MX$ +616,$\mathsf{E}_\mathsf{Q}[\lambda X] = \lambda \mathsf{E}_\mathsf{Q}[X]$ +617,"$\displaystyle\int_0^{1-g(S(a))} \kappa_i(q(1-g^{-1}(1-p)))\,dp + a\beta_i(a)g(S(a))$" +618,$X(\omega)=\exp(10 + 2\Phi^{-1}(\omega))$ +619,$g(s)=\nu s + \delta$ +620,$\mathsf{E}[W\tilde X] \le \rho(\tilde X)$ +621,$W$ +622,$\mathsf{var}nothing$ +623,$f=f_x=f_{xx}$ +624,$1_A$ +625,$\wedge$ +626,$g'(s)$ +627,$a$ +628,$\mathsf{E}[Y]$ +629,$\rho(X)=\rho(\mathsf{E}[X]+X-\mathsf{E}[X])=\mathsf{E}[X] + \rho(X-\mathsf{E}[X])$ +630,$\mathsf{E}_\mathsf{Q}[X]$ +631,$X\wedge l$ +632,"$X_{t-d,d}$" +633,$\alpha(\mathsf Q)=0$ +634,$\bar q_{X_1+X_2}(s) \approx \bar q(s/2)$ +635,$X_2$ +636,"$(s,g(s))=(0.2,0.36)$" +637,$\mathsf{E}[kX]=k\mathsf{E}[X]$ +638,$ \& $ +639,$\inf_x\{ x + c{(X-x)^+} \}$ +640,$P(X\wedge a)$ +641,$x_2(S(x_1)-S(x_2))=x_2\mathsf{P}(X=x_2)$ +642,$1-g(S(a))$ +643,$\mathsf{E}_\mathsf{Q}[X_i \mid X]=\mathsf{E}[X_i \mid X]$ +644,$\| Z \|^*= \sup\ \{ \mathsf{E}[YZ] \mid \| Y \| \le 1 \}$ +645,"$Y_{1,0}$" +646,$\nu^{\ast}$ +647,$A(\lambda X)=A(\lambda X)$ +648,$dF$ +649,$\downarrow\downarrow$ +650,$\rho_2(X_1)=1$ +651,$-X$ +652,"$[x_1, x_2]$" +653,$v_f(\mathsf{E}_\mathsf{Q}[X_i] - \mathsf{E}_\mathsf{Q}[X_i/X(X-a)^+])$ +654,$\kappa_i(x)$ +655,$\mathbf {g_2(s)=s^{0.5}}$ +656,$r-r_L$ +657,$\mathbf {S\Delta X}$ +658,$\alpha_i(x) S(x)$ +659,$(g(s_0)-g_0)/s_0 = g'(s_0)$ +660,"$\mathbb{Q} = \left \{ q:I(q,p) \le I^* \right \}$" +661,$\rho=0$ +662,$\mathsf{E}_{\mathsf Q}[\cdot]$ +663,$\mathbf {Q}$ +664,$s=f'(x_0)$ +665,$\rho(X)=\sup(X)$ +666,$g(0+)>0$ +667,$S(x)=e^{-\beta x}$ +668,"$s_g, s_b$" +669,$1000$ +670,$da>0$ +671,$\mathbf {\beta_{2}g(S)\Delta X}$ +672,$\mathsf{P}(X=0)=0.4$ +673,$u'''\ge 0$ +674,$0\le \lambda_1 \le 1$ +675,$\rho(X+tY)\ge \mathsf{E}_{\mathsf Q_X}[X+tY]=\mathsf{E}_{\mathsf Q_X}[X]+\mathsf{E}_{\mathsf Q_X}[tY]=\rho(X)+t\mathsf{E}_{\mathsf Q_X}[Y]$ +676,$P_X$ +677,$x_1+x_2=x$ +678,$=\mathrm{MV}(X\wedge a)$ +679,$M_i(x)+Q_i(x)=\alpha_i(x)F(x)$ +680,$\delta = \iota/(1+\iota)$ +681,$a_1'=a_0-X_1$ +682,$X=\sum X_i$ +683,$\mathbf {S\Delta X'}$ +684,$X\le b$ +685,$\delta=\iota/(1+\iota)$ +686,$(\delta_p - il_p)/(\nu_p-l_p)$ +687,$x=\mathsf{VaR}_p(X)$ +688,$\mathbf {\alpha_2}$ +689,$1200/1800=0.667$ +690,$\sigma_0=\sigma_1$ +691,$a(f + (1-f)/q) -1$ +692,$g \cdot dX$ +693,$\beta_i(a)/\alpha_i(a) < 1$ +694,$\xtext$ +695,$Q_{1}\Delta X$ +696,$X_g$ +697,"$X=X(x_1,\dots,x_n)=x_1X_1 + \cdots + x_nX_n$" +698,$s\leftrightarrow 1-s$ +699,$\mathcal Q_i(X)$ +700,$\mathsf{E}[X] +\lambda\mathsf{E}[(X-\mathsf{E} X)^+]$ +701,$V_j$ +702,$X'=X\wedge a$ +703,$20+8t$ +704,$\mathsf{Pr}(X < x)\le \mathsf{Pr}(X\le x)$ +705,$\Delta_{2}$ +706,$\alpha_{2}$ +707,"$(1,1)$" +708,$4$ +709,"$Q_{i,j} = M_{i,j}/\iota_j$" +710,$L^\infty$ +711,$f(1)=1$ +712,"$0,10,40$" +713,$\rho(X+c)=\rho(X)+c$ +714,$H[Y_j]$ +715,$Z=(1-p)^{-1}1_A$ +716,$\beta_i(x)g(S(x))$ +717,"$A_3=[0, \epsilon-k]$" +718,$\mathsf{TVaR}_{0.95}$ +719,"$dx,dt,ds$" +720,$f(\omega)\ge 0$ +721,$\beta=0.57$ +722,$(X\wedge a)$ +723,$X < a$ +724,$\lambda<1$ +725,"$X_{0,1}$" +726,$\omega'\not=\omega$ +727,$X_0< X_1 < \dots < X_m$ +728,$\tilde X_1 + \tilde X_2 = X_1 + X_2$ +729,$\mathbf {X_1(a)}$ +730,$\mathsf{VaR}\_p(X\_0)$ +731,$-(1-s)g''(1-s) + g(0+)\delta_1 + \sum_s s(g'(s-)-g'(s+))\delta_{1-s} + g'(1)\delta_0$ +732,$a>a_{ro}$ +733,$g'(0)=\infty$ +734,$(X\wedge a)/X$ +735,$P_g\ll P_X$ +736,$Z\le (1-p)^{-1}$ +737,$F_g$ +738,$\bar P(x)$ +739,$d^*=(\log(A/L) + (r_h-\mu_L + \sigma^2/2))/\sigma\sqrt{t}$ +740,"$g(s)= \displaystyle\int_0^s g'(t)\,dt = (s/(1-p)) \wedge 1$" +741,"$(s_j=0,g_j>0)$" +742,$P'<\rho(W_1\wedge a_1)$ +743,$\mathsf{COHERENT}$ +744,$\Delta g(S_0)=1-g(S_0)$ +745,$\rho_g(V)$ +746,$X_t$ +747,$X_1+X_2=X=x$ +748,$m=1$ +749,$X_n\uparrow X$ +750,$v_1$ +751,$a\ge 10$ +752,$\mathbf {X_{1}}$ +753,$\gamma=0.633$ +754,$r=0.038$ +755,$1000(1+t)$ +756,$\mathbf {x_2}$ +757,$f(0)=0$ +758,$\mathcal M(\mathsf{P})$ +759,$p(\nu(p)-l(p))$ +760,$B(X)$ +761,$h(0.9)/0.9 = 0.76$ +762,"$\int_{[0,p]} \dfrac{\mu(dt)}{1-t}$" +763,$\mathsf{TVaR}_{0.5}(X_1)=9$ +764,${}^nS(t)$ +765,$Q(a)=\nu F(a)$ +766,$\rho(X_i)$ +767,$S(x_5)$ +768,$h_x$ +769,$\mathbf {Z_1}$ +770,$Y\le 0$ +771,$\mathsf{E}[X] + \pi \mathsf{E}[(X-\mathsf{E}[X])^+]$ +772,$(I/a + U/R)$ +773,$v=1/1.1<1$ +774,$0 < r \le 1$ +775,$\{ p \mid q^-(p) \le x \}=\{ p \mid p \le F(x) \}$ +776,"$(s,g(s))$" +777,$R_f=0$ +778,$\alpha_i'(x)>0$ +779,$\lim_{s\downarrow 0} g_\tau(s) = \tau / (1+\tau)$ +780,$\mathit{NPV}_1=0$ +781,$X\wedge a\Delta S$ +782,$\mathsf{TVaR}_{0.75}(X_2)=90$ +783,$K = A-P$ +784,$A\in\mathcal F'$ +785,$\le 0$ +786,$Z'(g(s))g'(s)=Z'(s)$ +787,"$\sum_i a(X_i, p^*)=a(X)$" +788,$a_{gc}:=\mathit{VaR}_{p}(X)=18000.0$ +789,$v=1/(1+i)$ +790,"$\alpha, \beta, \kappa$" +791,$S_{X\wedge a}(x) = S_X(x)$ +792,$W_0=Y_{0} + W_1$ +793,"$s_0, s_1, s_2$" +794,$AR$ +795,$S_j:=S(X_j)$ +796,$f'_-$ +797,"$ is average invested assets, equal to $" +798,$\mathsf{VaR}_{0.99}(X_2)=100$ +799,$X_t:=\mathsf{E}[X\mid \mathcal F_t]$ +800,$q(F(x))$ +801,$a_i$ +802,$X_1=t$ +803,$q=ps_g$ +804,$X>Y$ +805,$M=g(S)-S$ +806,$X=1800$ +807,$g_2(s)=s^{0.5}$ +808,$xS(x)|_0^\infty$ +809,$x_h(1-p)$ +810,$\nu+\delta=1$ +811,$\rho_i$ +812,$\mathbf {Q_{2}\Delta X}$ +813,$\mathsf{SSD}$ +814,$X_i\dfrac{X\wedge a}{X}$ +815,$r(X)=g'(S(X))$ +816,$X\wedge d$ +817,$1_{X>x_1}$ +818,"$\int g(S(x))\,dx$" +819,"$c(1,3)-c(3)$" +820,$0.5$ +821,$A(\lambda X)=\lambda A(X)$ +822,$\mathsf{Pr}(X=y_j)$ +823,$\mathsf{E}[u(R - X)]=0$ +824,$\rho(X\wedge a)=\mathsf{E}_\mathsf{Q}[X\wedge a]$ +825,$\mathbf {Z_5}$ +826,$c=(1-\alpha)^{-1}$ +827,$\mathsf{TVaR}_p(X)=1=\mathsf{E}_\mathsf Q[X]$ +828,$M_2dX$ +829,$\mathit{EGL}_{ro}(a)=P(X_{-1}\wedge a) - P(X_{-1}\wedge a_{ro}) \ge 0$ +830,$2\le x\le 8$ +831,$\mathsf{CTE}_p$ +832,$f(\mathsf{VaR}_p(X))$ +833,$\mathsf{E}_{\mathsf Q_k}[X'']=\mathsf{E}_\mathsf{P}[X'']$ +834,$X_n=X$ +835,"$Y_{t',d}$" +836,$\mathsf{E}[F_2]=\mathsf{E}[F_0]$ +837,$\mathsf{E}[e^{hX}] = \exp(h\mu+\sigma^2h^2/2)$ +838,$D\rho_X(X_i)=D\rho_i = x_i\dfrac{\partial\rho}{\partial x_i}$ +839,$a(X)\le a(Y)$ +840,$g'(s)<1$ +841,$\beta > \alpha$ +842,$\bar\iota=\iota$ +843,$\int_a^{a+y} S(x)dx$ +844,$0.125 \cdot 8 = 1$ +845,$h\left(\displaystyle\int_\Omega g(X(\omega))\mathsf{Pr}(d\omega)\right)$ +846,$\bar\delta(x)$ +847,$P_{act}-P$ +848,"$\rho(X, p^\star)=a(X)$" +849,$q(0.75)$ +850,$s=S_X(y)$ +851,$\rho l = \iota C$ +852,$\alpha(1-\alpha)(1-s)^{\alpha-1} + \alpha\delta_0$ +853,$Y_s$ +854,$\eta\nu$ +855,$(g_j-s_j)/(1-g_j)$ +856,$Z=g'(S_X(x))$ +857,$\mathsf{E}_{\mathsf Q}[Y]=\mathsf{E}[Yg'(S(X))]$ +858,$\Delta S_5$ +859,$F(x)$ +860,$D=(X-a)^+$ +861,$\sigma^2/2$ +862,$i=1$ +863,$h(p)\le p$ +864,$b = g/(1-g)$ +865,"$d=d(X_1,\dots,X_n)$" +866,$X=\max(X)$ +867,$v$ +868,$F(q(p))=p$ +869,$\mathsf{E}[Z]=1$ +870,$g(0+)=\mu(\{1\})$ +871,$\mathsf{E}[X\mid \mathcal F_{\tau}]$ +872,$X_i(a)$ +873,$p=0.999$ +874,$m\ge 1$ +875,$X_1(a)$ +876,$\Delta_s=g'(s-)-g'(s+)$ +877,$\mathsf Q \ll \mathsf P$ +878,$k/n$ +879,$L(X)=w(X)/\mathsf{E}[w(X)]$ +880,"$X_{t-1,2}$" +881,$\mathsf{Pr}(X\ge x)\ge 1-p\ge \mathsf{Pr}(X> x)$ +882,$d=1-v$ +883,"$f(t)=a(tx_1,\dots, tx_n)=ta(x_1,\dots, x_n)$" +884,$\partial a/ \partial v_i$ +885,$-g''$ +886,$g'(1)=0$ +887,$\mathsf{E}[X_ih(X)]=\mathsf{E}[\kappa_i(X)h(X)]$ +888,$\mathsf{E}[XZ(X)]$ +889,$P(a)=g(S(a))\ge S(a)$ +890,$x\mapsto x$ +891,$x^{\ast}=\mathsf{VaR}_p(X)$ +892,$\mathsf{E}[X] \le \bar P \le \sup X$ +893,"$(1,\dots,1)$" +894,$\mathsf{Pr}(X=x_i)=\lambda_i/\lambda$ +895,$Y=-X$ +896,$\lim_{y\downarrow x} f(y)$ +897,$\iota=0.1$ +898,$A_Y = 2.155$ +899,$g(S)=1$ +900,$X:=Y$ +901,$0.05$ +902,"$\mathbf {j, p, S, \kappa_1, \Delta X, \Delta(X\wedge a)}$" +903,$\mathsf{Pr}(M=m)=\frac{r}{1+r}\frac{1}{(1+r)^m}$ +904,$xS(x)\vert_0^\infty =\lim_{x\to\infty} xS(x)=0$ +905,$k!$ +906,$\kappa_i(x)=\mathsf{E}[X_i\mid X=x]$ +907,$602.6 billion and converted to net premium based on $ +908,$q(p)\phi(p)\times dp$ +909,$B_t$ +910,$ABC$ +911,$\lim_{x\to-\infty}F(x)=0$ +912,$\mathsf{E}[X_2\mid X=20]=6$ +913,$\mathbf {M_2\Delta X}$ +914,$a = 0.6565$ +915,$\mu(ds)$ +916,$p<\infty$ +917,$X_n(2/3)$ +918,$X_s$ +919,$x=q(p)$ +920,$q_X(p)=\mu+\sigma z_p$ +921,"$Y_{0,t}:=\sum_{d>t} X_{0,d}$" +922,$\mathsf{E}[X1_A] / \mathsf{E}[1_A]$ +923,$Z_{a}(a)$ +924,$\le p$ +925,$dx$ +926,$A = 8.14864$ +927,$L(X)=1_{X=x_p}(X)/f(x_p)$ +928,"$\{0, 8, 10\}$" +929,$\mathcal D(X)=c\mathsf{TVaR}_p(X-\mathsf{E}[X])$ +930,$P = \mathsf{TVaR}_\pi(X)$ +931,$w=w f(1)=w f(1)+(1-w)f(0) \le f(w 1 + (1-w)0)= f(w)$ +932,$Z_\mathit{lin}$ +933,$X_t=\mu t + \sigma W_t$ +934,$\alpha S$ +935,$f(x)=\sin(x)$ +936,$\mathbf {X_{2c}}$ +937,"$\Omega=\{\omega_1,\dots,\omega_n\}=\{\text{Ada}, \text{Bernhard}, \dots, \text{Zeno} \}$" +938,$\alpha(1+fg/(1-g))$ +939,$s > s_1$ +940,$t=2/3$ +941,$\int_0^s \phi(1-t)dt$ +942,$\rho(U)=\mathsf{E}_\mathsf Q[U]$ +943,$H_k(X) \le H_k(Y)$ +944,$X\preceq Y$ +945,$1-1/c$ +946,$0 < s < 1$ +947,$-\rho(-X)\le \mathsf{E}[X]$ +948,$\infty$ +949,$q(\hat p)$ +950,$Z=g'(S(X))$ +951,$n+1=N$ +952,$P=L/(1+R_L)$ +953,$\rho(X_n)\not\to \rho(X)$ +954,$X'\Delta g(S)$ +955,$\mathbf {x_1}$ +956,$\beta_i(X_4)$ +957,$s>0.2$ +958,$q_{X+c}(p)=c+q_X(p)$ +959,$X=q(F(X))$ +960,$0.2 < s < 1$ +961,$\mathsf{E}[X\mid \mathcal F'](\omega)$ +962,$t>0.5$ +963,$0 \le t \le 1$ +964,"$\mathsf{TVaR}_p(X(x_1,x_2))=(x_1 + x_2)\mathsf{TVaR}_p(Y)$" +965,$X_1\le X_2\implies a(X_1;X)\le a(X_2;X)$ +966,$\rho(X_j)=\max_k \mathsf{E}_\mathsf{Q_k}[X_j]$ +967,$\rho_c(X)=\mathsf{TVaR}_{0.8}(X)=8.5$ +968,$\mathbf {\Delta S}$ +969,$V_X$ +970,$\mathsf{E}[g'(S(X))]=\int_0^\infty g'(S(x))f(x)dx=\int_0^\infty -\frac{d}{dx}g(S(x))dx=g(S(0))-g(S(\infty))=g(1)-g(0)=1$ +971,$\rho(1)=1$ +972,"$(3,2)$" +973,$a_2'$ +974,$x_{i-1}\le x'_i\le x_i$ +975,$\mathsf{E}[ X_i \mid X(x) = q_{x}(p)]$ +976,$\mathsf{TVaR}_p(X)=(12(0.9-p) + 2.5)/(1-p)$ +977,$V$ +978,"$D^f\rho_{W_t\wedge a, W_t}(Y_{0})$" +979,$\mu$ +980,$\beta_i(x) =\mathsf{E}_{\mathsf Q}[X_i/X\mid X>x]$ +981,$y=(\log(x)-\mu)/\sigma$ +982,$\sup(X)<\infty$ +983,$+\infty$ +984,$F^{-1}(p)=q(p)$ +985,$Z(y_j)$ +986,$\bar Q_{d}=a_{d}-\bar P_{d}$ +987,$\rho(X_n) \uparrow \rho(X)$ +988,$\bar P_0>\mathsf{E}[Y_{0}]$ +989,$S(a)$ +990,$(1-g(s))(1-q)$ +991,$\Delta \mathit{MV}_{gc}(a)$ +992,"$X_1,\dots,X_m$" +993,$da1_{X>x}$ +994,$g_1F$ +995,"$\mathsf{E}[X_i(a)\,g'(S_{X\wedge a}(X\wedge a))]$" +996,"$\bar P_{0,t}:=\rho(Y_{0,t})$" +997,$x_0+x_1+x_2$ +998,$\bar S(a)=\displaystyle\int_0^a S(x)dx$ +999,$S(X_j)>0$ +1000,$f(s)=\alpha(1-\alpha)(1-s)^{\alpha-1}$ +1001,"$1_A:\Omega\to \{0,1\}$" +1002,$g(S(\infty))=0$ +1003,"$\alpha_i(a) = \dfrac{\sum_{j:X_j>a} (X_{i,j}/X_j)p_j}{\sum_{j:X_j>a} p_j}$" +1004,"$P_i,M_i, Q_i$" +1005,$C'_i$ +1006,$l_i$ +1007,$A(c)=c$ +1008,$I$ +1009,$X\preceq_m Y$ +1010,"$(-\x, 2)$" +1011,"$\rho(X),\rho(Y)\le 0$" +1012,$a_{d} = \mathsf{E}[Y_{d}]+4\sigma(Y_{d})$ +1013,"$X_{0,t}$" +1014,$a-X\le 0$ +1015,$m_3=0$ +1016,$\mathsf{E}[Z]\ge 1$ +1017,$\mathsf{E}[X_iZ_j]$ +1018,$\rho(W_1\wedge a_1 \wedge a_1')$ +1019,$\mathsf{E}[XZ]$ +1020,"$\mathsf{CONVEX,LI}$" +1021,$1_{X>x}$ +1022,$\tau a$ +1023,$E\in\mathcal F$ +1024,$a/Q = 1 + R/Q$ +1025,$F_Y$ +1026,$\mathbf {\Delta g(S)}$ +1027,$X(T(U))$ +1028,$\esssup(X)=\sup\{x\mid \mathsf{Pr}(X>x)>0 \}$ +1029,$\le 1/(1-p)$ +1030,$0\le \lambda\le 1$ +1031,$r\times 1$ +1032,"$(0,1,2,3,4,5,6,7,8,9)$" +1033,"$(3,1)$" +1034,$M=\mathsf{var}nothing$ +1035,$\mathcal F_0\subset\mathcal F_1\subset \cdots\subset \mathcal F_N$ +1036,$v_f\mathsf{E}_\mathsf{Q}[X_i]$ +1037,$\mathsf{Pr}(X=2)=0.5$ +1038,$\dots$ +1039,$R_C$ +1040,$k = 3.3 s^{0.82}$ +1041,"$X_n=1_{\{0,1,\dots,n-1\}}$" +1042,$X(\omega)=x$ +1043,$R_L$ +1044,$X=10$ +1045,$Q_i$ +1046,$P(a)$ +1047,$\mathsf{E} X + c{(X-\mathsf{E} X)^+}_p$ +1048,$\rho(X)\ $ +1049,$U(1)=1$ +1050,$g(S_{X\wedge a'}(x))$ +1051,"$ occurs, i.e., those with the value 1 in the $" +1052,$\Delta X_m$ +1053,"$(0,0,0,0,0,0,0,5,0,5)$" +1054,$D=1$ +1055,$\rho(X)=\max_i \rho_i(X)$ +1056,$a_h=2-a_l$ +1057,$0 < \alpha \le 1$ +1058,"$i=1,\dots,N$" +1059,$-norm equal to 1. (Note that $ +1060,$g(0.1)=\sqrt{0.1}=0.316$ +1061,$\rho_g(X)=\mu+\lambda$ +1062,$0.5 + U/2$ +1063,$-g'(S(x))f(x)$ +1064,$\mathsf{E}[Y \mid U]$ +1065,$1-(p_R+p_Y)$ +1066,$(1+\epsilon)v_1$ +1067,$\Vert X-Y\Vert := \sup_{\omega\in\Omega} |X(\omega) - Y(\omega)|$ +1068,"$(\partial a/\partial x_1)(tx_1,tx_2)= 3tx_1 /a(tx_1, tx_2) = 3x_1 /a(x_1, x_2)=\partial a/\partial x_1$" +1069,$\mathsf{TVaR}_p$ +1070,$U\le u$ +1071,$-dS=f(x)dx$ +1072,$\mathsf{E}_\mathsf{Q}\left[\dfrac{X_i}{X}(X\wedge a)\right] + \tau a \mathsf{E}_\mathsf{Q}[X_i/X\mid X > a]$ +1073,$\mathsf{COM}$ +1074,$1_\omega$ +1075,$\alpha=0.5$ +1076,"$\mathsf{biTVaR}_{p_0,p_1}^w(X)=\mathsf{TVaR}_{p^\ast}(X)$" +1077,$\mathbf{x}=\mathbf{1}$ +1078,$\beta_i(x)/\alpha_i(x)$ +1079,$d^*$ +1080,"$\mathbf {\omega_1},\dots,\mathbf {\omega_n}$" +1081,$X_2-X_1$ +1082,$q_{X_i}(p)=\Phi^{-1}(p)$ +1083,$\mathsf{Q}\in\mathscr{P}$ +1084,$Z_a$ +1085,$\mu(\{p_0\}) = 1-w$ +1086,$Z(\omega)> 0$ +1087,$r=0.045$ +1088,$h(s)=s^m$ +1089,$X\_{1}$ +1090,$cv=0.557$ +1091,$du = -g'(S(x))dF(x)$ +1092,$g(0)=r_0$ +1093,$\sup_\mathsf{Q} (\mathsf{E}_\mathsf{Q}[X] - l(Q))$ +1094,$M_i=\beta_ig(S)-\alpha_iS$ +1095,$j$ +1096,$g-s$ +1097,$\mathsf{E}[X_i \mid X=q(1-g^{-1}(1-p))]$ +1098,$w_u=1+c(1-\gamma)$ +1099,$a:=\rho(X)$ +1100,$g\Delta X \wedge a$ +1101,$\beta_i(a)g(S(a))=\mathsf{E}_{\mathsf{Q}}[(X_i/X) 1_{X>a}]$ +1102,$M=rQ$ +1103,"$X,X_i$" +1104,$Y_c$ +1105,$($ +1106,$S_{X\wedge a}$ +1107,$\rho(1_A)$ +1108,$g_4(s)=s^{0.9}$ +1109,"$(4,1)$" +1110,$f(L)=0$ +1111,$\mathsf{Q}'(\Omega_a) =\mathsf{Q}(\Omega_a)$ +1112,$E[(X-qp)^+]$ +1113,$I/a + U/R > 0$ +1114,$g'(S(x))=(1-p)^{-1}$ +1115,$a\le 1$ +1116,$a-b_h<0$ +1117,$\mathcal V(X)=\frac{1}{1-p}\mathsf{E}[X^+]$ +1118,$\mathsf{TVaR}_p(X) := (1-p)^{-1}(T_1+T_2)/N$ +1119,$0.417 < p < 0.791$ +1120,$1-\nu p$ +1121,$\sqrt{0.9}=0.95$ +1122,"$c(1,2)-c(2)$" +1123,$\lambda X$ +1124,$r_A$ +1125,$\dfrac{\iota}{1+\iota} p$ +1126,$a = a(W) = \mathsf{E}[W]+4\sigma(W)$ +1127,$\rho_g(X)=452.98$ +1128,$a < \infty$ +1129,$\alpha(\mathsf Q) = 0$ +1130,$\mathsf{VaR}_p$ +1131,$X_1=X_2=Y$ +1132,$P = 1.5$ +1133,$\mathsf{E}_\mathsf{P}[X']$ +1134,$S(x)dx$ +1135,$L_a^{a+y}$ +1136,"$\mathsf P,\mathsf Q_2,\dots,\mathsf Q_r$" +1137,$F(t)$ +1138,"$P((1+\epsilon)v_1, v_2, a+da)=P^a((1+\epsilon)v_1, v_2)$" +1139,$\mathsf{E}[X] + \pi\mathsf{var}(X)$ +1140,$\tau=0+d$ +1141,$Y=f(X)$ +1142,$a_1 = 5.991$ +1143,$\mathbf {\iota=M/Q}$ +1144,$X=X\wedge a + (X-a)^+$ +1145,"$s\wedge p=\min(s,p)$" +1146,$a=30$ +1147,$1_{U_X\ge p}$ +1148,$g(s)\ge s$ +1149,$\mathsf Q(A)>0$ +1150,$\mathsf{COH}$ +1151,$D f(x_0)$ +1152,$r_H$ +1153,$d=iv$ +1154,$U>p$ +1155,$p<0.1$ +1156,"$\mathsf{biTVaR}_{0,0.9}^{0.3138}$" +1157,$(g(s)-s)/(1-s)$ +1158,$P/L$ +1159,$\mathsf{E}_Q[X]$ +1160,$j=7$ +1161,$\mathbf{v}'$ +1162,$0< p <1$ +1163,$\psi(u)=\mathsf{Pr}(Y > u)$ +1164,$\mathsf P(A)=0$ +1165,$X_{-1}=x$ +1166,$x=q^-(p)$ +1167,$(\lambda S(x))$ +1168,$Q=1-g(S)$ +1169,$\mathsf{E}[X_i/X|X>a]$ +1170,$1^+$ +1171,$X \wedge a$ +1172,$\mathsf{E}[Y_i\mid X_n]$ +1173,$\delta(s)$ +1174,"$[x, y]$" +1175,$\omega>0$ +1176,"$t \in (0,1)$" +1177,$1=1_{X\le a}+1_{X>a}$ +1178,$\rho(X_n)$ +1179,$Y\equiv 1$ +1180,$(dt)^{3/2}$ +1181,$m_0=0$ +1182,$\iota=\dfrac{M}{Q}$ +1183,$X\circ f$ +1184,$\rho_c(X)=\mathsf{E}[X]+c\sigma(X)$ +1185,$g(s)=s^\lambda$ +1186,"$\mathsf{MON,\ NORM}$" +1187,$\sum_i \kappa_i'(x)=1$ +1188,$ax$ +1190,$p'\ge p$ +1191,"$H_k(X):=\mathsf{E}[\max(X_1\dots, X_k)]$" +1192,$\bar P_i(a)$ +1193,$\sum_\omega \mathsf Q(\omega) =\mathsf{E}[Z] / \mathsf{E}[Z]=1$ +1194,$\mathsf{P}(X=0)$ +1195,$B$ +1196,$Np=67.45$ +1197,"$X_n,X$" +1198,$(1-p)\gamma(dp)$ +1199,$X'=X$ +1200,$(1-p)/(p(\nu_p-l_p)^2)$ +1201,$0.33$ +1202,$\mathsf{E}[X] = \mathsf{E}[\mathsf{E}[X\mid Y]]$ +1203,$\mu_U = 1-p = 0.995$ +1204,$j+1$ +1205,$q_{X+Y}=q_X+q_Y$ +1206,$\mathsf Q_{X}$ +1207,"$u_{X,r}(p)=\psi_{X,r}^{-1}(p)$" +1208,$L_a^{a+da}=L_0^{a+da}-L_0^a$ +1209,$c(X(\mathbf{v}))=c(\mathbf{v})$ +1210,$\mathsf{MRM}$ +1211,$^{*}$ +1212,"$s=0,1$" +1213,"$X(x,-x)\equiv 0$" +1214,$F(x):=\mathsf{P}(X\le x)$ +1215,$\mathbf {X_{g}}$ +1216,$\max X$ +1217,"$\{\mathsf{E}[X_i\,Z] \mid \rho(X)=\mathsf{E}[XZ] \}$" +1218,$\rho(X)=\mathsf{E}_{\mathsf Q_X}[X]$ +1219,$q=q(p)$ +1220,$\rho(\mathsf{E}[X_2\mid X_1])\le \rho(X_2)$ +1221,$1/m>0$ +1222,"$B\subset [0,1]$" +1223,$g(S(x))=1-p$ +1224,"$f:(0,1)\to (0,1)$" +1225,$\mathbf {S}$ +1226,"$p_0,\dots, p_{n'}$" +1227,$X_1-X_0$ +1228,$\bar P = \bar S + \bar M$ +1229,$\mathbf {X_1}$ +1230,$\rho(\tilde X)=\rho(X) + \rho(\tilde X-X)$ +1231,$\mathsf{E}[(X-a)^+]$ +1232,"$u\in D_n=\{ u \mid u^{(k)} \ge 0, k=1,\dots,n-1, u^{(n-1)}\text{ nondecreasing} \}$" +1233,$l(\mathbf X)=(\sum_i X_i^2)^{0.5}$ +1234,$\mathsf{E}_{\mathsf{Q}}[\tilde X-X] \le \rho(\tilde X-X)$ +1235,$s=S(x)$ +1236,$\mathsf{E}_{\mathsf Q}[Y]=\mathsf{E}[YZ]$ +1237,$s_j < 1$ +1238,$\bar S(a+da)-\bar S(a)\approx \bar S'(a)da = S(a)da$ +1239,$t-1$ +1240,$\mathcal D(X+c)=\mathcal D(X)$ +1241,"$s\in[0,1]$" +1242,$\mathsf{E}[Yg'(S(X))]$ +1243,$p=1-1/n$ +1244,$X(\omega)=X_1(\omega)+X_2(\omega)$ +1245,"$S(x) + d\,F(x) + (\delta^{\star}-d)\sqrt{S(x)F(x)}>1$" +1246,$\bar S_i(a) := \mathsf{E}[X_i(a)]$ +1247,$S(x_#4)$ +1248,$1-e^{-\lambda S(x)}$ +1249,$\mathcal V$ +1250,$\beta>1$ +1251,$X_n=n1_A$ +1252,$d-1$ +1253,$g(S(x))\approx S(x)\approx 1$ +1254,$t_0$ +1255,$D_1$ +1256,$\mathcal E$ +1257,$s\uparrow 1$ +1258,$Mg(0+)$ +1259,$S/L\ge A/L-1$ +1260,$\succeq$ +1261,$2\mathsf{VaR}_p(X_1) - \mathsf{VaR}_p(X)$ +1262,$Y = X + Z$ +1263,$)$ +1264,$\rho(X)=\mathsf{VaR}_{0.995}(X)-\mathsf{E}[X]$ +1265,$\tilde X_2 = X_2 -\mathsf{E}[X_2\mid X_1]$ +1266,$p\to 1$ +1267,$1-(1-s)^m$ +1268,$\mathsf P(T^{-1}(A))=\mathsf P(A)$ +1269,$-zf(x)=(d/dx)g(S(x))$ +1270,$\rho_X(X_i)$ +1271,$P=\rho(X \wedge a)$ +1272,$s=0.02$ +1273,$F(q^-(p_0))=p_+>p_0$ +1274,$\Delta g(S)$ +1275,$\Delta$ +1276,"$\mu=10, \sigma=2$" +1277,$t=3$ +1278,$0\le q\le 1$ +1279,$L_a^y$ +1280,$l=\sum_i l_i$ +1281,$X=30$ +1282,$f:I\to\Omega$ +1283,$\mathsf{E}[X_2\mid X=x]$ +1284,"$f(x,y)=x^3/(x^2+y^2)$" +1285,$g(0+)=\delta$ +1286,$S_i(x)$ +1287,$h=2$ +1288,$g'_\tau(s) = g'(s)/(1+\tau)\ge 0$ +1289,$1-\mathsf{P}(X=0)$ +1290,$t \ne 0$ +1291,"$\mathbf {D^f\rho_{X\wedge 30,X}(X_1)}$" +1292,$\rho=\mathsf{TVaR}_p$ +1293,$\kappa_j(x)\approx \mathsf{E}[X_j]$ +1294,$\tilde M_i(a) = \bar M_i(a)-\tau_i a_i$ +1295,$a>10$ +1296,$x^+$ +1297,$\pi^{-1}\log\mathsf{E}[e^{\pi x}]$ +1298,$A(-X)=-A(X)$ +1299,$g(s)=s^{1/3}$ +1300,$\{X = x\}$ +1301,"$p_1,p_1$" +1302,$0\le x \le 1000$ +1303,$U_s$ +1304,$\mathsf{Pr}(X< x)\le 0.75 \le \mathsf{Pr}(X\le x)$ +1305,"$\{1,2,3\}$" +1306,"$i=0,1$" +1307,$\mathsf{Var}(\Pi)$ +1308,$\mathsf{TVaR}_{0.75}(X_1)=10$ +1309,$g_k(s)=1-(1-s)^k$ +1310,$g'(S_{X}(X))$ +1311,$(8t+10t)/2$ +1312,$g(S(x_i-))=g(S(x_{i}))$ +1313,$\nu + \delta = 1$ +1314,$1-1/n$ +1315,$\Omega_1$ +1316,$\mathsf{Pr}(A\cup B)=\mathsf{Pr}(A)+\mathsf{Pr}(B)$ +1317,$\Delta g(S_j)$ +1318,$x\leftrightarrow u(x)$ +1319,$\eta=0.49$ +1320,$X=q(p)$ +1321,$\log(\mathit{EER}) = \gamma + \eta \log(\mathit{PFL}) + \beta \log(\mathit{LGD})$ +1322,$Y=-X_0$ +1323,$g'\circ S_{X\wedge a}$ +1324,$s_2 - s_1$ +1325,$y < q_A(p)$ +1326,$\Delta\mathit{MV}$ +1327,$g'(s+)$ +1328,$\mathsf{Q}(A)=\mathsf{E}[1_AZ]=0$ +1329,$w=E[w|s=0.1]=0.06405$ +1330,$f'_+$ +1331,$f_x=1/S_t$ +1332,$S(X(\omega))$ +1333,$\rho_2(X)$ +1334,$\mathsf{E}[X\mid \mathcal F_t](\omega)=\sum_{i \le t} \omega_i/2^i+2^{-(t+1)}$ +1335,$L$ +1336,$\partial a/\partial x_1=3x_1/a$ +1337,$g(s)\ge 0g(0) + sg(1)=s$ +1338,$T:\Omega\to\Omega$ +1339,$t>x$ +1340,$L^1$ +1341,$(a-X_{\mathsf{j}(a)})$ +1342,$\alpha=d_i$ +1343,"$A=\mathbb Q\cap [0,1]$" +1344,$\mathsf{E}[F_1] > \mathsf{E}[F_0]$ +1345,$Q_1\Delta X$ +1346,$f(L) \ge 0$ +1347,$\rho(X_1)=\rho(X_2)$ +1348,$\rho(\tilde X)$ +1349,$\mathsf{E}[X] + \pi \mathsf{Var}(X)$ +1350,$F_3$ +1351,$\mathsf{CTE}_p(X)$ +1352,$1_{U < s}$ +1353,$Q_2dX$ +1354,$p\to S\to gS \to \Delta gS$ +1355,$P\ge (\mathsf{E}[X] + \iota a)/(1 + \iota)$ +1356,$\Delta Q_{gc}(a)$ +1357,$g(s) = s^a$ +1358,$\mathsf{P}(X=1)=0.6$ +1359,$d^\ast = 1-(1-g^\ast)/(1-s^\ast)$ +1360,$g(s)=g(1-p)$ +1361,$\alpha_{Cat}$ +1362,"$D^f\rho_{X\wedge a,X}(X_i(a))$" +1363,$\mathsf{E}[e^{kX}]$ +1364,$\tilde{\mathsf{Q}}$ +1365,$r_f$ +1366,$X = \sum_i X_i$ +1367,$x_3(S(x_2)-S(x_3))=x_3f(x_3)$ +1368,$\preceq_2$ +1369,$\Delta \bar Q$ +1370,$m_0$ +1371,$(\alpha_i S)'(x)=-\mathsf{E}[X_i\mid X=x]f(x)/x=-\kappa_i(x)f(x) / x$ +1372,$Q(a)=1-g(S(a))$ +1373,$\bar P_i(x)$ +1374,$S\subset T$ +1375,$f(L)$ +1376,$D_n$ +1377,"$\{1+\lambda(f-\mathsf{E} f) \mid f\ge 0, \|f\|_q\le 1 \}$" +1378,$R_M$ +1379,$Z_5$ +1380,$\mathbf {s_1}$ +1381,$q^-=q^+$ +1382,$\mathsf{E}[X_i \mid X = x]$ +1383,$-\int xd(g\circ S)=\int g(S(x))dx$ +1384,$y\not=z$ +1385,$1-g_\tau(s)$ +1386,$a=\mathsf{E}[X \mid X > q(p)]$ +1387,$\rho(aX+bY) = a\rho(X) + b\rho(Y)$ +1388,$\rho L = \iota Q$ +1389,$W \equiv T_{(1)}=min_k{T_k}$ +1390,$\lambda \rho(X)$ +1391,$Y=h(Z)$ +1392,$y^{\ast}-x^{\ast} < \epsilon$ +1393,$\mathsf{E}[X] + \pi \mathsf{E}[((X-\mathsf{E}[X])^+)^p]^{1/p}$ +1394,$U/4$ +1395,$D\rho(X_0)=\{Z \}$ +1396,$X > A$ +1397,$\pi-\lambda\mathsf{E}[X]$ +1398,"$\mathsf{Pr}(A)\in [0,1]$" +1399,$1=\mathsf Q(\Omega)\not=\sum_n \mathsf Q(\{n\})=0$ +1400,$\sigma=0.25$ +1401,$\Delta \mathit{MV}_{gc}(a)$ +1402,$G(x)=\mathsf{Q}(\{X\le x\}) = 1-g(1-F(x))$ +1403,$\Phi'(Z(s))Z'(s)=1$ +1404,$\bar q_{X_1+X_2}(s) \ge \bar q(s/2)$ +1405,$K = 5.029$ +1406,$1_{X>x_2}$ +1407,$S\Delta X$ +1408,$\mathsf{Pr}(X > x)$ +1409,"$G(X_1,\dots, X_n)'=(Y_1,\dots, Y_r)'$" +1410,$\mu_L=r_L +\pi$ +1411,$X=20$ +1412,$\mathsf P(X=\max(X))=0$ +1413,$r_a+r_l$ +1414,$S_1$ +1415,$\mathbf X / l(\mathbf X)$ +1416,"$w, 1-w$" +1417,$\mathcal D$ +1418,"$ (range.south)+(0, -1) $" +1419,$\mathsf{P}$ +1420,$X=\sum_{i=1}^n X_i$ +1421,$X_j=x$ +1422,$\Omega_a$ +1423,$S_j$ +1424,$\beta>\alpha$ +1425,"$f(W_t,t)$" +1426,$Z=d\mathsf{Q}/d\mathsf{P}$ +1427,$\mathbf {Q_{1}\Delta X}$ +1428,$p\le S(x^*)$ +1429,$\phi(t)$ +1430,$S(x)=p$ +1431,$U/2$ +1432,$\int Zd\mathsf P=1$ +1433,$1+t$ +1434,$a_{1}'$ +1435,$r_h=-0.025$ +1436,$\mathsf{E}_{\mathsf{Q}}[\cdot]$ +1437,"$(x_A,g(S(x_A)))$" +1438,$p(1-\nu(p))=p\delta(p)$ +1439,$\beta_i$ +1440,$1-S$ +1441,$p_{\mathit{pr}}$ +1442,$g(0+)=\lim_{t\downarrow 0} g(t)\ge 0$ +1443,$0\le \pi\le 1$ +1444,$\mathsf{E}[cZ]=c\mathsf{E}[Z]=c$ +1445,$Z=Z(X)$ +1446,$r_a$ +1447,"$\int_a^\infty g(S(x))\,dx$" +1448,$\prec X$ +1449,"$\{2, 3\}$" +1450,"$(0,1,2,3,4,8,8,8,8,9)$" +1451,$n\ge 3$ +1452,$=\mathrm{MV}(a-X)^+$ +1453,$g(s)/(1-g(s))$ +1454,$\bar P_i(a)=\mathsf{E}_{\mathsf{Q}}[X_i(a)]=\mathsf{E}[X_i(a)g'(S(X))]$ +1455,"$E[Y\,dG/dF]$" +1456,$g(S_X(x))=1$ +1457,$q(p)=\inf\{x \mid F_X(x)\ge p \}$ +1458,$\mathit{NPV}_{\infty}$ +1459,$E[X_1 | X]$ +1460,$\beta_D$ +1461,$\mathbf {X_{2}}$ +1462,$\alpha_i(x)S(x)=\mathsf{E}[(X_i/X)1_{X>t}]$ +1463,$\sigma=0.1246$ +1464,$F(x;\alpha)$ +1465,$D_\infty$ +1466,"$(1,3)$" +1467,"$X, Y$" +1468,$\mathsf{E}_{\mathsf{Q}}$ +1469,$q^-(p)=\mathsf{VaR}_p(X)$ +1470,"$i=1,\ldots,n$" +1471,$P/l-1 =\rho= \iota Q / l = \iota(C/l + g)$ +1472,$c(x)=\rho(\sum_i x_iX_i)$ +1473,$\omega_1=0$ +1474,$E_{\mathsf{Q_X}}$ +1475,$M_{2}\Delta X$ +1476,$S(x_#5)$ +1477,"$(\nu,\nu,\dots,\nu,\nu+10\delta)$" +1478,$\mathsf{E}[X\wedge a(X)]$ +1479,$\mathcal F'\subset \mathcal F$ +1480,$\Delta S_0$ +1481,$a_{d}$ +1482,$\tilde X(x) = x$ +1483,$A/L<1$ +1484,$X_n(\omega)$ +1485,$\mathsf{E}[X_{d}]$ +1486,$\bar P^a(\mathbf{v})$ +1487,$\int_0^1 f(s)ds = 1 - \alpha < 1$ +1488,$\mathcal{N}_{X}(X_i(a))$ +1489,$a-P$ +1490,$\rho(X)=\sup_{\mathsf Q\in\mathcal Q} \mathsf{E}_\mathsf{Q}[X]$ +1491,$\mathsf{Q}(A)\le g(\mathsf{P})(A))$ +1492,$d=0$ +1493,$x\mapsto g(s)+g'(s)(x-s)$ +1494,$\mathsf{VaR}_{1-s}$ +1495,$\rho_g(X\wedge a)=(\bar L + ra)/(1+r)$ +1496,$(a-X)$ +1497,$\omega'=1$ +1498,$1/6 + 2 /6 + 4/2 + 9/6$ +1499,$\rho_a(kX) = \rho(kX \wedge a(kX)) = \rho(kX \wedge ka(X)) = \rho(k(X\wedge a(X))) = k\rho(X\wedge a(X)) = k\rho_a(X)$ +1500,"$500mm, enough to materially impair their franchise, is judged to be 0.4%. This has a corresponding risk-neutral value of 2.5%. However, they believe that a loss over $" +1501,$(a_1'-a_1)^+$ +1502,$X\wedge a=\sum_i X_i(a)$ +1503,$\mathbf {a}$ +1504,$\int_0^a g(S(x))dx$ +1505,"$Q,\iota,M$" +1506,$\mathsf{E}[p]=1$ +1507,$p>p^*$ +1508,$\{X\ge q(p)\}=\{X \ge 12\}$ +1509,$g(1)-g(0)=1$ +1510,$g(s)(1-q)$ +1511,$(g(S(x^-)-g(S(x)))/(S(x^-)-S(x))$ +1512,"$\sum_j X_{i,j}(a)\Delta g(S_j)$" +1513,"$\mathsf{P}(a,b]=b-a$" +1514,"$j=1,\dots,d$" +1515,$Z(\omega)=0$ +1516,$l(p)= \nu(p)-\sqrt{(1-p)/p}$ +1517,$\int_0^1 g(s)ds - 0.5$ +1518,$\rho_{g}$ +1519,$\prec_1$ +1520,$S\ge (1-\epsilon)\mathsf{E}[X]$ +1521,$\alpha(\mathsf{Q})$ +1522,$\mathsf{E}[\mathsf{E}[Z\mid X]]=\mathsf{E}[Z]$ +1523,$\epsilon v_1$ +1524,"$\phi(p) = (1-\alpha)^{-1}1_{[1-\alpha, 1)}(p)$" +1525,$S(M)=0$ +1526,$c\ge 0$ +1527,$p_1=1$ +1528,"$x_{1,i}+x_{2,k(i)}$" +1529,"$(x_1, x_2)$" +1530,$\alpha_i'(x) \to 0$ +1531,"$\displaystyle\int_0^{F(a)} \kappa_i(q(p))\,dp + a\alpha_i(a)S(a)$" +1532,$\bar P(a)$ +1533,$q(U)$ +1534,$\iff\rho$ +1535,$F_g(x)$ +1536,$Q(a) = 1-P(a)= \nu F(a)$ +1537,$\mathsf P(\{x\})=0$ +1538,$\mathsf{E}[X_2]=22.75$ +1539,$ = \mathsf{E}_{\mathsf{Q}}[X_i\mid X= x]$ +1540,$1_V$ +1541,$R_Q$ +1542,$\mathcal D:=\{X\mid X\preceq_2 Y \}$ +1543,"$X_{j,i}$" +1544,$g(1-F(x))=1-\tilde p$ +1545,$p'$ +1546,$\beta_i(a)g(S(a))$ +1547,"$A\subset[0,\infty)$" +1548,$X_1/X$ +1549,$x$ +1550,$q_{\mathbf{v}}(p)$ +1551,$\rho(X) = \rho(X\wedge a) + \rho((X-a)^+)$ +1552,$q^-(p)=\sup\ \{ x\mid \mathsf{Pr}(X < x) < p \}$ +1553,$1\not\in S$ +1554,$\mathsf{VaR}_{0.99}(X)=1100$ +1555,$X_n=1/n$ +1556,$\rho_g(X)=\mu/b>\mu$ +1557,$<1$ +1558,$S(X)$ +1559,$a=kP+Q$ +1560,$X\wedge a = \sum X_i(a)$ +1561,$\mathsf{TVaR}_{p_0}(X)=\mathsf{E}[X \mid A]$ +1562,$A\subset \{ Z=0 \}$ +1563,$Z\circ T_i$ +1564,$a(X_i; X)\le \sup(X_i)$ +1565,"$Y_{1,2}$" +1566,$M_{2}$ +1567,$x \le 300$ +1568,$\implies c_i\ge 0$ +1569,$F(x)=1-s$ +1570,$h(0.9) = 1-\sqrt{0.1} = 0.684$ +1571,"$\alpha = 1, \kappa = 0.2$" +1572,$(8)(0.25)+(10)(0.25)=4.5$ +1573,$W_0=0$ +1574,$Q=S$ +1575,$X^{(d)}_i(a):=(X_i-d)^+$ +1576,${\mathcal{M}}$ +1577,$X = X_1 + X_2$ +1578,$V_t$ +1579,"$\mathsf P(\{ \omega\mid X(\omega)=X(\omega_0), \omega \le \omega_0 \})$" +1580,$m_3 := m_2$ +1581,$g(s)=(s+\iota)/(1+\iota)$ +1582,$\iota = \delta/\nu$ +1583,$r_X= r_f + \beta_X(r_m-r_f)$ +1584,$Z\circ T\in \mathcal Q$ +1585,$\mathbf {s}$ +1586,$Z\succeq_2 \mathsf{E}[Z\mid X]$ +1587,$\rho(X_1) \ge P_1$ +1588,$a-X$ +1589,$P(A)=1-p$ +1590,$10+0$ +1591,$\phi'(p)=-g''(1-p)>0$ +1592,"$\mathsf{TI,\ MON,\ SA,\ PH}$" +1593,$\Delta_1=a_1'-a_1$ +1594,$\mathit{RDS}_k$ +1595,$t=-ln(1-p)$ +1596,$C_i=c_i$ +1597,$\lim_{s\to 1} (g(s)-s)/(1-s) = \lim_{s\to 1} 1-g'(s)$ +1598,$\rho_i(X)$ +1599,$v(A\cap B) + v(A\cup B)\le v(A)+v(B)$ +1600,$\mathsf{TVaR}_{0.5}$ +1601,"$X_1, X_2$" +1602,$\rho=\sup$ +1603,$\mathsf{E} X + c\mathsf{E}[((X-\tau)^+)^p]^{1/p}$ +1604,$m_i$ +1605,$\mathsf{E}[g(X_n)]\to \mathsf{E}[g(x)]$ +1606,$k\in\mathbb{R}$ +1607,$g'(s) = as^{a-1}$ +1608,$q(p)=F^{-1}(p)$ +1609,$E_4$ +1610,"$\psi_{X, m}(u)$" +1611,$f=(1-p)^{-1}1_A$ +1612,$<0$ +1613,$X=X_1 + X_2$ +1614,$G=g$ +1615,$-q_{-Y}^-(1-p)$ +1616,"$\rho(\lambda P,\lambda R,\lambda a)=\lambda\rho(P,R,a)$" +1617,$1+bf$ +1618,$Y_j$ +1619,$dP_g/dP_X$ +1620,$\mathsf{E}[X|X>x]=x+\mathsf{E}[X]$ +1621,$M=g-S$ +1622,$FL$ +1623,$\int gS(x)dx=\int xg'(S(x))P_X(dx)$ +1624,$\mathit{MV}_{ro}(a) = a-\rho(X_{-1}\wedge a)$ +1625,$\mathcal V(X)=\mathsf{E}[X]+c\mathsf{E}[X^2]$ +1626,$n+1$ +1627,$g'(s)=\phi(1-s)$ +1628,$X_i(a)\not= X_i\wedge a_i$ +1629,$\lim_{x\downarrow x_0} F(x)=F(x_0)$ +1630,$F(w) = 1-\exp(-w)$ +1631,$\mathsf{E}[X(1_{U_X\ge p}-B)]=\mathsf{E}[(X-m)(1_{U_X\ge p}-B)]$ +1632,$B_i^c$ +1633,$\Omega_a := \{\omega\in \Omega \mid (X\wedge a)=a \}$ +1634,$1/10$ +1635,$\mathsf{Q}_k$ +1636,$Q_i(a)$ +1637,$Q>0$ +1638,$r_h-\mu_L$ +1639,$s_j$ +1640,$\beta g(S)$ +1641,$\rho(W)=\mathsf{E}[W]+\lambda\sigma(W)$ +1642,$\ge 0$ +1643,$E[u_j(W_j - X_j)]$ +1644,$\phi((x-\mu)/\sigma)/\sigma$ +1645,$X_{2}$ +1646,$E[X \wedge x+a]-E[X \wedge a]$ +1647,$\mathsf{TVaR}_p(X)=25$ +1648,$X-(1+r)T$ +1649,"$\int_0^1 a'(tx)\,dt=\int_0^1 a(1)\,dt = a(1)=a'(x)$" +1650,$ (#1)+(#3) $ +1651,$g=F_G^{-1}(p_{\mathit{pr}})-1$ +1652,$X_{2}(a)$ +1653,$g(s)=s(1-s)$ +1654,$\mathsf{VaR}_{0.995}(U)-0.5=0.495$ +1655,$\kappa_2(10)$ +1656,$\lambda < 0$ +1657,$\mathit{ROE}(s) = fs/(1-f-s)$ +1658,$p_i$ +1659,$X_m$ +1660,$g(t) = r_0 + (1-r_0)t$ +1661,"$Y_{1,1}$" +1662,$s > s^*$ +1663,$\theta$ +1664,$g(s)=s^{1/2}$ +1665,$X\wedge a=a$ +1666,$\mathsf{Pr}(X < x)=1/6=\mathsf{Pr}(X\le x)$ +1667,$P=l + \iota Q$ +1668,$X-Y$ +1669,$\log(\mathit{ROL}) = a + b \log(\mathit{EL}) + b X$ +1670,$q_{X_1+X_2}(p) \le q_{X_1}(p) + q_{X_2}(p)$ +1671,$k\ge 0$ +1672,$\Phi'(z)=\phi(z)$ +1673,$q^-(p)=\inf \{ x \mid F(x) \ge p \}$ +1674,$\rho_X(X_i) \ge \mathsf{E}[X_i]$ +1675,"$g'(s)=(1-p)^{-1}1_{[0,1-p]}$" +1676,$X(\mathbf{v})=\sum_i v_iX_i$ +1677,$s_0$ +1678,"$t=0,1$" +1679,$d^\ast = 2g^\ast-1$ +1680,"$(s_1,g(s_1))$" +1681,$g(s)=s$ +1682,$0\times\infty=0$ +1683,"$\bar Q_{0,t}:=a_{0,t}-\bar P_{0,t}$" +1684,$q_X(p)$ +1685,$\rho_c$ +1686,$\mathbf {X\wedge a}$ +1687,$M(a)=g(S(a))-S(a)$ +1688,$\rho(X_n)=\rho(0)=0$ +1689,$\mathbf {X}$ +1690,"$\displaystyle\int_0^a \kappa_i(x) f(x)\,dx + a\alpha_i(a)S(a)$" +1691,$\bar\iota = 0.12$ +1692,$\mathsf P(X=\sup(X))=0$ +1693,$\mathsf{E}[Y\mid\mathcal F']=\mathsf{E}[Y]$ +1694,$\alpha_2(98)=0.9$ +1695,$p\delta(p)/p\nu(p)=\iota(p)$ +1696,$g_\tau(1)=1$ +1697,"$H(A, L, t)=LH(A/L, 1, t)$" +1698,$g_2F$ +1699,$X=X_0+X_1$ +1700,"$697.6 billion in 2016, $" +1701,$\bar Q=53.031$ +1702,$\mathsf{P}(\{n\})>0$ +1703,$c(S\cup\{i\})=c(S\cup\{j\})$ +1704,$\mu_L=0.03$ +1705,$Q_0=\rho(V_0)=\rho(X_1)$ +1706,$g'(s-)=g'(s+)$ +1707,$U = X + Y$ +1708,$B=B(p)$ +1709,$9+1=10+0$ +1710,$n=67$ +1711,$a(X(\mathbf{v}))$ +1712,$v(\Omega)=1$ +1713,$p_Y=1-p_R$ +1714,"$p\,da$" +1715,$t\mapsto \rho(X+tY)$ +1716,$Y^S$ +1717,$g'(S(x)) = (1-p)^{-1}1_{x >\mathsf{VaR}_p(X)}$ +1718,$E_{\mathsf{Q_X}}[X_i(a)]$ +1719,$\rho(X)\le \rho(Y)$ +1720,$1-\tilde p=g(1-p)$ +1721,$R_f-R_L>0$ +1722,$P = \log(\mathsf{E}[e^{\pi X}])/\pi$ +1723,$\rho_c(X)$ +1724,$X^\star$ +1725,$X\wedge a'$ +1726,$\mathsf{E}[\Pi]$ +1727,$0.675=(6.258/7.613)^2$ +1728,$q<1$ +1729,$\alpha_1(90) = (0.0909 \times 0.0625 + 0.1 \times 0.0625)/(0.0625+0.0625)=0.0955$ +1730,$g(Q)$ +1731,"$X_2=0,0,0,0,1,1,1,4,24, 500$" +1732,$\bar P_i$ +1733,$Z=\mathsf{E} Z$ +1734,$a(X)=3.769$ +1735,"$\rho(P,R,a)=\sqrt{(0.4P)^2+(0.25R)^2+(0.1a)^2}$" +1736,$\exp(x)$ +1737,$X_j$ +1738,"$(anch.west |- lee.north)+(-0.125,0.25)$" +1739,$\ge\mathsf{E}[X_i]$ +1740,$g(s)=20s\wedge 1$ +1741,$f(x_p)$ +1742,$\{X=q_X(p) \}$ +1743,$\mathsf{E}[X_i\mid X=x]$ +1744,$EL(a)$ +1745,$30-11=19$ +1746,$x\in\mathbb{R}$ +1747,$p_R<0.5$ +1748,$\beta_{1}$ +1749,$g(S(a))\ge S(a)$ +1750,$r=16$ +1751,$\beta_i(a)$ +1752,$N=71$ +1753,$\rho(X_1+X_2)\le \rho(X_1)+\rho(X_2)\le 0$ +1754,$a_{gc}$ +1755,"$1 between any of the layers, then $" +1756,$\mathcal{M}$ +1757,"$\sum_i \rho(X_i, p^*)=a$" +1758,$\int_0^\infty g(S(x))dx$ +1759,$t=1-p$ +1760,$\rho'(x)=U'(-x)$ +1761,$D\rho_X(X_i) \ge \mathsf{E}[X_i]$ +1762,$\mathsf{Pr}(B)=\mathsf{Pr}(A)$ +1763,$x=\mathsf{VaR}_{0.99}(X)$ +1764,$\alpha_i(x)-\kappa_i(x)/x=0$ +1765,$x\mapsto |x|$ +1766,$\mathsf{Pr}(X_{-1}\sigma(Y)=0$ +1771,$D\rho_X(X_2)$ +1772,$\beta_i(a)g(S(a))=\mathsf{E}_{\mathsf{Q}}[(X_i/X) \mid X>a]g(S(a))=\mathsf{E}_{\mathsf{Q}}[(X_i/X) 1_{X>a}]$ +1773,$\rho_g(X)=\mathsf{E}[X]$ +1774,$L_d^l(x)$ +1775,$\beta_1g(S)dX$ +1776,$p_j=\Delta S_j$ +1777,$x1$ +1779,$E[s|t]$ +1780,$\mathsf{Q}(A)=\mathsf{E}_\mathsf{Q}[1_A]$ +1781,"$C(a)=\int_a^\infty S(x)\,dx + \tau a$" +1782,$\beta=d^\ast-d$ +1783,$-0.00002$ +1784,$y=0$ +1785,$L_X$ +1786,$\lambda=0.5$ +1787,$g(s)=(1-p)^{-1}s\wedge 1$ +1788,$\sum M_i\Delta X$ +1789,$1\le x \le 2$ +1790,$f(x) \ge f(x_0) + f'(x_0)(x-x_0)$ +1791,"$1,\dots,m$" +1792,$X\in L_p$ +1793,$n\mathsf{Pr}(Y\le y_c)$ +1794,$x=1.5$ +1795,$u^{iv} \le 0$ +1796,$1_{X > x}$ +1797,$S_{X_i}$ +1798,$xS(x)\to 0$ +1799,$(a-X)^+=a-(X\wedge a)$ +1800,"$j=0,1,\dots, n'$" +1801,$\mathsf{P}(\omega)$ +1802,$\bar Q=a-\bar P$ +1803,"$\mathbf {X\,p}$" +1804,$SdX$ +1805,$\sqrt{p}$ +1806,$L^p$ +1807,$\mu<0$ +1808,$\mathsf{E}[Y_{d}]=\sum_{s>d} \mu_s$ +1809,"$X_{i,i}(a)=X_{i,j}\dfrac{X_j\wedge a}{X_j}$" +1810,$\mathscr{M}$ +1811,$ so $ +1812,$1/4$ +1813,$\mathsf{E} X+\lambda\sigma(X)$ +1814,$\lambda\ge 0$ +1815,$d\bar S(a)/da=S(a)$ +1816,$(\alpha S)'(x)=-\kappa_i(x)f(x)/x$ +1817,$\sup f=1$ +1818,"$X_{t-2,3}$" +1819,$\beta_i(x)/\alpha_i(x) 0$ +1821,$\bar\nu a$ +1822,$a(1-f)$ +1823,$X\succeq Y$ +1824,$p_R$ +1825,$\mathsf{E}[p]\not=1$ +1826,$s_1 < s_2$ +1827,$1$ +1828,$\mathbb{Q}$ +1829,$a_x=1/\lambda$ +1830,$f:\mathbb{R}\to\mathbb{R}$ +1831,$\mathsf{E}[1_{U < s}]=s$ +1832,"$I=[0,1]$" +1833,$\rho(X)\le 0$ +1834,$B(0.5)$ +1835,"$i=1,2,\dots$" +1836,$r_D=1-D/L$ +1837,"$\min(X,a)$" +1838,$\mathbf {t-1}$ +1839,$\Delta S$ +1840,$ is the total return on invested assets and $ +1841,$\mathsf{E}[(A-L)^+]/\mathsf{E}[L]$ +1842,$X(\psi)=X(\omega)$ +1843,$X_j\ge 0$ +1844,$\mathcal{S}$ +1845,"$i=1,\dots, n$" +1846,"$\rho_{a,\tau}(X)=v\rho(X\wedge a) + da$" +1847,"$(brR15 |- lee.south)+(-0.125,-0.25)$" +1848,$n\ge N$ +1849,$x_1 \wedge x_2$ +1850,$X_s = X_{s_1} + X_{s_2}$ +1851,$0$ +1909,$x_0 \in \{ x \mid F(x) \ge p \}$ +1910,"$\bar P(\mathbf{v}, a)$" +1911,$x\mathsf{E}[X_i/X\mid X>x]$ +1912,$x_2(S(x_1)-S(x_2))=x_2f(x_2)$ +1913,$r_h=0$ +1914,"$S=[0,2\pi]$" +1915,$gn$ +1916,$p=F(x)$ +1917,$1/g'(s)$ +1918,$z(x)$ +1919,$-\sigma^2u''(w)\approx -cu'(w)$ +1920,$r=0.1$ +1921,$\mathsf{CTE}_p(X) := \mathsf{E}[X \mid X \ge \mathsf{VaR}_p(X)]$ +1922,$\beta_1$ +1923,"$i=1,\dots, M$" +1924,$\mathsf{E}_\mathsf{P}[X]$ +1925,$S^{-1}(g_i)$ +1926,$\mathbf {\Delta X'}$ +1927,$d =\iota/(1+\iota)$ +1928,"$\mathsf{E}[X_{t,d}\mid \mathcal F_{\tau}]$" +1929,$Z=g'(S_X(X))$ +1930,$E_i\cap E_j = \mathsf{var}nothing$ +1931,$i\not\in S$ +1932,$s+\delta p$ +1933,"$X_1=1+cos(X_3), X_2=1-cos(X_3)$" +1934,"$\mathcal F'=\{\mathsf{var}nothing, \Omega \}$" +1935,$(1-p)^{-1}1_A$ +1936,$\rho=P/L-1=M/L$ +1937,$F(X)$ +1938,$\lambda=$ +1939,$\rho_g(X)=352$ +1940,$x=0.5$ +1941,$A = -\log(p) = 5.298$ +1942,$\rho(X_{-1}\wedge a)$ +1943,$g'(S)dF(x)$ +1944,$-norm by integrating against a function with $ +1945,$(X-d)^+$ +1946,"$x=1000,2000,\ldots$" +1947,$\mathsf{E} X +\lambda {(X-\mathsf{E} X)^+}_1$ +1948,$\int_0^\infty S(x)dx$ +1949,$a=100$ +1950,$+ \mathit{PV}_{r_f}(\text{Inv Inc tax})$ +1951,$S(x_1)(x_2-x_1)$ +1952,$\mathsf{E}[(X-m)(1_{U_X\ge p}-B)] = 0$ +1953,$m=q(p)$ +1954,$wx + (1-w)y\in C$ +1955,$m_X$ +1956,$A(\text{Bernoulli})$ +1957,"$X,Y$" +1958,$\tilde Q$ +1959,"$Y_{0,2}$" +1960,"$\mathbf {X\,\Delta g(S)}$" +1961,$E[T]=s$ +1962,$\max(X)<\infty$ +1963,$\rho(Z_2)$ +1964,$\alpha_2SdX$ +1965,$\mathbf {x_0}$ +1966,$c\ge 1/2$ +1967,$g(s)=\dfrac{s+\iota}{1+\iota}$ +1968,"$X_i(\mathbf{v}, a)$" +1969,$X \prec_n^* Y$ +1970,"$X\wedge a'=\min(X, a')$" +1971,$d=2$ +1972,$s^\alpha$ +1973,$X(x)=\sum_i x_iX_i$ +1974,$Z(\omega):=(d\mathsf{Q}/d\mathsf{P})(\omega)$ +1975,"$\{\, (\mathsf{E}_\mathsf{Q}[X_i], \mathsf{E}_\mathsf{Q}[X]) \mid \mathsf Q\in\mathcal Q \, \}$" +1976,$1/6\le x < 2/6$ +1977,$p\ge r\ge 1$ +1978,$\mathbf{B}(0)=\mathbf{P_0}$ +1979,$Q=(a-EL)/(1+\iota)$ +1980,"$\rho(P,R,a)$" +1981,$t\mapsto v^t$ +1982,$\{ X=x\}$ +1983,$\omega \in \Omega$ +1984,"$j, p, S, \kappa_1, \Delta X, \Delta(X\wedge a)$" +1985,$0.375/1.5 = 0.25$ +1986,"$a(v_1(1+\epsilon),v_2)=a(v_1,v_2)+da$" +1987,$M_i$ +1988,$\alpha_i$ +1989,$p=1-\exp(-t)$ +1990,$\rho(X - b)=\rho(X)-b\le 0$ +1991,"$\boldsymbol{j, p, S, \kappa_1, \Delta X, \Delta(X\wedge a)}$" +1992,$x\ge 0$ +1993,$\rho(\lambda X) \le\lambda\rho(X)$ +1994,"$(1,1,\dots,1,1)$" +1995,$\mathbf {\Delta X}$ +1996,"$1-p, p$" +1997,$\mathsf{Pr}(X_n\in A)\to\mathsf{Pr}(X\in A)$ +1998,$S(x)=(k/(k+x))^\beta$ +1999,$p = 0$ +2000,$x_1$ +2001,$x=X(1-g^{-1}(1-\tilde p))$ +2002,$s < 1$ +2003,$\cdot$ +2004,$a'=a(1+r)$ +2005,$\phi(\cdot)$ +2006,"$i \in \{1,\dots,4\}$" +2007,$\gamma=r_f$ +2008,$\Delta A$ +2009,$P(X_{-1}(a))$ +2010,$0\le\lambda\le 1$ +2011,$\max$ +2012,$\Omega_0$ +2013,$0\le v\le 1$ +2014,$Y(\omega)=1$ +2015,$Q=A-P$ +2016,$0.75$ +2017,$a+y$ +2018,$\mathsf{Pr}$ +2019,$0.25$ +2020,$s=\mathit{EL}$ +2021,"$(1-g(S(x)),x)$" +2022,$\nu+10\delta$ +2023,$1=ps_g + (1-p)s_b$ +2024,$U(1)=2$ +2025,$\Phi(-d^*)>0$ +2026,$P = \mathsf{E}[X] + \pi \mathsf{E}[|X-\mathsf{E}[X]|^p]^{1/p}$ +2027,$x\to\infty$ +2028,$g(pq)=g(p)g(q)$ +2029,$\frac{d}{dp}(1-p)^{-1}=(1-p)^{-2}=q^{-2}$ +2030,$\rho(X)<\infty$ +2031,$X_0=\mathsf{E}[X]$ +2032,$\mu_L=r_L + \pi$ +2033,$\mathsf{E}[X_i(v_i)]=v_i\mathsf{E}[X(1)]$ +2034,"$k=(0.04, 0.4)$" +2035,"$A,B$" +2036,$\Delta S=p$ +2037,$N(1-p)$ +2038,"$(\omega'=1, \omega'')\in B_k$" +2039,$\sum\mathsf{E}[C_i^2]=\sum m_i(1+v_i^2)$ +2040,"$p_0,\dots, p_m$" +2041,$\tilde Z$ +2042,$\tilde X+X$ +2043,$dF(x) = dp$ +2044,$x_0 < \mathsf{TVaR}_{p_0}$ +2045,$\lambda\sigma$ +2046,$Z_j$ +2047,$m'(1) \to -1$ +2048,$g(S_j)$ +2049,$g(s(t)) = m(t)+s(t)$ +2050,$A\subseteq \mathbb{R}^N$ +2051,$f(x)\ge f(x_0) + s(x-x_0)$ +2052,$p=0.9982$ +2053,$a=10$ +2054,$\mu + \lambda\sigma$ +2055,$\beta<\alpha$ +2056,$Z\ge 0$ +2057,$\mathsf{E}_{\mathsf Q}[X]=\mathsf{E}[XZ]$ +2058,$\bar\nu(x)$ +2059,$6.258$ +2060,$\rho(X)=-\rho(-X)$ +2061,$-\sigma^2/2$ +2062,$k>0$ +2063,$r = 0.12$ +2064,$\mathsf{E}[Z \mid X]\preceq_2 Z$ +2065,"$(3,4)$" +2066,$dG/dF=r(x)$ +2067,$F_0=2.5$ +2068,$F_g(b)-F_g(a)=g(S(a)) - g(S(b))$ +2069,$P_g$ +2070,$\bar S$ +2071,$p=F(a)=1-s$ +2072,$Z(\omega)<1$ +2073,$\alpha\equiv 0$ +2074,$Var(G)=c^2$ +2075,$a = a(X)$ +2076,"$x\in\Omega=[0,1]^N$" +2077,$1_{U_X\ge p}=1$ +2078,$r_h<0$ +2079,$g(S(x_i)-g(S(x_i-))$ +2080,$F(a)$ +2081,$\mathbf {q}$ +2082,$\mathbf {d}$ +2083,$L_d^{d+l}(x)=(x-d)^+ \wedge l$ +2084,$\psi(0)=1-\mathsf{Pr}(Y=0)=1-\mathsf{Pr}(M=0)=\frac{1}{1+r}$ +2085,$X_3$ +2086,$\mathsf{E}[XB]$ +2087,$\bar P(a+y) - \bar P(a)$ +2088,$\bar P$ +2089,$x_{i+1}$ +2090,$-X_2$ +2091,$M_2\Delta X$ +2092,$(1+r)\mu$ +2093,$\bar P^a$ +2094,$\ge p$ +2095,$\displaystyle\int_0^\infty u(x) g'(S_X(x)) dF_X(x)$ +2096,$\mathsf{E}_{\mathsf{Q}}[X_i \mid X]$ +2097,"$\omega\in [k2^{-m}, (k+1)2^{-m}]$" +2098,$p=2$ +2099,$X=98$ +2100,"$0\le U, V\le 1$" +2101,$Y'$ +2102,$\mathbf {\mathsf{P}(X_1)}$ +2103,$\displaystyle\int_0^\infty xf(x)dx$ +2104,$(1-g(s))/(1-s)$ +2105,$00$ +2113,$a/X$ +2114,$q_1(t)=t$ +2115,$\mathbf{B}'(1) = -3\mathbf{P_2}+3\mathbf{P_3}$ +2116,$k\ge 1$ +2117,$X_{1}(a)$ +2118,$\Delta(X\wedge a)$ +2119,$P = S + M$ +2120,$(0.304-0.2)/(1-0.304) = 15$ +2121,$\omega_2$ +2122,$\mathsf{E}[h(X_i)L(X)]$ +2123,$P/S-1$ +2124,$g(s)/s$ +2125,$C(t)$ +2126,"$h(x)=\sup_{s\in[0,1]} g(s)-sx$" +2127,$t=4$ +2128,$\rho(X)=\max_\mathsf{Q} \mathsf{E}_\mathsf{Q}[X]$ +2129,$i^*$ +2130,$g(1)=1$ +2131,$C'_1+\cdots + C'_n$ +2132,$s_1$ +2133,$BY \succ AR$ +2134,$0.8 \times 1.2 = 24/25$ +2135,$(g(s)-s)/(1-g(s))$ +2136,$a = 8.1484$ +2137,$Y\circ T_i$ +2138,$p=0.9999$ +2139,$Z_X$ +2140,"$X_{0,1},X_{0,2},\dots, X_{0,N}$" +2141,$Z=0$ +2142,$-k$ +2143,$v(A)=g(\mathsf{P}(A))$ +2144,"$\bar P_i(\mathbf{v}, a)$" +2145,$B_p$ +2146,$a_i=x_i(\partial a/\partial x_i)$ +2147,$N$ +2148,$\sup$ +2149,$q_X(p)\le q_Y(p)$ +2150,$S(x)=s$ +2151,$X\preceq_n Y$ +2152,"$y,z\in X$" +2153,$\Omega_0 \times \Omega_1$ +2154,$P = \mathsf{E}[X] + \pi \mathsf{E}[((X-\tau)^+)^p]^{1/p}$ +2155,$df/dx=f$ +2156,$\mathsf{TVaR}_p(X)$ +2157,$X=8$ +2158,$\mathbf {\mathsf{P}(X_2)}$ +2159,$Q\in\mathcal{Q}$ +2160,$0.125$ +2161,$s < p$ +2162,$P(X_{-1}\wedge a)$ +2163,"$n=1,2,\dots, m-1$" +2164,$S(x)\approx 1$ +2165,"$X_2=(0,1,2,3,4,8,6,4,0,9)$" +2166,$1.5$ +2167,$q_X(p) = X(T(p))$ +2168,$1-m\le 1$ +2169,$v_f(\mathsf{E}_Q[X_i] - \dfrac{\mathsf{E}_Q[X_i]}{\mathsf{E}_Q[X]}\mathsf{E}_Q[(X-A)^+])$ +2170,"$k=1,\dots, n-1$" +2171,$\rho_g(X)=\mathsf{E}_{\mathsf{Q}}[X]$ +2172,$X_{-1}+X_{0}$ +2173,$p<0.05$ +2174,$\delta$ +2175,"$\gamma([0,p])=C(p)$" +2176,$10$ +2177,$T(U)$ +2178,$\rho_a(X+c) = \rho((X+c)\wedge a(X+c)) = \rho((X+c)\wedge (a(X)+c)) = \rho((X\wedge a(X))+c) = \rho((X\wedge a(X))) + c=\rho_a(X)+c$ +2179,$\bar M_t$ +2180,"$x~\text{Unif}[0,1]$" +2181,$g'(S(X))$ +2182,$\tilde Z=\mathsf P(X=\sup(X))^{-1}1_{X=\sup(X)}$ +2183,$\bar P(a+da) -\bar P(a)$ +2184,$X(x)=1/x$ +2185,$x=\mathsf{VaR}$ +2186,$\beta_2g(S)dX$ +2187,$\sigma(X_d)$ +2188,$\mathsf Q(X>a)/\mathsf P(X>a)$ +2189,$\mu(dp)$ +2190,$c=(g-s)/(g(1-g))$ +2191,$X\wedge a=a=90$ +2192,$\sigma(W)$ +2193,$1\le p\le \infty$ +2194,$X=4$ +2195,"$\sigma(L^\infty, L^1)$" +2196,$\mathsf{E}[X^n]$ +2197,$p_0\not= p_1$ +2198,"$a_{0,0}'=a_{0,0}$" +2199,$\mathsf{E}_{\mathsf{Q}}[X_i \mid X=x] = \mathsf{E}[X_ig'(S_X(X)) \mid X=x]/\mathsf{E}[g'(S_X(X)) \mid X=x] = \mathsf{E}[X_i \mid X=x]$ +2200,$\mathbf {K}$ +2201,$\{\omega\mid X(\omega) > x\}$ +2202,$P_i$ +2203,$\lambda_2\not=1$ +2204,$p>0.9$ +2205,$E(X^k)=E(Y^k)$ +2206,$\mathsf{E}[X_i \mid X=q(p)]$ +2207,$\mathsf{E}[Z \tilde X]$ +2208,$\bar P_t$ +2209,"$\Omega=\{ 1,2,3,4,5,6 \}$" +2210,$p<0.7$ +2211,"$a=10,20,40,50,60$" +2212,$-\infty+\lambda=-\infty$ +2213,$\mathbf {g(s)}$ +2214,$x=y$ +2215,$d=0.1/1.1$ +2216,$\beta_2>\alpha_2$ +2217,$k(h):=\log\mathsf{E}[e^{hX}]$ +2218,$=\displaystyle\int_0^\infty x dF(x)$ +2219,$\mathcal Q=\{\mathsf Q_k\}$ +2220,$a(f + (1-f)/q)$ +2221,$\lfloor x \rfloor$ +2222,$A\in\mathcal F$ +2223,$v(A)=\lambda(\pi_1(A))$ +2224,$\mathsf{E}_{\mathsf{Q}}[(X - a)^+] = \rho((X - a)^+)$ +2225,$n\to\infty$ +2226,$\mathsf{EPD}$ +2227,$\Longleftarrow$ +2228,"$\eta_{p,\alpha}$" +2229,$\Omega$ +2230,$\mathsf{QCX}$ +2231,$\omega=\omega'$ +2232,$\mathbf {M_{1}\Delta X}$ +2233,$g(S_{\mathsf{j}(a)})(a-X_{\mathsf{j}(a)})=(0.5)(80-11)=34.5$ +2234,$z_p=\Phi^{-1}(p)$ +2235,$g_1(s)=s^{0.4}$ +2236,"$1-e^{-\lambda S(\mathsf{PML}_{n, \lambda})}=1/n$" +2237,$q^-(U)$ +2238,$s=\exp(-a/b)$ +2239,$F(x)\ge p\iff q^-(p)\le x$ +2240,$(P-L)/L=P/L-1$ +2241,"$[p,1]$" +2242,$F_2$ +2243,"$\{H,T\}$" +2244,$a(1-p) + \mu p - \sigma\phi(z_p)$ +2245,$\rho(b-X)=b+\rho(-X)$ +2246,$s<1$ +2247,$g''(s)=-s^{3/2}/4$ +2248,$D^n\rho_X(X_1)=6.2048$ +2249,$\Delta X\wedge a$ +2250,$v=1/(1+r)$ +2251,$(1-p)^{-1/2}/4$ +2252,$T(X):=y\wedge (X-r)^+$ +2253,$x=S^{-1}(g^{-1}(u))$ +2254,$\mathsf{E}_\mathsf{Q}[X\mid A]$ +2255,$\mathsf{Q}(\{\omega_i\})=0$ +2256,$A(X+c)=A(X)+c$ +2257,$P \le \dfrac{S}{\lambda} \approx \dfrac{\mathsf{E}[X]}{\lambda}$ +2258,$\mathit{EGL}_{gc}(a)$ +2259,"$c\in[0,1/2]$" +2260,$\mathbf {X_{1c}}$ +2261,$\sigma=2.58$ +2262,$dp=\exp(-t)dt$ +2263,$a_x=4$ +2264,"$\beta_i(a) = \dfrac{\sum_{j:X_j>a} (X_{i,j}/X_j) \Delta g(S_j)}{\sum_{j:X_j>a} \Delta g(S_j)}$" +2265,$X = X\wedge a + (X - a)^+$ +2266,$(1-p)/(p\nu_p^2)$ +2267,$u$ +2268,$\omega$ +2269,$\mathsf{TVaR}_{0.8}(X+tX_1)$ +2270,"$\rho_g(X)= \sum_j X_j\,\Delta g(S_j)$" +2271,"$X_1,\dots,X_n$" +2272,$D\rho_{X}(Y) \subset D\rho_{X\wedge a}(Y)$ +2273,$\lambda=\dfrac{1}{1+\rho}$ +2274,$q^-(s)=\mathsf{VaR}_s(X)$ +2275,$v_i$ +2276,$\mathsf{E}_{\mathsf{Q}}[X\wedge a] \le \rho(X\wedge a)$ +2277,"$p=0.01, 0.02, \dots, 0.99$" +2278,$\mathsf{VaR}\_p(X)$ +2279,$a_0$ +2280,$0\le b\le 1$ +2281,"$A=(a,b]$" +2282,$a(\mathbf{v}) =\mathsf{TVaR}_p(X(\mathbf{v}))= (1-p)^{-1}\int_p^1 q_{\mathbf{v}}(s)ds$ +2283,$-g$ +2284,$q^-(p) := \sup\ \{x \mid F(x) < p \} = \inf\ \{ x \mid F(x) \ge p \}$ +2285,$p(\omega)\ge 0$ +2286,$D/L>1$ +2287,$-m_2/(1-s_2)$ +2288,$g(1-F(x))=1-p$ +2289,$h(1_{X\le a})$ +2290,$E(\pi)$ +2291,$\mathsf{TVaR}_{0.95}(X)$ +2292,$b-X\ge 0$ +2293,$Z = \sum_j X_j$ +2294,$X+Z$ +2295,$\mathsf{E}_{\mathsf{Q}}[X\wedge a] = \rho(X\wedge a)$ +2296,$\mathsf{VaR}_{0.75}(X)=90$ +2297,$\mathsf{E}[X_0] + \mathsf{VaR}_p(X_1)$ +2298,$QR_Q = aR_A + PR_L$ +2299,$x=\lambda y + (1-\lambda)z$ +2300,$dS=-dF$ +2301,$s \to 1$ +2302,$\tilde M(a)=\bar M(a)-\tau a$ +2303,$\kappa_i(x)=\mathsf{E}[ X_i \mid X = x]$ +2304,"$(ccc.south |- mcc.south)+(0,-0.5)$" +2305,"$[0,1]\to[0,1]$" +2306,$p=\infty$ +2307,$\bar P(a) = \rho_g(X\wedge a)$ +2308,$0\rho_2(X)$ +2310,$s(t)$ +2311,$\rho(W_1\wedge a_0)$ +2312,$0.8 \le p < 0.9$ +2313,$\epsilon_2$ +2314,$k=0$ +2315,$1-2c\mathsf{Pr}(Z>\mathsf{E} Z)$ +2316,$\Delta X_j=X_{j+1} - X_j$ +2317,${X}_p=\mathsf{E}[|X|^p]^{1/p}$ +2318,$\iota:1$ +2319,"$x_{2,1}$" +2320,$Y_{d}=\sum_{s>d} X_{s}$ +2321,"$\phi(x_1,...,x_n)$" +2322,$Z\in\mathcal Q$ +2323,$\iota^\ast$ +2324,$X-P$ +2325,$g(s)q=0.1839$ +2326,$X_2=x-t$ +2327,"$X_{t+2,1}$" +2328,$\mathsf{MON}$ +2329,$G(x)= 1-g(1-F(x))$ +2330,$g'(s)\to\infty$ +2331,"$\mathbf {g(S)\, \Delta X}$" +2332,"$j \in \{5,\dots,8\}$" +2333,"$\mathbb{R}=(-\infty, \infty)$" +2334,$e^{-r_Dt}$ +2335,$\rho((X-a)^+)$ +2336,$Q_t$ +2337,$X_0 < \dots < X_{N-1}$ +2338,$\mathbf {Z_6}$ +2339,"$B_4 = [\epsilon_1, \epsilon_2]$" +2340,$a(w_1X_1+w_2X_2;X)=w_1a(X_1;X)+w_2a(X_2;X)$ +2341,$(P-L) / (A-P)=$ +2342,$AR\succ BR$ +2343,$a(x)=xa(1)$ +2344,$X(\mathbf{v})$ +2345,"$x_{1,1}$" +2346,$\mathsf{E}_{\mathsf{Q}}[X_i\mid X\le a](1-g(S(a))) + a\mathsf{E}_{\mathsf{Q}}[X_i/X\mid X >a]g(S(a))$ +2347,"$d, r>0$" +2348,"$\phi(s)= g'(1-s) = \frac{1-w}{1-p_0}1_{[p_0, 1)}(s) + \frac{w}{1-p_1}1_{[p_1, 1)}(s)$" +2349,"$S\subset \Omega=\{1,\dots,N\}$" +2350,$x\le 0$ +2351,$S_0=1$ +2352,$0=\mathsf{Pr}(X<1)<\mathsf{Pr}(X\le 1)=1/6$ +2353,$f(x)=|x|$ +2354,$\mathbf {\mathsf{P}(X)}$ +2355,$S_t \ge 0$ +2356,$p=F(a)$ +2357,$\Psi^{-1}(t)=\log(-\log(t))$ +2358,$\mathsf{E}[e^{X_t}]=e^{\mu t + \sigma^2t /2}$ +2359,$q(U_X) > m$ +2360,$\mathsf{var}(\sum C_i)=\sum (m_i v_i)^2 = n(mv)^2$ +2361,$Y_s=(Y\mid Y\le y_c)$ +2362,$\mathsf{P}(d\omega)$ +2363,$h(0)$ +2364,$P_i/v_i$ +2365,$\mathsf{E}[X_1\mid X_1+X_2=x]=mx/(m+n)$ +2366,$\mathsf{E}_\mathsf{Q}[X_1]$ +2367,$\lambda > 0$ +2368,"$c(1,2) - c(2)$" +2369,"$(0,1]$" +2370,$t<0$ +2371,$\mathsf{COMON}$ +2372,$\beta_i(x)/\alpha_i(x)> 1 > g(S(x)) / S(x)$ +2373,$\int_0^\infty (1-F(x))dx=\int_0^\infty xdF(x)$ +2374,$(dW_t)^2=dt$ +2375,"$\mathbf {\Delta\,g(S)}$" +2376,$\mathsf{TVaR}_{0.95}(X)=3699$ +2377,$g(0^+) = r/(1+r)$ +2378,$x\mapsto 1/x$ +2379,$m\in\mathbb{R}$ +2380,$\mathsf{VaR}_{0.7}(X_i)=-\log(0.3)=1.204$ +2381,$-S(a)+\tau=0$ +2382,$\rho(c)\ge c$ +2383,$\beta_i(X)$ +2384,$0.8\le p<0.9$ +2385,$\mathsf P(X \le q_X(p)) > p$ +2386,$1/X$ +2387,$\displaystyle\int_0^1 X(p)dp$ +2388,$\mathsf{E}[X\tilde Z]$ +2389,$\rho_c\leftrightarrow\mathcal Q$ +2390,$U(X)\ge U(Y)$ +2391,$\lambda X_1 +(1-\lambda) X_2$ +2392,$\mathbf {a_{1}'}$ +2393,$\mathsf{E}[X\mid t+d]$ +2394,$MV = \bar Q + \mathit{NPV}_{\infty}$ +2395,$g(s)=1-(1-s)^m$ +2396,$g(0.05)=0.05\nu + \delta=0.1364$ +2397,$\mathbf {pK}$ +2398,$\min_{\eta\in \mathbb{R}} \eta + \alpha \mathsf{E}[(X-\eta)^+] -\beta\mathsf{E}](X-\eta)^-]$ +2399,$g(S(x)) = 1 - h(F(x))$ +2400,$\mathsf{E}[X]=k/(k+\beta)$ +2401,$g(s)\le s$ +2402,$L_1$ +2403,$X_1=1000$ +2404,$S$ +2405,$x < y$ +2406,$\mathsf{Pr}(E)$ +2407,$p>0.5$ +2408,$\mathsf{E}[p] \le 1$ +2409,$\mathsf{E}[1_A]$ +2410,$x=(y-\mu)/\sigma$ +2411,$a\to\infty$ +2412,$X+tX_1$ +2413,$M = \beta g(S)-\alpha S$ +2414,$0 < \nu = 1-\delta < 1$ +2415,$d=(\log(a/S_0)-(r-\sigma^2/2)t)/\sigma\sqrt{t}$ +2416,$X(\omega)=1/\omega$ +2417,$1/n$ +2418,$H(X)>-H(-Y)$ +2419,$s/(1-p) \wedge 1$ +2420,$\mathbf {\beta_{1}}$ +2421,$\Phi$ +2422,$\lambda y=x$ +2423,$\mathsf{MON'}$ +2424,$g'(S_X(X))$ +2425,$b<1$ +2426,$X\mapsto \mathsf{E}[XZ]$ +2427,$w < s$ +2428,$m_2$ +2429,$\mathsf{Pr}(X\in A)=0$ +2430,$\le c$ +2431,$n-1$ +2432,$qX$ +2433,$\bar P_2$ +2434,"$(4,3)$" +2435,$(X_i)_i$ +2436,$20+10t$ +2437,$s=1-\alpha$ +2438,$Z=d\mathsf Q / d\mathsf P\ge 0$ +2439,$X_i(a) = aX_i/X$ +2440,"$c(1,2,3)-c(2,3)$" +2441,$\sum_i q_iX_i$ +2442,$\mathsf{Pr}({\omega})=1/6$ +2443,"$\mathbf {X'\,\Delta g(S)}$" +2444,$\kappa_j(x)/x > \alpha_j(x)$ +2445,$a_i'$ +2446,$-\int xdS=\int Sdx$ +2447,$c\ge 1$ +2448,$f(P)=\mathsf{E}[f(X)]$ +2449,$\mathbf{B}(1)=\mathbf{P_3}$ +2450,"$\bar Q_{0,0}:=a_{0,0}-\bar P_{0,0}$" +2451,$p_- < p_0 < p_+$ +2452,$\mathbf {\Delta gS}$ +2453,$g'(t)=1-r_0$ +2454,$q(p)=\mathsf{VaR}_p(X)$ +2455,$g(0+):=\lim_{s\downarrow 0}g(s)$ +2456,$z\ge 0$ +2457,$\mathsf{E}[W]$ +2458,$ \& $ +2459,$A\setminus B$ +2460,$(k_1!)(k_2!)\dots$ +2461,$Q(x)=1-P(x)$ +2462,$\sup(X)$ +2463,$\mathbf {F(x)=\mathsf{Pr}(X\le x)}$ +2464,$1=\delta+\nu$ +2465,$=1/\lambda-1=(1-\lambda)/\lambda$ +2466,$U_X$ +2467,$\mathsf{Pr}(X_n=0)=1-1/n$ +2468,$q_X$ +2469,$\mathit{EGL}_{ro}(a)$ +2470,$\mathsf{E}[\cdot\mid X]$ +2471,"$i=1,2,\dots,10000$" +2472,$Z=z(X)$ +2473,$\{X > x \}$ +2474,$X_{\mathsf j(a)+1}>a$ +2475,$g_j<1$ +2476,$\rho(X)=0$ +2477,$\sum_i x_iX_i$ +2478,$Xq$ +2479,$\phi(p)=g'(1-p)=b(1-p)^{b-1}$ +2480,$N=1000$ +2481,$A\subseteq \mathbb{R}^n$ +2482,$a=90$ +2483,$S_m=\mathsf{P}(X>X_m)=0$ +2484,"$g:[0,1]\to [0,1]$" +2485,$q(p)$ +2486,$g(s)=\nu s+\delta$ +2487,$m=$ +2488,"$q(p)\phi(p)\,dp$" +2489,$q^+(p)=\sup\ \{ x\mid \mathsf{Pr}(X < x) \le p \}$ +2490,$x>\mathsf{VaR}_p(X)$ +2491,$a(\mathbf{v})=\mathsf{TVaR}_p(\mathbf{v})=\mathsf{E}[X\mid X > q_{\mathbf{v}}(p)]$ +2492,$\hat x > x$ +2493,$\text{VaR}_{0.99}$ +2494,$P_X\{X=M\}=0$ +2495,$X=X_0+X_{-1}+X_{-2}+X_{-3}$ +2496,$x>0$ +2497,"$X_{i,j}$" +2498,$a_1=\int_0^1 (\partial a/\partial x_1)dt=\partial a/\partial x_1$ +2499,$1=\bar\nu+\bar\delta$ +2500,$(1-p)/p=1$ +2501,$s=S(a)$ +2502,$\partial\rho(Z)$ +2503,$\mathbf X$ +2504,$\rho(W_1\wedge a_1 \wedge (a_0-X_1))=\rho(W_1\wedge a_1)$ +2505,$\sum_i \kappa_i(x)=x$ +2506,$g(s)=s^{0.4}$ +2507,$(g(s_0)-g_0)/s_0 \ge g'(s_0)$ +2508,$X_n(0)=1$ +2509,"$X_{t,2}$" +2510,$W=Z$ +2511,$\phi(x):=(2\pi)^{-1/2}\exp(-x^2/2)$ +2512,$g(s)=\sqrt{s}$ +2513,$1-p=S(x)$ +2514,$p(\delta_p-il_p)$ +2515,$\alpha(X)$ +2516,$=1$ +2517,$g''$ +2518,$f=f_X$ +2519,$dW_t\approx W_{t+dt}-W_t$ +2520,$X(\omega_1) > Y(\omega_1)$ +2521,$H_g(X) \le H_g(Y)$ +2522,$M:=\max(X)$ +2523,"$0,10,20$" +2524,$1/9=0.11\dot 1$ +2525,$a=80$ +2526,$n-2$ +2527,"$((0, x), (1-p, p))$" +2528,$P=D=L/(1+R_L)$ +2529,$w(A)\le v(A)$ +2530,$2^{20}\approx 1$ +2531,$^{**}$ +2532,$\mathsf{LI}\iff\mathsf{SSD}$ +2533,$p_j$ +2534,$P$ +2535,$s_00$ +2608,$i= \alpha/(1-\alpha)$ +2609,$\rho(X_0) = \mathsf{E}[X_0Z]$ +2610,$X\ge x$ +2611,$Z(x)=g'(S(x))$ +2612,"$c = 1.0, 1.5$" +2613,$a_{d}=a(Y_{d})$ +2614,$\mathsf{SD}(X)$ +2615,$-A(-X)$ +2616,$t\ge 0$ +2617,"$\Omega=\{0,\dots,99\}$" +2618,$g'(S(x))\ge 0$ +2619,"$p~\text{Unif}[0,1]$" +2620,$R_A=R_f$ +2621,$\mathsf{VaR}_p(X)=q^-(p)$ +2622,$E(u(X)) \le E(u(Y))$ +2623,$\rho_g(V)= g(F(x^*)) \ge F(x^*)=\mathsf{E}[V]$ +2624,$\mathsf{E}[X]+\lambda\sigma(X)$ +2625,$\mathsf{Pr}(\{\omega \})= 1/100$ +2626,"$(2,-\x*0.75)$" +2627,$\mathsf{Pr}(A\le t)= 1/2 + \mathsf{Pr}(U\le t) /2 = 1/2 + t/2$ +2628,$a \ge 1$ +2629,"$\mathsf{biTVaR}_{0,p}^w(X)$" +2630,$\iota^\ast = (g(s^\ast)-s^\ast) / (1 - g(s^\ast))$ +2631,$\mathsf{E}_{\mathsf{Q}}[X_i]$ +2632,$g'(s)\ge 0$ +2633,$\mathsf{E}[X]+k\mathsf{Var}(X)=a(X)$ +2634,"$X:\Omega\to [0,\infty]$" +2635,"$\mathsf{TVaR}_{0.95}(X)=\int_0^{1000}g(S(x))\,dx$" +2636,$\rho(X_n)\to \rho(X)$ +2637,$\lambda_{obj}$ +2638,$W_0$ +2639,$cv=0.287$ +2640,$0=q(0)=q(Y+(-Y))\le q(Y) + q(-Y)$ +2641,$g_\tau(0)=0$ +2642,$\mathsf{P}(X=1)$ +2643,$\mathsf{Pr}(X\le y) < p$ +2644,$0.41$ +2645,$\mathsf P(X=q_X(p))>0$ +2646,$p=0.8$ +2647,$\kappa_1(10)$ +2648,$\mathsf{E}_\mathsf{Q}[X+tY]$ +2649,"$[0,p)$" +2650,$a_1'$ +2651,$S(x)=1-\Phi((x-\mu)/\sigma)=\Phi(-(x-\mu)/\sigma)$ +2652,$X_{t+1}$ +2653,$X=0$ +2654,$p\mapsto g(1-p)$ +2655,$\downarrow$ +2656,$X\wedge 20$ +2657,$\mathsf{TVaR}_1( X )$ +2658,"$x_1, x_2$" +2659,$\bar P_{2}$ +2660,$\Sigma$ +2661,$B\subset A$ +2662,$\bar P=\mathsf{TVaR}_{p^\ast}(X)$ +2663,$\bar P^a_g(X_i\subseteq X)$ +2664,"$\mathcal{M} = \{ f \mid \|f\|_q\le c, f\ge 0 \}$" +2665,$X \preceq_m Y$ +2666,$qX_i$ +2667,$\mathsf{E}[X_i\tilde Z]=\rho_g(X)/2$ +2668,$X \prec_n Y$ +2669,$\bar\iota=0.10$ +2670,$a=18000.0$ +2671,$\mathsf{TVaR}_{0.95}(X)=1000$ +2672,$s_0/2^{n+1}$ +2673,$\delta(x)$ +2674,$H[X]$ +2675,"$x_1,x_2$" +2676,$>100$ +2677,$\mathsf{Pr}(X\ge q(p))>1-p$ +2678,$dh - h_x dx = (r_h-\mu_L)(h-h_x x)dt$ +2679,$\alpha<1$ +2680,$2.576\times 6.258$ +2681,$\mathsf{TVaR}_{p*}(X)=a$ +2682,$\kappa_i(X) = X_i$ +2683,$a_i = a(X_i; X)$ +2684,$\rho(X_n(t))+t\pi$ +2685,$\mathsf{TVaR}_{p}$ +2686,$g(A)/p=59.142$ +2687,$Z(S_X(x))=-(x-\mu)/\sigma$ +2688,$\nu=1-\delta$ +2689,$\{\omega\in\Omega\mid X(\omega)\le x\}$ +2690,$X_n= X_g-X_c$ +2691,$s^*$ +2692,$\bar P_{0}$ +2693,$P = \mathsf{E}[X] + \pi \mathsf{SD}(X)$ +2694,$\rho(X_0+\epsilon Y)=\mathsf{E}[(X_0+\epsilon Y)Z_\epsilon ]$ +2695,$X\wedge 30$ +2696,$k+1/2$ +2697,$X'=\mathsf{E}[X\mid A]$ +2698,$\mathsf{Pr}(X_i>\bar q(s))=s$ +2699,$\lambda=5$ +2700,$D_3$ +2701,$\ge c$ +2702,$\kappa_i(X)$ +2703,"$\mathsf{PH,SA,CX}$" +2704,$\phi:=\rho\circ F$ +2705,$u_j(x) = 1 - exp(-\lambda_j x)$ +2706,$M_{1}$ +2707,$X-V$ +2708,"$\bar P_i = \sum_{j} X_{i,j}\Delta g(S_j)$" +2709,$f(t)$ +2710,"$[0, \epsilon_1]$" +2711,$\pi=1$ +2712,$a_l \le 1$ +2713,$\rho=\mathsf{E}$ +2714,$P_g\not\ll P_X$ +2715,$\delta_i=\delta$ +2716,$\mathbf {X_1pK}$ +2717,$p_Y>0.5$ +2718,$-g'(1-p)<0$ +2719,$\rho(X+Y) = \rho(X) + \rho(Y)$ +2720,"$(0.5,1]$" +2721,$F(x_0)=p_+$ +2722,"$(X_i, X)$" +2723,$\mathsf{E}[X_i\sum_j w_jZ_j]=\sum_iw_j\mathsf{E}[X_i Z_j]$ +2724,$\beta_i(x) =\mathsf{E}_{\mathsf{Q}}[X_i/X\mid X>x]=\mathsf{E}[(X_i/X)g'S(X))\mid X>x]$ +2725,$l(kX)=k\rho(X)$ +2726,$\int udv = uv - \int vdu$ +2727,$\mathbf {Q_2\Delta X}$ +2728,"$\mathcal F_0=\{\mathsf{var}nothing, \Omega\}$" +2729,$r_P-\mu_L$ +2730,$\bar P(a)=\displaystyle\int_0^a g(S(x))dx$ +2731,$g(x) = (x-\mu)^2$ +2732,$\mathsf{biTVaR}(Y)=\mathsf{TVaR}_{p^\ast}(Y)$ +2733,$\mathsf{E}[(X_i/X)g'(S(x)) \mid X > x]$ +2734,$0$ +2735,$p=1-g(1-F(x))$ +2736,$\bar S_i(3463)$ +2737,$X=X(\omega)$ +2738,$\int_0^1$ +2739,$D/L=\mathsf{E}[A\wedge L]/\mathsf{E}[L]$ +2740,$\esssup(X)g(0-)$ +2741,$\tilde p$ +2742,$\bar P'$ +2743,$\sum_i E[X_i|anything]\le _{cx} \sum X_i \le_{cx} F_{X_i}^{-1}(U)$ +2744,$\bar{\mathbf M}$ +2745,$\lambda = \lambda_0+\lambda_1$ +2746,$X_{-2}=C_1 + \cdots + C_n$ +2747,$X_{0}$ +2748,$\rho_g = \int g(S)$ +2749,$a={{break_even}}$ +2750,$x=q(1-g^{-1}(1-\tilde p))$ +2751,$\mathsf x\mathsf{TVaR}_p(X):= \mathsf{TVaR}_p(X)-\mathsf{E}[X]$ +2752,$0.5\le p^* \le 0.75$ +2753,$X(\omega)=1$ +2754,$P(a)=S(a)+\delta F(a)$ +2755,$\mathsf{TVaR}_{1-c\epsilon}(X) = \mathsf{VaR}_{1-\epsilon}(X)$ +2756,$q(p)=S^{-1}(1-p)$ +2757,$d_i= i/(1+i)$ +2758,$P=\sum_i P_i$ +2759,$B_i$ +2760,$a_1=a(W_1)$ +2761,$\rho=\esssup=\mathsf{TVaR}_1$ +2762,$\sigma(X)^2$ +2763,$g'(s)=1/(1-p)$ +2764,$0.4$ +2765,$f'_+(x)=\lim_{h\downarrow 0} (f(x+h)-f(x))/h$ +2766,$g'(1)=1$ +2767,$\mathcal Q_1$ +2768,$X\le m$ +2769,$dt^2$ +2770,$q=p$ +2771,$\sigma_d = \mu_d/5$ +2772,$Q_0$ +2773,$X_t=X_{t+1}$ +2774,$g(s)=0.1995$ +2775,$\log(0)=-\infty$ +2776,$\mathsf{VaR}_p(X_1)$ +2777,$W_t$ +2778,$Z_{\tilde X}$ +2779,$U0$ +2792,"$\{4,5\}$" +2793,$h(x)=f(x)/S(x)$ +2794,$S\Delta X\wedge a$ +2795,$r_U$ +2796,$(c(S\cup \{i\})-c(S))$ +2797,$\mu(dp)=f(p)dp$ +2798,$X\preceq_2 Y$ +2799,$\mathbf {\min a}$ +2800,$\mathsf{E}_{\mathsf{Q}}[X]=\infty$ +2801,$v(E)$ +2802,$\{Z\circ T\mid T:\Omega\to\Omega\text{\ PPT}\}$ +2803,$6/6$ +2804,$\Phi(\Phi^{-1}(s) + \lambda)$ +2805,"$[0, -k]$" +2806,$\rho(X)=\mathsf{E}[f_X X]$ +2807,$\mathsf P(X=\mathsf{VaR}_p(X))>0$ +2808,$\rho(W_0\wedge a_0)=\bar P_0 +\bar P'$ +2809,$U(t)$ +2810,"$\sum_i a(X_i, p^*)=a$" +2811,$p=1-s$ +2812,$q_{X_1}(p)+q_{X_2}(p)=q_{X_1+X_2}(p)$ +2813,$\mathsf{E}[X_i\mid X\le a]F(a) + a\mathsf{E}[X_i/X\mid X >a]S(a)$ +2814,$8.5$ +2815,$M_{1}\Delta X$ +2816,$\bar P_i(a)$ +2817,$T_i$ +2818,$L_0^y$ +2819,"$\mathbf {g(S)\,\Delta X'}$" +2820,$\mathsf{E}_Q\left[\dfrac{X_i}{X}(X\wedge A)\right] + \delta A \mathsf{E}_Q[X_i/X\mid X > a]$ +2821,$2$ +2822,$\rho(c)=\rho(0+c)=\rho(0)+c$ +2823,$U_X(\omega)=F(X(\omega)-) + V(\omega)(F(X(\omega)) - F(X(\omega)-))$ +2824,$s = 1-10^{-15}$ +2825,$s/g(s)$ +2826,$\bar F(a)=\int_0^a F(x)dx = a-\bar S(a) = \bar Q(a) + \bar M(a) = \mathsf{E}[(a-X)^+]$ +2827,$a(W)=\mathsf{E}[W] + 4\sigma(W)$ +2828,$\alpha f$ +2829,$\{\omega\in\Omega \mid X(\omega)=x\}$ +2830,"$[0,1]$" +2831,$1/(1-p)$ +2832,"$(0,0),\ (1,0),\ (1,1)$" +2833,$\omega_i\in B$ +2834,$g'(1)=\alpha$ +2835,$\mathsf{E}[Xe^{\pi X}]/\mathsf{E}[e^{\pi X}]$ +2836,$\le 1$ +2837,$-\rho(-X)$ +2838,$f_{opt} = 1-s/g$ +2839,$g(S(x_i-))-g(S(x_{i-1}))$ +2840,$\Delta \mathit{MV}_{ro}(a)$ +2841,$Q_i=a_i-P_i$ +2842,$0.0476/(1-0.0476)=0.05$ +2843,$q=0.9215$ +2844,$c\ge \mathsf{E}[cZ]$ +2845,$f(x)dx$ +2846,$\mathcal F'$ +2847,$p=\Phi((a-\mu)/\sigma)$ +2848,$\nu (1-s)$ +2849,$g'(1-s)$ +2850,$P(X_{-1}\wedge a_{ro})=9196.39$ +2851,$\omega_0$ +2852,$g_2(s) = 2s/3 + 1/3$ +2853,"$\mathbf {X_{i,j}}$" +2854,$x^\ast$ +2855,$2/3$ +2856,$\iota(s)=w/(1-w)$ +2857,$\kappa_i(x) = \mathsf{E}[X_i \mid X=x]=\mathsf{E}_{\mathsf Q}[X_i \mid X=x]$ +2858,$\phi(0)=0$ +2859,$\log(S) =\mu t$ +2860,$a\le (P(1+\iota)-S)/\iota$ +2861,$\mathsf{E}_\mathsf{Q}[\cdot]$ +2862,$g'(s_1) \ge (1-g(s_1))/(1-s_1)$ +2863,"$U, V$" +2864,$s^{0.642}$ +2865,$\kappa_i(x)=mx/(m+n)$ +2866,$\mathbf {n}$ +2867,$C\mathsf X$ +2868,$s_0=1$ +2869,"$\Omega=\{1,2\}$" +2870,$\kappa_i(x) = \mathsf{E}[X_i \mid X=x]$ +2871,$x' - \mathsf{E}_\mathsf{P}[X]$ +2872,$\rho(X_0)=\mathsf{E}[X_0Z]$ +2873,$X\mapsto\int X(\omega)Z(\omega)\mathsf(d\omega)$ +2874,$\rho(X_0+\epsilon Y)-\rho(X_0)$ +2875,"$\sigma_A,\sigma_L$" +2876,$P(a)=g(S_X(a))$ +2877,$Z_\epsilon\to Z$ +2878,$a\beta_1g(S)$ +2879,"$\mathsf{biTVaR}_{p,1}^w$" +2880,"$(s^\ast, g(s^\ast))$" +2881,$a^\star$ +2882,$\beta_i(x)$ +2883,$\mathbf {\alpha_2S\Delta X}$ +2884,$\rho(X)=1.169$ +2885,$U(\omega)=\omega$ +2886,${X}$ +2887,$D^f\rho_{X;\tilde X}(X_i)$ +2888,"$d(g(S(x)))/dx=-g'(S(x))\,dF/dx$" +2889,$S(x+a)$ +2890,$\rho(0)=0$ +2891,$\succeq^2$ +2892,$\rho(\lambda X)=\lambda \rho(X)$ +2893,$q(p) \times \phi(p)dp$ +2894,$E(X_{-1}(a))=\bar S_0(a)$ +2895,$q_2(t)=t^2$ +2896,$\sigma^2=\sigma_A^2 + \sigma_L^2 - 2\rho\sigma_A\sigma_L$ +2897,"$(0.5, 0.5)$" +2898,$a_lp}$ +2920,$\int X=0$ +2921,$\mathsf{j}(0)=0$ +2922,$0<\alpha\le 1$ +2923,$g'(S(x))>1$ +2924,$r_f /(1+ r_f)$ +2925,$X_c$ +2926,$-q(-Y)$ +2927,"$[1,\infty)$" +2928,$4.75$ +2929,$D_c$ +2930,"$X_{t-2,1}$" +2931,$L\mathsf{VaR}_p(X))>1-p$ +2940,$\mathsf{E}[X\wedge a] = (1-e^{-a\beta})/\beta$ +2941,$(r-i)\sum_t Q_t$ +2942,$\gamma(ds)$ +2943,$Z=20\cdot1_A$ +2944,$X_n(\omega)= 1$ +2945,$F_0 = P_{act}-\mathsf{E}_{rn}[U]$ +2946,$\mathsf{Var}(X)$ +2947,$f=(1-p)^{-1}1_{W}$ +2948,$\rho(X_n)=0$ +2949,$1_{X\le a}$ +2950,$af\le 1$ +2951,$ for estimates $ +2952,$X+W$ +2953,"$\mathsf{biTVaR}_{0,1}^w(X)=(1-w)\mathsf{E}[X]+w\sup(X)$" +2954,$\mathsf{TVaR}_p(X)=\mathsf{E}[X\mid X >\mathsf{VaR}_p(X)]$ +2955,$gS$ +2956,$-\rho(-X)\le \mathsf{E}[X] \le \rho(X)$ +2957,$\Delta X_7$ +2958,$Z=\tilde X_2$ +2959,$a\alpha_i(a)=\kappa_i(a)$ +2960,$\mathsf{Pr}(X < x)\le 0.99 \le \mathsf{Pr}(X\le x)$ +2961,$\mathsf{E}[(X-m)(1_{U_X\ge p}-B)]\ge 0$ +2962,$B-p(\nu(p) + il(p))$ +2963,"$(0,0,0,0,0,0,5,0,0,5)$" +2964,$\mathsf{VaR}_p(X)=q_X^{-}(p) = \sup \{ x\mid F_X(x) < p \}$ +2965,$\omega\in\Omega$ +2966,$g=0$ +2967,$\bar P(a)=\mathsf{E}_\mathsf{Q}(X\wedge a)$ +2968,$L_0^a$ +2969,$-5.91$ +2970,$\bar q_{X_1+X_2}(s)=q_{X_1+X_2}(1-s)$ +2971,$\Pi=B-p\nu(p)$ +2972,"$Y_{2,1}$" +2973,$U(a)=-s$ +2974,$\rho(X+Y) = \rho(\lambda(X/\lambda) + (1-\lambda)(Y/(1-\lambda))))$ +2975,$\mathsf{E}_G(X)$ +2976,$\mathbf {\beta_{2}}$ +2977,$P(x)$ +2978,$r=(1+\bar\iota)/(1+\tau)-1$ +2979,$\rho(-X)=-\rho(X)$ +2980,$R_L=-k R_f + \beta_L(R_M-R_f)$ +2981,$\mathbf {\Sigma}$ +2982,$g(t)$ +2983,$N := \lceil (1-p)M \rceil$ +2984,"$a_i=\mathsf{E}[X_i] + k\mathsf{cov}(X_i, X)$" +2985,$\mathsf{Pr}(X\le x)=0$ +2986,$\{2\}$ +2987,"$(\nu,\delta)$" +2988,$p\to\infty$ +2989,$1\le\lambda$ +2990,$P_1=\mathsf{E}[X_1g'(S_X(X))]$ +2991,$\rho E/(1-\tau) - rA$ +2992,$x\mapsto x^{1/2}$ +2993,"$j=0,\dots,m=8$" +2994,$\beta_i(a)/\alpha_i(a) > 1$ +2995,$=\displaystyle\int_0^\infty x f(x)dx$ +2996,$a_l-1<0$ +2997,$F_X(x):=\mathsf{Pr}(X\le x)$ +2998,$F_Y^{-1}(V)=q_Y(V)$ +2999,$Z_{\mathit{lin}}$ +3000,$\mathsf{E}[f(X-\pi P)] = f((1-\pi)P)$ +3001,$0\le p_0\le p^*\le p_1\le 1$ +3002,$log(x)$ +3003,$\mathsf{P}(A) = \mathsf{E}[1_A]$ +3004,$\rho(X)=\mathsf{E}_{\mathsf{Q}}[X]$ +3005,$\nu=\nu(a)<1$ +3006,$X_1$ +3007,$X(\cdot)$ +3008,"$Z=(0,0,0,0,0,0,0,0,5,5)$" +3009,$Z_{\mathit{lift}}$ +3010,"$\mathbf{B}:\left [0,1 \right ] \ni t \mapsto (x(t),y(t)) \in \mathbb{R}^2$" +3011,$\mathcal Q(X)=\{ \mathsf Q\in\mathcal Q\mid \rho(X)=\mathsf{E}_\mathsf{Q}[X] \}$ +3012,"$(\Omega, P)$" +3013,$0 \le p<1$ +3014,$\mathsf Q_X$ +3015,$\mathbf {M_{1}}$ +3016,$\mathsf{E}[X]=1/\beta$ +3017,$n < N-1$ +3018,$\bar P(x)=\int_0^x P(t)dt$ +3019,$\mathsf{E}[g'(S(X))]=1$ +3020,$F(x_0)\ge p$ +3021,$X-a$ +3022,$Z\not=0$ +3023,$\rho(\cdot)$ +3024,$p = (1-s)$ +3025,$p=0.417$ +3026,$(j)$ +3027,$\mathsf{E}[X_1\mid X=x]$ +3028,"$\int_{[0,1]}$" +3029,$y^{\ast}:=\min(y)$ +3030,$\mathsf{E}|X|<\infty$ +3031,$\mathsf{E}_{\mathsf{Q}}[X] = \rho(X)$ +3032,"$ (MA.south)+(0, -1) $" +3033,$q^-(U(\omega))$ +3034,$Q_2\Delta X$ +3035,"$\mu=0.1, \sigma=0.15$" +3036,$\mathsf{Pr}(X > q_{\mathbf{v}}(p))=1-p$ +3037,$P_X(dx)$ +3038,$Y_{2}$ +3039,$Q_1dX$ +3040,${}^nS_X(t)\le {}^nS_Y(t)$ +3041,$(0.5)(20)+(0.5)(30)=25$ +3042,$\rho(X)\not=\sum_i\rho(X_i)$ +3043,"$M\subset \{1,\dots, n\}\setminus \{i, j\}$" +3044,$u(x)=(1-e^{-\pi x})/\pi$ +3045,$55+0.675\times 3.807=57.572$ +3046,$D \rho(X_0)$ +3047,$\alpha(1-f)$ +3048,$90$ +3049,$L_{250}^{\infty}(x)$ +3050,$\mathsf{Pr}(X < x) \le 0.1 \le \mathsf{Pr}(X\le x)$ +3051,$d\mathsf{Q}/d\mathsf{P} = g'(S(X))$ +3052,$\mathsf{E}[XZ_\epsilon]\to \mathsf{E}[XZ]$ +3053,$q(1)=\infty$ +3054,"$(p,q(p))$" +3055,$\prec_2^*$ +3056,$1/r$ +3057,"$\mathbf{v}=(v_1,\ldots,v_n)$" +3058,$\rho(X+X_i)=\rho(X)+\rho(X_i)$ +3059,$\displaystyle\int$ +3060,$\mathsf{E}[X1_{U_X\ge p}]\ge \mathsf{E}[XB]$ +3061,$\alpha_i(x)<\kappa_i(x)/x$ +3062,$\mathbf {\mathsf{E}[X_i\wedge a_i]}$ +3063,"$\{1,2,3,4,5,6\}$" +3064,"$(X_1,\dots, X_n)'$" +3065,$1_A(x)=0$ +3066,$X_{-4}=x$ +3067,$\mu-\sigma^2/2$ +3068,$s(1)=s_3=1$ +3069,$g(x)=0$ +3070,$L_0^{a+y}=L_0^a+L_a^{a+y}$ +3071,$x_{#4}$ +3072,$\mathsf Q(A)=\mathsf{E}[Z1_A]$ +3073,$n\ge 1$ +3074,$\mathsf{E}[X_m\mid X_{m+n}=x]=mx/(m+n)$ +3075,$\mathsf{E}[X]=28$ +3076,$2.576$ +3077,$Q_j=1-g(S_j)$ +3078,"$B_2=[0,0]$" +3079,$\sum c_i^2$ +3080,$X_i(v_i)$ +3081,$\alpha_i(a)S(a)$ +3082,$X(\omega)=0$ +3083,$\lambda=(1-\alpha_p)^{-1}$ +3084,$g_i$ +3085,$U \ge U_s$ +3086,$\bar P = a - \bar Q$ +3087,$p(1-\nu(p)-il(p))$ +3088,$Z_{a}(x)=g(S_X(a))/S_X(a))$ +3089,$X(\omega_1)a'$ +3093,"$0.1, 0.4, 0.5,\dots, 0.9$" +3094,"$I(q,p) \ne I(p,q)$" +3095,$k=-\log(p)/u$ +3096,$S(x_{i-1})-S(x_{i})=S(x_i-(x_i-x_{i-1}))-S(x_i)=-S'(x'_i)(x_i-x_{i-1})=f(x'_i)(x_{i}-x_{i-1})$ +3097,"$x,y\in C$" +3098,$g^{-1}(x)\le s$ +3099,$\mathsf{Q}(A)=2\mathsf{P}(A\cap B)$ +3100,$f(x) < f(y)$ +3101,$\bar S(a):= \mathsf{E}[L_0^a(X)]=\mathsf{E}[X\wedge a]$ +3102,$\iota^{\star}$ +3103,$\zeta_{s} = \Phi^{- 1}(s)$ +3104,$Z(\omega)=\dfrac{1}{1+r}\dfrac{\mathsf Q(\omega)}{\mathsf{P}(\omega)}$ +3105,$\iota$ +3106,$\mathsf{E}[X_ih(X)]$ +3107,$d^* = D/L^*$ +3108,"$\rho(X) = \max\{\rho_c(X), \mathsf{TVaR}_{0.8}(X) \}$" +3109,$S(x)=1$ +3110,$g(s)=\Phi(\Phi^{-1}(s)+\lambda)$ +3111,$\rho_g(X)<\infty$ +3112,$M=\iota Q$ +3113,$q + 2pq + 3p^2q+\cdots=q(1+2p+3p^2+\cdots)=1/q$ +3114,$g'(1-p^* )=1$ +3115,$-1$ +3116,"$ In general, define $" +3117,"$(4,2)$" +3118,$\alpha=1$ +3119,$\mathsf{E}[X \mid X \ge x] = \mathsf{E}[X 1_{X \ge x}] / \mathsf{Pr}(X \ge x)$ +3120,$\alpha_{Cat} \le \beta_{Cat}$ +3121,$\rho(X)=\mathsf{E}_\mathsf{Q}[X]$ +3122,$R_S$ +3123,$dt$ +3124,$\mathsf{Pr}(X\ge x_0)=p_-$ +3125,$E_i\in\mathcal F$ +3126,"$\bar P_{0,0}:=\rho(Y_{0,0})$" +3127,$a_1 < a_0-X_1$ +3128,$ is different from the contact function $ +3129,$t < 2/3$ +3130,"$\omega\in[0,1]$" +3131,"$h(x):=H(x, 1, t)$" +3132,$g(s)=3s$ +3133,$\mathbf {X_1/X}$ +3134,$\mathsf{E}[(X-\mathsf{E} X)^+]$ +3135,$\{ Z\mid \rho(X)=\mathsf{E}[XZ] \}$ +3136,$\mathsf{E}[X_iZ]=\rho_g(X)/2$ +3137,$\rho(X) \ge \mathsf{E}[X]$ +3138,$p=0.5$ +3139,"$\lambda\rho(X) + (1-\lambda)\rho(Y) \le \max(\rho(X),\rho(Y))$" +3140,$\{n_s\}$ +3141,$S(X_0)$ +3142,$r_m$ +3143,$X_i=X_i(a)$ +3144,$\bar Q_{act} = \bar Q - F_0$ +3145,$\beta_i/\alpha_i$ +3146,$\mathsf{E}[X_i/X \mid X > x]$ +3147,$\bar P(a)= (1-e^{-a\alpha\beta})/(\alpha\beta)$ +3148,$S(x)=e^{-x/\mu}$ +3149,$m_i\ge0$ +3150,$M(x)/(1-S(x))$ +3151,$\mathsf{E}[X \mid \mathcal F_0]$ +3152,"$(r,c)$" +3153,$r\le 0$ +3154,$\mathsf Q\in\mathcal Q$ +3155,"$X\,\Delta S$" +3156,$Z=(1-p)^{-1}1_{X>q_X(p)}$ +3157,$\mathsf{E}[X]=\sum_{\omega\in\Omega} X(\omega)\mathsf{Pr}(\omega)$ +3158,$d(g(S(x))/dx=g'(S(x))f(x)$ +3159,$p={{p}}$ +3160,"$\mathsf{E}_{\mathsf{Q}}[Y]=\mathsf{E}[Y\,g'(S(X))]$" +3161,$v\in V$ +3162,$\rho(\tilde X_1)=\rho(X_1) + \mathsf{E}[X_2]$ +3163,$\mathsf{E}[B]=p$ +3164,$\iota=0.10$ +3165,$\hat p > p$ +3166,"$C(S_0, a, t)$" +3167,$c = 0.5(0.5)2.5$ +3168,$M = 0.603$ +3169,$A/(A-P)$ +3170,"$A,B,C,D$" +3171,$h=\sin(77 s)$ +3172,$g(s)=1-(1-s)^3$ +3173,$\mathsf{CTE}_{p_0}=\mathsf{E}[X \mid X \ge x_0]$ +3174,$\phi\in \mathcal E$ +3175,$F_1 \prec_1 F_0$ +3176,$\mathsf{E}[u(w-X)] = u(w-c)$ +3177,$\lim_{s \downarrow 0}1/g'(s)$ +3178,$\Delta_j =g'(s_j-)-g'(s_j+)=\phi((1-s_j)+)-\phi((1-s_j)-)$ +3179,$\Omega_0:=\{\omega\in \Omega\mid X(\omega)=\max(X)\}$ +3180,$f(s)\le s$ +3181,$\bar\iota(a)$ +3182,$h(X)=(X-\mathsf{E} X)$ +3183,$j>0$ +3184,$n=1$ +3185,$S_0$ +3186,$g(S(x))=g(S(x-))=1$ +3187,$\mathsf{E} X + c\mathsf{E}[((X-\mathsf{E} X)^+)^p]^{1/p}$ +3188,$\mathsf{E}[hY]$ +3189,$A\subset\mathbb{R}$ +3190,$f(p)=(1-p)\phi'(p)=-(1-p)g''(1-p)$ +3191,$\mathsf{TVaR}_0( X )=\mathsf{E}[X]$ +3192,$ then $ +3193,$\epsilon_1$ +3194,$i>0$ +3195,"$0, 1, 90$" +3196,$\beta_1<\alpha_1$ +3197,$\nu p$ +3198,"$n=1, p=1/{{p}}={{pf}}$" +3199,$q(U)=F^{-1}(U)$ +3200,$\sqrt{0.1}=0.316$ +3201,$L(X)=e^{kX}/\mathsf{E}[e^{kX}]$ +3202,$\ge 0.95$ +3203,$\mathsf{E}[X_i \mid X=x]$ +3204,$vL + da$ +3205,$g'(s)=\nu$ +3206,$b=0.5$ +3207,$a < b_h$ +3208,"$(-\x*.8, 2*2)$" +3209,$L>d$ +3210,"$a_{0,2}$" +3211,$\mathbf {s_0}$ +3212,$\mathsf{E}[Z\mid X>a]=g(S(a))/S(a)$ +3213,$a_i=a(X_i; X)$ +3214,$\dot f(t)=a(x)$ +3215,$A^c$ +3216,$P_i \ge \mathsf{E}[X_i]$ +3217,"$\mathsf P((a,b])=b-a$" +3218,$\mathbf {g_1(s)=s^{0.4}}$ +3219,$1-p$ +3220,$\lim_{s \downarrow 0} s/g(s) = \lim_{s \downarrow 0}1/g'(s)$ +3221,$\rho_\mu$ +3222,$\mathsf{EPD}_\pi(X)$ +3223,$\bar F(a)$ +3224,$\mathsf{E}[Z_i\mid X] \ne \mathsf{E}[Z_j \mid X]$ +3225,$\mathbf {Q_1\Delta X}$ +3226,$P(X_{0}(a_{gc}))$ +3227,$20+8t>20+10t$ +3228,$b=1$ +3229,$p_0 = p^\ast = p_1$ +3230,$Z\in L^1$ +3231,$Y$ +3232,$g(S(x))=u$ +3233,$\phi'(s)\ge 0$ +3234,$x\mapsto (x-d)_+^{n}$ +3235,$\{X \le x^*\}$ +3236,"$X_1=0,0,0,0,1,1,2,3,20, 400$" +3237,$m_1=m_2$ +3238,"$\dfrac{\partial\rho}{\partial P} = \dfrac{0.4^2 P}{\rho(P,R,a)}$" +3239,$\mathbf {t+3}$ +3240,$u = g(S(x))$ +3241,$\mathsf{E}[X_2Z]$ +3242,$\rho(X)=g(q)$ +3243,$\bar M=\bar P-\bar S$ +3244,$(1-p)^{-1} \min_x x(1-p) + \mathsf{E}[(X-x)^+]$ +3245,$q_Y(1-U)$ +3246,$h(s)$ +3247,$f^{-1}(A)\in\mathcal B$ +3248,$(1-p)^{-1}\mathsf{E}[X_i1_{X\ge x_p}(X)]$ +3249,$\beta_1g-\alpha_1S$ +3250,$X_2(a)$ +3251,$g'(s)=bs^{b-1}$ +3252,$\mathsf P(A)=1-p$ +3253,$dF(x)$ +3254,"$(0,g_0)$" +3255,$\kappa_1(X)$ +3256,$x \mapsto -x$ +3257,$A(1_{X>x_1} + 1_{X>x_2})= A(1_{X>x_1}) + A(1_{X>x_2})$ +3258,${Z}_p \le c$ +3259,$X:\Omega\to\mathbb{R}$ +3260,$C_1+\cdots + C_n$ +3261,$\mathsf{E}[Y_d]$ +3262,$\mathbf {\alpha_1}$ +3263,$\tilde X\wedge a$ +3264,$d+v=1$ +3265,"$\Omega=[0,1]$" +3266,$q_Y$ +3267,$D\rho_X(\cdot)$ +3268,$\mathbf {X_2}$ +3269,$g^{-1}(u)$ +3270,$\sum_{i}X_{i} = X$ +3271,$g_{ROE}$ +3272,$>1-p$ +3273,$\mathsf{Pr}(X < x)$ +3274,$a=\mathsf{VaR}_{1-\tau}(X)$ +3275,$\mathsf{Pr}(X < x) \le 1/6 \le \mathsf{Pr}(X\le x)$ +3276,$h(x):=f(x)/S(x)$ +3277,$X_n(\omega)=1$ +3278,$\mathbb{R}$ +3279,$S_Y$ +3280,$\chi^2$ +3281,$X=X' + X''$ +3282,$(X\wedge a)\Delta g$ +3283,$\rho(X)=\mathsf{E}[h(X)L(X)]$ +3284,$f(x)\approx 0$ +3285,$ but if $ +3286,$Q=(a-EL)/(1+r)$ +3287,$a\ge 0$ +3288,$N=5$ +3289,$\mathsf{Pr}(E')=1-\mathsf{Pr}(E)$ +3290,"$D_n,D_n^*$" +3291,$\{ X=x \}$ +3292,$X_d$ +3293,$P=g(s)$ +3294,$\int xdF(x)=\int xf(x)dx$ +3295,$X({\mathbf{v}})$ +3296,$g(s)=s^{0.9}$ +3297,"$X_{t,1}$" +3298,$x=X(p)$ +3299,$\mathsf{E}_{\mathsf Q}[X_i]=\mathsf{E}[X_ig'(S(X))]$ +3300,$\hat s$ +3301,$\sigma_i^2$ +3302,"$(1-s, 1-g(s))$" +3303,$\ge$ +3304,$h(p)$ +3305,$\kappa_1(x)=\mathsf{E}[N_1/(N_1+N_2)]x$ +3306,$\max(X)=1$ +3307,$R_f$ +3308,$\phi(s)=0$ +3309,$\mathsf{E} X + c{ X-MX }$ +3310,$\mathsf{Var}(X+c)=\mathsf{Var}(X)$ +3311,$\mathsf{TVaR}_{0.975}$ +3312,$l^\infty$ +3313,$x_p=\mathsf{VaR}_p(X)$ +3314,$\sum v_iX_i$ +3315,$\mathsf{E}[X] + c\mathsf{E}[(X-\mathsf{E} X)^21_{X>\mathsf{E}[X]}]$ +3316,$R$ +3317,$s=0.5$ +3318,"$(1-S(x),x)=(p,q(p))$" +3319,$0!=1$ +3320,$\rho(U)=1$ +3321,$x=1000$ +3322,$\mathsf{E}[YZ]\le 0$ +3323,$\mathsf{Pr}(X<2)=1/6<\mathsf{Pr}(X\le 2)=1/3$ +3324,$m(1)=0$ +3325,$a_{t} = a_{t-1}$ +3326,$\mathsf{E}[\phi] = 1$ +3327,$A=\{X(\omega) > x\}$ +3328,$\mathbf {X_2(a)}$ +3329,$0.1 < s < 0.2$ +3330,$p < 1$ +3331,$g(0+)\ge 0$ +3332,"$3.129=\lambda \sigma(Y_{0,0})$" +3333,$\beta_i(x)/\alpha_i(x)> 1 > S(x) / g(S(x))$ +3334,$S(x)\approx k x^\alpha$ +3335,"$\mathit{EGL}_{gc}(a)>\max(0, \mathit{EGL}_{ro}(a))$" +3336,$\alpha_1(99)=0.1$ +3337,$\mathsf{TVaR}_p(X)=80$ +3338,$m\ge n$ +3339,$(a-X)^+$ +3340,$M_1dX$ +3341,$(X\wedge l)(\omega)=X(\omega)\wedge l$ +3342,$a=a[X]$ +3343,$\mathsf Q_k(B_k)=\mathsf{P}(B_k)/\mathsf{P}(B_k)=1$ +3344,$\mathsf{E}_\mu[\phi(\mathsf{E}_\pi u\circ f)]$ +3345,$a^{\star}(X)-a(X)$ +3346,$\mathit{PV}_{r_X}(X) + \mathit{PV}_{r_f}(\text{UW profit tax})$ +3347,$A-A\Phi(d^*)=A\Phi(-d^*)$ +3348,"$j=0,\dots, m-1$" +3349,$(P-S)/(a-P)\ge \iota$ +3350,$S=\mathsf{Pr}\{X>x\}$ +3351,$\mathsf{E}[p]$ +3352,$r^*$ +3353,$F(x)=\P(X\le x)$ +3354,$\mathsf{P}_X$ +3355,"$\int |X_n(\omega) - X(\omega)| \,\mathsf{P}(d\omega)\to 0$" +3356,$\bar Q(x)$ +3357,"$(a,b] \subset [0,1]$" +3358,$\mathbf {\omega_i}$ +3359,$\mathsf{Pr}(X< q(p))\le p \le \mathsf{Pr}(X\le q(p))$ +3360,$\iff$ +3361,$\exp$ +3362,$D>L$ +3363,"$\mathsf{biTVaR}_{0,1}^{0.0476}$" +3364,$\iota(0.5)=\iota^{\star}$ +3365,$n \ge 1$ +3366,$\kappa_{2}$ +3367,$Y\in L^\infty$ +3368,$\{ X=\mathsf{E}[X] \}$ +3369,$\mathsf P(f^{-1}(A))=\mathsf{Pr}(A)$ +3370,$0 \ge \rho(-X+a)=\rho(-X) + a \ge -\rho(X) +a$ +3371,$\mathsf{E}[X^k]$ +3372,$a>0$ +3373,$1\le p \le \infty$ +3374,$Z>\mathsf{E} Z$ +3375,$\mathit{PFL}$ +3376,$X_i(a)=X_i\dfrac{X\wedge a}{X}$ +3377,$g'(1)$ +3378,$0\le \alpha\le 1$ +3379,$g(S(x))=0$ +3380,$\rho\ge 0$ +3381,$\nu(p)=1/(1+\iota(p))$ +3382,"$[0,\infty)$" +3383,$\uparrow$ +3384,$a_i + b_i\ \mathit{EL}$ +3385,$\mu t + \sigma dW_t -\sigma^2 dt /2 +o(dt)$ +3386,$F(x):=\mathsf{Pr}(X\le x)$ +3387,$h$ +3388,$4/6$ +3389,$X_2=c_2+2Y$ +3390,$-Y\ge 0$ +3391,$S(x_2)(x_3-x_2)$ +3392,$0\le\lambda \le 1$ +3393,$x \ge x^\ast$ +3394,$1/4 < s\le 1$ +3395,$A_X = 5.976$ +3396,$\rho(X+Y)\ge$ +3397,$M = r K$ +3398,$X_n(\omega)=n$ +3399,$r = 0.6565$ +3400,$\nu^{\star}$ +3401,$-\rho(-X) =b-\rho(b-X)$ +3402,$\mathsf{E}_{\mathsf Q}$ +3403,$\alpha_1SdX$ +3404,"$a(\cdot, p)$" +3405,$\tau \ge t+d$ +3406,$\mu(\{p\})=1$ +3407,$c\approx -\sigma^2u''(w)/u'(w)$ +3408,$\|Z\|_p = \mathsf{E}[| Z|^p]^{1/p}$ +3409,$X\wedge a=\sum X_i(a)$ +3410,$\kappa_i(x)=E[X_i \mid X=x]$ +3411,$\lambda_0$ +3412,$\epsilon /2^{n+1}$ +3413,$\nu(x)$ +3414,$S(x)=\exp(-\int_x^\infty h(t)dt)$ +3415,$g(P)$ +3416,$2x$ +3417,$P(a) = g(S(a))$ +3418,$[F(x)](\cdot)$ +3419,"$\Omega=\{\omega_1, \ldots, \omega_6\}$" +3420,$\mu-\sigma^2/2=0.0992$ +3421,$F(p)=0.6$ +3422,$\rho(X_j)$ +3423,$S(a)=\mathsf{E}[1_{X>a}]$ +3424,$\mathsf{E}[X_ie^{kX}]/\mathsf{E}[e^{kX}]$ +3425,$y=a$ +3426,"$\mu,\sigma$" +3427,$g_i=g^{-1}(u_i)$ +3428,$u=0.1$ +3429,$1_{U>s}$ +3430,"$\rho(X)=\int g(S(t))\,dt$" +3431,$\mathbf {t}$ +3432,$\{ x \mid F(x) \ge p \}$ +3433,$g(s)q$ +3434,$\mathsf{VaR}_1(X)$ +3435,$\sigma_L$ +3436,$\bar S_i(a)=\mathsf{E}[X_i(a)]$ +3437,$Q=1-g$ +3438,$L_a^{a+y}(X)$ +3439,$\rho(X)=\mathsf{SD}(X)$ +3440,"$\int_{[a,b]} h(x)dF(x)$" +3441,$\bar\nu(a)=1/(1+\bar\iota(a))$ +3442,$-g''(1-p) = \phi'(p) = (1-p)^{-1}f(p)$ +3443,$g(S_X(X))$ +3444,$d\mathsf{Q}/d\mathsf{P}$ +3445,"$(\Omega, \mathcal F, \mathsf P)$" +3446,$\mathsf{E}_{\mathsf{Q}}[Y \mid X] = \mathsf{E}[Y \mid X]$ +3447,$0\le p\le 1$ +3448,$1/(1+r_f) = \mathsf{E}[p]$ +3449,"$D^f\rho_{X\wedge a,X}(\cdot)$" +3450,$V^{\ast}(1)=p/(1+r-p)$ +3451,$H_k(X)=H_{g_k}(X)$ +3452,$f(x)/S(x)$ +3453,$\partial\bar P/ \partial a$ +3454,"$X_{t,d}$" +3455,$\Delta S=0$ +3456,$\mathsf{E}[X_i(1) \mid X(\mathbf{v}) = q_{\mathbf{v}}(p)]$ +3457,$s_3=1$ +3458,$0< a\le 1$ +3459,$B(1_{X\le x})$ +3460,$2^{-t+1}$ +3461,$\beta < \alpha$ +3462,"$\bar P_i(\mathbf{v},a)$" +3463,$\sum \Delta g(S)_jX_j$ +3464,$\rho(0) = \rho(0+0)\le \rho(0)+\rho(0)$ +3465,$a<\infty$ +3466,$X=Y/\lambda$ +3467,$a\alpha_i(a)$ +3468,$q(1-s)$ +3469,$a_2 = 2.157$ +3470,$\mathsf{TVaR}_p = 20(0.55x_{67}+x_{68}+x_{69}+x_{70})/71$ +3471,$\mathsf{TVaR}$ +3472,$q(\psi)$ +3473,$a_{ro}:=\mathit{VaR}_{p}(X_{-1})={{a_x0}}$ +3474,$( x_{(j)}-x_{(j-1)} )$ +3475,$l(\mathbf X)$ +3476,$p\nu(p)$ +3477,$w_{0.75}$ +3478,$0.7 \ge p < 0.8$ +3479,$\omega_1=1$ +3480,"$(1-g(S(x)),x)=(p,q(1-g^{-1}(1-p))$" +3481,$v=1/(1+\iota)$ +3482,$f$ +3483,$a(X_i)=2.665$ +3484,$\mathbf{B}'(0) = -3\mathbf{P_0}+3\mathbf{P_1}$ +3485,$g_3(s)=s^{0.7}$ +3486,$1-\hat p$ +3487,$P(A\cup B)\le P(A)+P(B)$ +3488,$\iff \rho$ +3489,$0\le s\le \epsilon$ +3490,$q(p)=25$ +3491,$\rho(X)\le c$ +3492,$X_n(\omega)\to 0$ +3493,"$(0,3)$" +3494,$g(s)=sv+d$ +3495,$a=P+S$ +3496,"$(x,y)\not=(0,0)$" +3497,$\bar P_0$ +3498,$S=1-F$ +3499,$-t$ +3500,$f(x) = \dfrac{dF}{dx}$ +3501,"$\mathbf {D^f\rho_{X\wedge 30,X}(X_2)}$" +3502,$\bar{\mathbf P}$ +3503,$-g''(s)=\alpha(1-\alpha)s^{\alpha-2}$ +3504,$p(x) = \mathsf{Pr}(\{\omega\mid X(\omega) = x\})=\mathsf{Pr}(X=x)$ +3505,$\sigma=1$ +3506,$P(a)=1-Q(a)=1-h(F(a))$ +3507,$\delta=\dfrac{\iota}{1+\iota}=\dfrac{M}{a}$ +3508,$s\le s^*$ +3509,$a' := (1-S)\Delta X$ +3510,"$\mathbf{j, p, S, \kappa_1, \Delta X, \Delta(X\wedge a)}$" +3511,$\mathbf {Q=1-g(S)}$ +3512,$w/s = g'(s-) - g'(s+)$ +3513,$e^{\mu_L}-1$ +3514,$X=m$ +3515,$k(s)$ +3516,$\mathsf Q(A)=\int_A f(\omega)\mathsf P(d\omega)$ +3517,$(g-S)dX$ +3518,"$k, b$" +3519,$\mathsf{E}_\mathsf{P}[X_j]$ +3520,$p^*$ +3521,$\int_0^\infty xf(x)dx$ +3522,$\Delta P$ +3523,$\alpha_i(x)=\mathsf{E}\left[\frac{X_i}{X}\mid X > x \right]$ +3524,$r$ +3525,$s+\delta p = 1-\nu p$ +3526,$\mathbf p$ +3527,$\mathsf{Var}(\lambda X)=\lambda^2\mathsf{Var}(X)$ +3528,"$m_0, s_1, m_1, s_2, m_2$" +3529,$=\displaystyle\int_0^\infty x \P_X(dx)$ +3530,$\mathit{NPV}_1 = \bar Q - \bar Q_{act} = F_0$ +3531,$\rho_g(X)=35.2$ +3532,$\max_\mathsf{Q} \mathsf{E}_\mathsf{Q}[X]$ +3533,$Z=Y-X$ +3534,$\mathbf r$ +3535,$\mathsf{TVaR}_{0.642}$ +3536,$g(S(x_B))-g(S(x_B-))$ +3537,$u = \alpha_i(x)S(x)$ +3538,$\alpha_1 < \alpha_2$ +3539,$Z(g(s))=Z(s)+\lambda$ +3540,$\mathbf {\rho(X)}$ +3541,$\mathit{NPV}_{\infty}=a_xF_0$ +3542,"$X_{1,0}=\cdots=X_{m,0}=X_0=0$" +3543,"$\Omega=\{\omega_1, \omega_2 \}$" +3544,$a(X_i+X_j) < a(X_i)+a(X_j)$ +3545,$m=0.25$ +3546,$\{Y\mid Y\preceq_2 Z\}$ +3547,"$(de.east |- lee.north)+(0.375,0.25)$" +3548,$\mathsf{E}[X] + \pi\mathsf{E}[X]$ +3549,$c(\{i\})=c(i)$ +3550,$\hat g(s)=1-g(1-s)$ +3551,$W_{s+t}-W_s$ +3552,$\mathsf{Pr}(X=x)=0$ +3553,"$(1,2)$" +3554,$1-s$ +3555,$D_2$ +3556,$x=200$ +3557,$\mathbf{v}$ +3558,"$(0,0,0,0,0,5,0,0,0,5)$" +3559,$P=l + \delta(a-l)$ +3560,$S/L$ +3561,"$\int_0^a F(t)\,dt$" +3562,$\\mathbf {\1}$ +3563,$\int_0^s \mu(dt)/(1-t)$ +3564,$Z\circ T$ +3565,$\mathbf {D^n\rho_{X\wedge 30}(X_2)}$ +3566,$=v_f \mathsf{E}_Q\left[\dfrac{X_i}{X}(X\wedge A)\right]$ +3567,$g(S(a))$ +3568,$\mathcal M_\rho$ +3569,$P=(1+r)\lambda\mathsf{E}[X]$ +3570,$F(a+)=\lim_{x\downarrow a} F(x)$ +3571,$f<1$ +3572,$\alpha>1$ +3573,$\mathcal F_0\times \mathcal F_1$ +3574,$\rho(Y)$ +3575,$\mathsf Q(\omega)\ge 0$ +3576,$\lim_{s \uparrow 1}g'(s)$ +3577,$k>2$ +3578,$S\to Y$ +3579,$\mathsf{E} X$ +3580,$s'(t)$ +3581,$g'\circ S_X$ +3582,$s=0.1$ +3583,$g = s/(1-f)$ +3584,$g(s)=A(1_{U < s})$ +3585,$\Delta g(S_j)=g(S_{j-1})-g(S_j)$ +3586,$A\wedge L$ +3587,$\mathbf {2\mathsf{VaR}_p(X_1)}$ +3588,$g'(1-s)+g(0+)\delta_1$ +3589,"$5^{-1},5^{-2},5^{-3},\dots$" +3590,$\mathsf{EPD}_p(X)$ +3591,$+$ +3592,$c(\alpha)x^\alpha g(x)$ +3593,$\mathit{NPV}_{\infty} = a_xF_0$ +3594,$\mathsf{E}[X]=\mathsf{E}[Y]$ +3595,$v/\sqrt{n}$ +3596,$X_h$ +3597,$\mathsf{E}[(a-X)^+]=\int_0^a F(x)dx$ +3598,$\mathsf{Pr}(X < x)\ge 1/6$ +3599,"$\mathsf{cov}(X_i,X)$" +3600,"$(p,t)$" +3601,$e^{-rt}S_t$ +3602,$9+1$ +3603,$(x-d)^+ \wedge l$ +3604,$\mathsf Q(B) = \mathsf P(A\cap B)/\mathsf P(A)=\mathsf P(A\cap B)/(1-p_0)$ +3605,$\mathsf{E}[S_t]=e^{\mu t}$ +3606,$Y_i$ +3607,$\sqrt{x}$ +3608,$\rho(X-X)=\rho(X)+\rho(-X)=0$ +3609,$dG/dF=g'(S(x))$ +3610,$D_m\subset D_n$ +3611,"$[0,1]\subset\mathbb R$" +3612,$r-1$ +3613,$d_f = r_f / (1+r_f)$ +3614,$\hat q(p)=q(1-g(1-p))$ +3615,$X=Y$ +3616,$\mathsf{Pr}(X_n=1)=1/n$ +3617,$U^{1/b}$ +3618,$X\preceq_1 Y$ +3619,$E(X-q(X))^+$ +3620,$X_{-2}$ +3621,$t=U_X(s)$ +3622,$\mathsf{E}_{\mathsf Q}[Y]$ +3623,$3^{30}=2.06\cdot 10^{14}$ +3624,$\rho(kX)\ge k\rho(X)$ +3625,$M(x)=P(x)-S(x)$ +3626,$H$ +3627,$a=\mathsf{VaR}$ +3628,$\int X_n=1$ +3629,"$\displaystyle\int_0^a \kappa_i(x)f(x)\,dx + a\alpha_i(a)S(a)$" +3630,$\kappa_i(x)\approx x -\sum_{j\not=i} \mathsf{E}[X_j]$ +3631,$\alpha_1(90) = (0.0816 \cdot 0.0625 + 0.1 \cdot 0.0625)/(0.0625+0.0625)=0.01135/0.125=0.0908$ +3632,$c_i$ +3633,$0 \le X_i(a) \le X_i$ +3634,$\sup_i f_i$ +3635,$D\rho_X(X_1)=6.2085$ +3636,$+\mathsf{NORIPOFF}$ +3637,"$(a,b)$" +3638,$t\downarrow 0$ +3639,$\rho_g(X)=\mathsf{E}_\mathsf{Q}[X]$ +3640,$\{\mathsf{P} \}$ +3641,$\mathsf{E}_{\mathsf Q}[Y] = \mathsf{E}[YZ]$ +3642,$\mathcal{G}=\sigma(X)$ +3643,$\pi$ +3644,$h(x)=-d/dx(\log(S(x)))$ +3645,$x=8$ +3646,"$\displaystyle\int_\Omega g(X(\omega), \omega)\mathsf{Pr}(d\omega)$" +3647,$X\_{2}$ +3648,$dS$ +3649,$\sum \alpha_i S\Delta (X\wedge a)$ +3650,"$g'(s) = \frac{1-w}{1-p_0}1_{[0, 1-p_0)}(s) + \frac{w}{1-p_1}1_{[0, 1-p_1)}(s)$" +3651,$\mathscr{E}$ +3652,$\mathsf{E}_{\mathsf Q}[X_i \mid X=x] = \mathsf{E}[X_iZ \mid X=x]/\mathsf{E}[Z \mid X=x] = \mathsf{E}[X_i \mid X=x]$ +3653,$pX$ +3654,$g(S(a))/S(a)$ +3655,$\sum X_i(a)\Delta g(S)$ +3656,"$(p,q(1-g^{-1}(1-p)))$" +3657,$0\le \pi\le 0.5$ +3658,$\bar\delta=\bar\iota/(1+\bar\iota)$ +3659,$q^-(F(x))=x$ +3660,$1-g(s)$ +3661,$P=L + d(a-L)$ +3662,$p\not=0.75$ +3663,"$a=0, b=\alpha$" +3664,$\mathsf{E}[X_i\mid X=q(p)]$ +3665,$\mathbf {\vert S\vert}$ +3666,"$\bar S(a)=\int_0^a S(x)\,dx$" +3667,$X=X\wedge a + (X-a)^+=\sum_i X_i(a) + (X-a)^+$ +3668,$r_f = 0.01$ +3669,$X_2=X-X_1$ +3670,$c_1$ +3671,$\max_\mathsf{Q} \mathsf{E}_\mathsf{Q}[X] - \alpha(\mathsf Q)$ +3672,$u^{(n-1)}$ +3673,$(r-\sigma^2/2)t$ +3674,$\tau_i=\tau$ +3675,$\tau=\tau_i=0$ +3676,$a=a(s)$ +3677,$\mathsf{E}[Z\mid X]=Z$ +3678,$\mathsf{E}[X_1\tilde Z]=\mathsf{E}[X_2\tilde Z]=500$ +3679,$f(L)=L$ +3680,$f(L) \le L$ +3681,$p=0.283$ +3682,$g'(s)=\alpha s^\alpha/s$ +3683,$n-4$ +3684,$xdF(x)$ +3685,$\mathsf{TVaR}_{0.8}(X)=25$ +3686,$X_0=X_1=0$ +3687,$Q_X$ +3688,$\mathsf{TVaR}_{p^\ast}(X)=\bar P$ +3689,$\mathsf{E}[(S_t-a)1_{\{S_t>a\}}$ +3690,$P_X(A)=0$ +3691,$L > a$ +3692,$f=0$ +3693,$f(x)dx=dp$ +3694,$P_X(A)=\mathsf P(X\in A)= F(b)-F(a)$ +3695,$Z(a')=g(S_X(a))/S_X(a))$ +3696,"$X_i(\omega), i=1,...,N$" +3697,$\alpha(\mathsf Q)$ +3698,$\phi(p)$ +3699,$\mu(\{p_1\})=w$ +3700,$G\mathsf X$ +3701,$\omega=0$ +3702,$\displaystyle\int_\Omega X(\omega)\mathsf{Pr}^*(d\omega)$ +3703,$P-D$ +3704,$X>a$ +3705,$\iota=$ +3706,$\lim_{t\to 0}a(X_1; X+tX_1)=a(X_1;X)$ +3707,$1+Z-\mathsf{E} Z$ +3708,$e = P/C$ +3709,$t^\star=1/2$ +3710,$t+1$ +3711,$1-B_p=B_{1-p}$ +3712,$\mathsf{Pr}(|X_n(\omega)-X(\omega)|>\epsilon)\to 0$ +3713,$\bar M(x)$ +3714,$X\not\preceq_n Y$ +3715,$0\le x < a$ +3716,"$Z_2:=\sum_{t+d=2} Y_{t,d}$" +3717,$ since the contact function $ +3718,"$c_1+c_2=(c(1) + c(1,2) - c(2) + c(2) + c(1,2) -c(1))/2=c(1,2)$" +3719,"$(-\infty, \infty)$" +3720,$a(X)\equiv a$ +3721,$\mathcal E(X)=c\mathsf{E}[X^2]$ +3722,$x^{**}$ +3723,"$D^f\rho_{X\wedge a,X}(X_i)$" +3724,$X(p)=q(T(p))$ +3725,"$(1-S(x), x)$" +3726,$\tilde X_1+\tilde X_2\succeq^2 \tilde X_1$ +3727,$\mathcal D(X)+\mathsf{E}[X]$ +3728,$S_{\mathbf{v}}$ +3729,$\mathsf{VaR}$ +3730,$\mathsf{E}[X]+\mathsf{SD}(X) \le \mathsf{E}[Y]+\mathsf{SD}(Y)$ +3731,$\bar S_i$ +3732,$\{X(\mathbf{v}) = q_{\mathbf{v}}(p)\}$ +3733,$\alpha_iSdX$ +3734,$c=0.5$ +3735,$K$ +3736,$g(p)/p-1$ +3737,$a(X_i; X)$ +3738,$\log(1+\mu t + \sigma dW_t)=\mu t + \sigma dW_t +o(dt)$ +3739,$\max(X)$ +3740,$\mathsf{E}[X_1\mid X < 2^{-m}]$ +3741,$x>\sup(X)$ +3742,$M=\inf\{ x\mid S(x)=0\}$ +3743,$\mathsf{VaR}_\pi(X)$ +3744,$0=\mathsf{Pr}(X<1)<1/6=\mathsf{Pr}(X\le 1)$ +3745,$-k<0$ +3746,$X_n=Y_1+\cdots +Y_n$ +3747,$^{}$ +3748,$\mathsf{CTE}_p(X)=(8+12+25)/3=15$ +3749,$p \ge 0.9$ +3750,$S_0=1000$ +3751,$\lim_{s \to 1}{\mathsf{E}[ r_{s} ] = - 1}$ +3752,$1_{U0}$" +3771,$\prod_{n\ge N}(1-\frac{1}{n})=0$ +3772,$X\le 0$ +3773,$g(s)=s^{0.8}$ +3774,$q \cdot X$ +3775,$p=0.1$ +3776,$\mathsf{E}[X_iX]$ +3777,"$(p, q(1-g^{-1}(1-p)))=(p, q(\hat p))=(p, \hat q(p))$" +3778,$\mathsf P(X\le q_X(p))=p$ +3779,"$\rho_1,\rho_2$" +3780,$P/(A-P)=P/Q$ +3781,$-\rho$ +3782,$\alpha_1(98)=0.1$ +3783,$\pi=1.2613$ +3784,$\gamma=0.421$ +3785,$8+11.1667=19.167$ +3786,$\mathsf{Pr}(Y_m > y) = 1 - (1 - \mathsf{Pr}(X > y))^n$ +3787,$\beta_i(x)/\alpha_i(x) < g(S(x))/S(x)$ +3788,$h(p)=s^3$ +3789,$\psi$ +3790,$\mathsf{VaR}_p(X)=\mu + \sigma \Phi^{-1}(p)$ +3791,$B_k$ +3792,$\bar P(\infty)=\mathsf{E}[q(U)\phi(U)]$ +3793,"$Binomial(s,N)$" +3794,$x=S^{-1}(g^{-1}(s))$ +3795,$e^{-rt}$ +3796,$\mathsf{VaR}_{p^*}$ +3797,$\mathsf{E}[X^2]$ +3798,$=\mathrm{MV}(y-T(X))^+$ +3799,$\mathsf{E}[YZ_\epsilon]\to\mathsf{E}[YZ]$ +3800,$p^{* }$ +3801,$\beta_Q=(a/Q)\beta_A + (P/Q)\beta_L$ +3802,$r\times n$ +3803,$F(2)=0.75$ +3804,$(80-11)\times 0.25$ +3805,"$S, S^{-1}$" +3806,$\mathsf{Q}'$ +3807,$q(0.1)=1$ +3808,"$k=0,1,\dots,n-1$" +3809,$q(1-g^{-1}(1-p))$ +3810,$\tilde X_j$ +3811,$\bar F$ +3812,$\pm\infty$ +3813,"$c\in[0,1]$" +3814,$dg$ +3815,$\rho_c(Y)=\mathsf{E}[Y]$ +3816,$p_Y<0.5$ +3817,"$\mathsf{E}[W]=\sum_{d\ge 0} \mathsf{E}[Y_{-d,d}]$" +3818,$\mathscr{P}$ +3819,"$(\mu,\sigma)$" +3820,"$(brR15 |- lee.south)+(-0.25,-0.25)$" +3821,$\pi=1.2497$ +3822,"$\mathsf{E}[(X-a)^+]= p\,\mathsf{E} X$" +3823,$\bar\iota$ +3824,$L(X)=(X-\mathsf{E} X)/\mathsf{SD}(X)$ +3825,$g(s) \ge 1$ +3826,$v(A\cup B) + v(A\cap B)\ge v(A) + v(B)$ +3827,$\bar P_\tau(a)=\bar P(a) + \tau(a-\bar P_\tau(a))$ +3828,$\nu>0$ +3829,$\mathsf{E}[X\mid\mathcal F_0]=\mathsf{E}[X]$ +3830,$P_g\{X=M\}=g(0+)>0$ +3831,$\mathsf{E}[X_i\mid X = x_p]$ +3832,$\Delta=a'-a$ +3833,$\alpha_i(x)S(x)$ +3834,$r_h-\mu_L=r-r_L$ +3835,$-0.0012$ +3836,$\rho(X)$ +3837,$\mathsf Q$ +3838,$+1$ +3839,$\implies\mathsf{FATOU}$ +3840,$\bar P(a)>\mathsf{E}[X\wedge a]$ +3841,$1-w$ +3842,$=1/(1-p)$ +3843,$Q_j = 1 - g(S_j)$ +3844,$A(X)$ +3845,$\mathsf Q(\omega)=Z(\omega)\mathsf{Pr}(\omega)$ +3846,$X\ge \mathsf{VaR}_p(X)$ +3847,$p_+-p_-$ +3848,$\mu(\{0\})=\phi(0)=g'(1)$ +3849,"$s\in (0,1]$" +3850,"$p\in (0,1)$" +3851,"$\lambda, \iota, \psi$" +3852,$h_{xx}$ +3853,$u=x$ +3854,$af + a(1-f)/q$ +3855,$\bar P_{act}$ +3856,$L_d^{d+l}(X)$ +3857,$r=0.06$ +3858,$x\mapsto x^{3/2}$ +3859,$L_0^{500}(x)$ +3860,"$A_3,B_3$" +3861,$\mathbf {\beta_{1}g(S)\Delta X}$ +3862,"$B_1=[0,0]$" +3863,$f(x)=x^2$ +3864,$E_\mathsf{Q}[1]$ +3865,$\mathsf{E}[(X-\mu)^2]$ +3866,$s\to 0$ +3867,$g_0=0$ +3868,"$\displaystyle\int_0^1 q(p)\,dp$" +3869,$P(X_{-1}(a_{gc}))=9094.25$ +3870,$\phi$ +3871,$S;g(S)$ +3872,$\kappa_{i^*}$ +3873,$\rho_a$ +3874,"$\bar P_{0,2}$" +3875,$x=e^{\mu + y\sigma}$ +3876,$f(w|s)$ +3877,$\mu_U = 15$ +3878,"$Y_{2,0}$" +3879,$r_{pq}:=\sqrt{p(1-p)}$ +3880,$A\subset B$ +3881,$g(s)=s^b$ +3882,$S_j=S_{j-1}-p_j$ +3883,$\{X = q_X(p) \}$ +3884,$X(\omega)=\omega$ +3885,$1_A(x)=1$ +3886,$g(s)=100s \wedge 1$ +3887,$(0.333...)(0.15)$ +3888,$\bar P_1$ +3889,$v(A\cup B)\le v(A)+v(B)$ +3890,$V(1)$ +3891,$x^2$ +3892,$a_{0}=a(Y_{0})$ +3893,$C=1-H$ +3894,$Y = NX$ +3895,$a_l < b_l$ +3896,"$1,9,10$" +3897,$g_0 \le 1-\alpha$ +3898,$(1+r)Z=\mathsf Q/\mathsf{P}$ +3899,"$\{0, 9, 10\}$" +3900,$\iota^*$ +3901,"$t=0,1,2,\dots$" +3902,$0.354 \cdot 8 = 2.83$ +3903,$x=z$ +3904,$(x+(X-x)^+)^n\not=x^n+((X-x)^+)^n$ +3905,$f(R) = \mathsf{E}[f(X)]$ +3906,$\delta_i$ +3907,$1-2/3=1/3$ +3908,$\mathsf{Amb}(X)$ +3909,$\rho^{ho}_c$ +3910,$X \le Y$ +3911,"$\mathsf{cov}(X,M)=\mathsf{cov}(X_i,M)$" +3912,$\mathbf {X'}$ +3913,$F(X(\omega))$ +3914,$\mathsf{VaR}_{p}( \cdot \mid \mathcal F_t)$ +3915,$d$ +3916,$\mathsf{E}[X\wedge a]= 2.4982$ +3917,$\mathsf{E} X + \inf_x \{\alpha_1\mathsf{E}[(x-X)^+] + \alpha_2\mathsf{E}[(X-x)^+] \}$ +3918,$M:=\esssup X$ +3919,$g(s)=1-\sqrt{1-s}$ +3920,$dF=-d(g\circ S)=$ +3921,$E_\mathsf{Q}[X_i]$ +3922,"$\mathsf{TVaR}_0,\mathsf{TVaR}_1$" +3923,$f_x$ +3924,$s^{-1}(\cdot)$ +3925,$X_{1c}$ +3926,"$(p, \mathsf{E}[X_i\mid X=q(1-g^{-1}(1-p))])$" +3927,$\mathsf{TVaR}_p(X)=\mathsf{TCE}_p(X)=\mathsf{E}[X\mid X \ge \mathsf{VaR}_p(X)]$ +3928,"$\omega\in [0,1]$" +3929,$\mathsf{PH}$ +3930,$N=X-L_{r_a}^{r_a+r_l}(X)$ +3931,$x_i-x_{i-1}=dx$ +3932,$P(\hat s)=\mathsf{E}[\hat s]=s$ +3933,$X_i(1)$ +3934,$0<\alpha_1<\alpha_2<1$ +3935,$v(B)$ +3936,$(dX_t)^2$ +3937,$L^2$ +3938,$g(s)=\Phi(Z(s)+\lambda)$ +3939,$|Y_n|\le 1$ +3940,$a=0$ +3941,$0.25 + U/4$ +3942,$F(q^-(p))\ge p$ +3943,"$[0,a)$" +3944,$0.909+0.273=1.182$ +3945,$\bar P=\bar S+\bar M$ +3946,$\rho(X)=\rho(X\wedge a) + \rho((X-a)^+)$ +3947,$X\wedge a=30$ +3948,$F$ +3949,"$(2,1)$" +3950,$X(\mathbf{v})=\sum_i X_i(v_i)$ +3951,$\Phi^{-1}(0)=-\infty$ +3952,$2/6$ +3953,$F_M$ +3954,$q_C(p)=\inf C$ +3955,$g(S)dX$ +3956,$\mu \cdot T_k$ +3957,$\rho:\mathcal{S}\to \mathbb{R}$ +3958,$F_I^{n*}$ +3959,$Q(x)$ +3960,$\mathsf{E}[X] + \pi\mathsf{E}[(X-\mathsf{E} X)^+]$ +3961,$X_1(v_1)$ +3962,"$g\in D_n^*=\{ g \mid (-1)^{k+1} g^{(k)} \ge 0, k=1,\dots,n-1, (-1)^n g^{(n-1)}\text{ non-increasing} \}$" +3963,$\mathbf {\mathsf{E}[X_i(a)]}$ +3964,$\bar Q_{1}$ +3965,$h=1+\lambda(f-\mathsf{E} f)$ +3966,$\Delta Q(a)$ +3967,$t\to 0$ +3968,"$i=1,2$" +3969,$\sigma=2.15$ +3970,$\mathsf Q\in \mathcal Q$ +3971,$\alpha \ge s_0 g'(s_0)/g(s_0)$ +3972,$g(s)=s^r$ +3973,"$t=1,2,...,\tau$" +3974,$Y(\omega)$ +3975,$Sdx$ +3976,$s=1/4$ +3977,$\int_0^\infty xg'(S(x))dF(x)=\int_0^\infty g(S(x))dx$ +3978,$\mathsf{E}[Xe^{\pi Z}]/\mathsf{E}[e^{\pi Z}]$ +3979,$X_2' = X_2+\cdots +X_n$ +3980,$1_{U>0.95}$ +3981,$g=u^2=0.01$ +3982,$100$ +3983,$X\wedge a \le X$ +3984,"$Y_{t,0}$" +3985,$s>s^\ast$ +3986,$g(s)-\hat g(s)$ +3987,$R:=\bar P_{act}-\bar S$ +3988,$Var[T]=s(1-s)/N$ +3989,$\sum w_i=1$ +3990,$\alpha_i(x) =\mathsf{E}[X_i/X\mid X>x]$ +3991,"$\{x_1,...,x_n\mid X < \max(X)-\epsilon\}$" +3992,$z=x$ +3993,$F_n(x)\to F(x)$ +3994,$c=2.5$ +3995,"$\rho(1000, 3000, 3500)$" +3996,$w(x)=e^{kx}$ +3997,$\mathsf{Pr}(X>0)$ +3998,$1_\omega(\omega')=1$ +3999,$g(s)=cs$ +4000,$f(t|s)$ +4001,$\mathbf {F}$ +4002,$\mathsf{E}[(a-X)^+]$ +4003,$\displaystyle\int_0^\infty u(x)dF_X(x)$ +4004,$\Lambda\dfrac{\mu_{U}}{\sigma_U} = \dfrac{E( r_{U} ) - r_{f}}{\sigma_{r_{U}}} \left(\dfrac{\mu_{U}}{\sigma_{U}}\right)$ +4005,$f'(x_0)$ +4006,$\mathsf{E}[X\mid X=x]\equiv x$ +4007,$y^{\ast}-x^{\ast} \ge \epsilon$ +4008,$(1+\rho)\mathsf{E}[C]$ +4009,$1-g$ +4010,$g(S(x_{i+1}-))-g(S(x_{i}))$ +4011,"$d\,F(X)$" +4012,$Q(a)$ +4013,$a_i=\mathsf{E}_\mathsf{Q}[X_i]$ +4014,$(1+c)\mu$ +4015,$\mathbf {\Delta(X\wedge a)}$ +4016,$\mathsf{E}[YZ]$ +4017,$\mathsf{VaR}_{0.99}$ +4018,$dG(x)=g'(S(x))dF(x)$ +4019,$n=100$ +4020,$\rho(c) = c$ +4021,$\delta_p$ +4022,$\sigma$ +4023,"$(s(t),m(t))$" +4024,$X>x$ +4025,$\mathsf{E}[X_i(a)]$ +4026,$\sigma_U = 1$ +4027,"$(p,q(p))=(1-S(x),x)$" +4028,$w_i\ge 0$ +4029,$\int X_n\to 0$ +4030,"$r_f\ge 0, r>0$" +4031,$Z$ +4032,"$X_1,X_2$" +4033,$r_L$ +4034,$\gamma=\mathsf{Pr}(X>\mathsf{E}[X])$ +4035,"$\Omega=[0,1]\times [0,1]$" +4036,$x_i$ +4037,$a(X)$ +4038,$g(s)+g'(s)(1-s)\ge 1$ +4039,$\mathbf {\alpha_1S\Delta X}$ +4040,"$\mathsf{cov}(X_i,\sum_j X_j)=\mathsf{cov}(X_i,X_i)=\mathsf{Var}(X_i)>0$" +4041,$\mathsf Q(A)=0$ +4042,$0-\rho(-H)$ +4064,"$Y_{0,1}$" +4065,$\mathsf{E}[X_ig'(S(X))]$ +4066,$a(X;X)=\rho(X)=\sum_i a_i$ +4067,$\displaystyle\int_0^\infty xg'(S(x))f(x)dx$ +4068,$A(X)\not= B(X)$ +4069,$\lim_{y\uparrow x} f(y)$ +4070,$\rho(X)= \mathsf{E}_{\mathsf{Q}_X}[X]$ +4071,$\psi^{-1}(p)$ +4072,$\mathcal Q\subset\mathcal M(\mathsf P)$ +4073,"$a(x_1,\dots,x_n):=a(X(x_1,\dots,x_n))$" +4074,$1-q$ +4075,$ds(t)/dt$ +4076,$X_{-3}=C'_1 + \cdots + C'_n$ +4077,$g(S(M-))/S(M-)$ +4078,$\mathsf{VaR}_{p_0}(X)=\sup X$ +4079,$p=1/2$ +4080,$y\not\in C$ +4081,$X_0=C_1 + \cdots + C_N$ +4082,"$2^0, 2^2, 2^4, ...$" +4083,$F_I$ +4084,$gdX$ +4085,$b_l \le 1 \le b_h=2-b_l$ +4086,$30+10t$ +4087,$m_1$ +4088,"$Y_{t,d}$" +4089,$F(x)=\sup\{ p\mid q(p) < x \}$ +4090,"$X_{i,j} \leftarrow \kappa_{i}(X_j)$" +4091,$1/g'(0)$ +4092,$1-g(1-p)$ +4093,$d\Pi = (r_h-\mu_L)\Pi dt$ +4094,$q(U_X) = m$ +4095,$\alpha_i(t)$ +4096,$\mathbf {gS}$ +4097,$U=X+Y$ +4098,$p^\ast$ +4099,"$0,0,0,1,2,5,8,12,23,40$" +4100,$0\le k < 2^m$ +4101,"$c=1,2,3$" +4102,$E[s]=0.1160$ +4103,"$\lambda([a,b]) = b-a$" +4104,$p^+$ +4105,$S_X(t)=S_{X\wedge a}(t)$ +4106,$h(X)=X$ +4107,$D_1\supset D_2\supset \cdots \supset D_\infty$ +4108,$g''(s)=-\phi'(1-s)\le 0$ +4109,$\prec_1^*$ +4110,$X=100$ +4111,$\mathsf{WCE}_p(X) = \mathsf{TVaR}_p(X)$ +4112,$X\wedge a(X)$ +4113,$\times$ +4114,$\bar M(a)$ +4115,$\mathsf{LI}$ +4116,$(p_0 < p^\ast < p_1)$ +4117,"$c = 0.5,1.0,\dots,2.5$" +4118,$\sup X_n=1\not=\sup X=0$ +4119,$IL$ +4120,$S(x)\leftrightarrow g(S(x))$ +4121,$\rho(X) = \mathsf{E}[X] + c\mathsf{E}[X-\mathsf{E}[X]]^+$ +4122,$\lambda=\sum_i \lambda_i$ +4123,$\mathsf{TVaR}_{0.8}$ +4124,$Q = M/\iota$ +4125,$\mathsf{Pr}(X>\mathsf{VaR}_p(X))=1-p$ +4126,$(a_i)_i$ +4127,$g(s)=d+sv$ +4128,$p\nu_p$ +4129,$f_i$ +4130,$\mathsf{P}(X=X_j)=\Delta S_j:=S(X_{j-1})-S(X_j)$ +4131,$P\approx \mathsf{E}[A(1)] + k\mathsf{Var}(A(1))/2$ +4132,"$X_{0,2}$" +4133,$aq_X(p) \}$ +4159,$X_i=\mathsf{E}[X_i\mid X]$ +4160,$S(x)>>0$ +4161,$q_B \le q_C$ +4162,$\mathsf{TVaR}_{0.75}$ +4163,$g'(s) < \infty$ +4164,$\hat p$ +4165,$\kappa_i(q(1-g^{-1}(1-\tilde p)))$ +4166,$q^-(p)$ +4167,$\rho(X-\rho(X))=\rho(X)-\rho(X)=0$ +4168,$g_0$ +4169,$\mathsf{TVaR}_p(X)=\mathsf{E}[X\mid X >\mathsf{VaR}_p(X)]=\sum_i\mathsf{E}[X_i\mid X>\mathsf{VaR}_p(X)]$ +4170,$dt\to 0$ +4171,$\{X\in L^\infty \mid \rho(X)\le c \}$ +4172,"$Y_{2,2}$" +4173,"$c_i=\displaystyle\int_0^1\dfrac{\partial c}{\partial x_i}(tx)\,dt$" +4174,$\rho(X_{-1}\wedge a_{ro})={{mvp_ro}}$ +4175,$\bar \iota = \dfrac{\bar M(a)}{\bar Q(a)}$ +4176,$\mathcal{N}_{X\wedge a}(X_i(a))$ +4177,$f'>0$ +4178,"$\bar M_{t,0}$" +4179,$E$ +4180,$p^\ast = 0.48732$ +4181,$r_P$ +4182,$\mathbf {t+1}$ +4183,$S=g(S)=1$ +4184,$\mu_d = (6-d)^2$ +4185,$g(s)=0.9s + 0.1$ +4186,$\left( g(S(x_{(j)}))-g(S(x_{(j-1)})) \right) / ( x_{(j)}-x_{(j-1)} )$ +4187,$t^\star$ +4188,$1_Z$ +4189,$\omega < p^-$ +4190,$q = s$ +4191,$\bar F(a):=\int_0^a F(x)dx=a-\mathsf{E}[X\wedge a]$ +4192,"$s\in (0,1)$" +4193,$\mathsf{E}[X] + \pi\mathsf{E}[((X-\mathsf{E}[X])^+)^2]^{1/2}$ +4194,"$\omega\in [0,0.1)\cup [0.25, 0.35) \cup [0.5, 0.6) \cup [0.75, 0.85)$" +4195,$80-11=69$ +4196,$g'$ +4197,$\rho(X)+c$ +4198,$S(x)=(1+x)^{-\alpha}$ +4199,$r_M$ +4200,$U(2)=0$ +4201,$\alpha_i(x)$ +4202,$\sup X\le \sup Y$ +4203,$\sigma(X)=\mathsf{E}[(X-\mathsf{E} X)^2]^{1/2}$ +4204,$S(x)=\Phi((-x+\mu)/\sigma)$ +4205,$\tilde X_1 + \tilde X_2 \succeq^2 \tilde X_1$ +4206,$p=F(a)=1-S(a)$ +4207,$v\mathrm{EL}+da\ge \mathrm{EL}$ +4208,$X=X_s + X_c$ +4209,"$\mathsf{VaR}_{0.995}=64,861$" +4210,$P = 3.1035$ +4211,$x=q(1-g^{-1}(1-p)))$ +4212,"$d=1,2,\dots$" +4213,"$(\x*1.2, 2)$" +4214,$h=1$ +4215,"$k_1, k_2$" +4216,$p=0.95$ +4217,"$s^{\ast}=1/2, \lambda^{\ast}=0$" +4218,$\esssup(X)=1$ +4219,$1-p \ge g^{-1}(1-p) \implies 1-g^{-1}(1-p) \ge p \implies q(1-g^{-1}(1-p))>q(p)$ +4220,"$x+y\wedge aX =\min(x+y,aX)$" +4221,$H(X)q(p) \}$ +4234,"$g'>0, g''<0$" +4235,$Y=c\in \mathbb R$ +4236,$h(u)=1$ +4237,$\lim_{\epsilon \downarrow 0} (f(x+\epsilon)-f(x))/\epsilon$ +4238,$\omega_1$ +4239,$r>0$ +4240,"$\alpha_i(\mathbf{v}, x)$" +4241,$\omega\ge 0.4$ +4242,$\mathsf{Pr}(B)=0$ +4243,$\bar q_{X_1+X_2}(s) \le 2\bar q(s)$ +4244,$\mathsf{E} X +\lambda_1 {(X-\lambda_2 \mathsf{E} X)^+}_1$ +4245,$f(t)=\rho(tX)$ +4246,$X_n\uparrow 1$ +4247,$\int S(x)dx$ +4248,$A\subset \Omega$ +4249,$(A-L)^+$ +4250,$P(x)/Q(x)$ +4251,$\mathsf{Pr}(X=x_i)=\mathsf{Pr}(X>x_{i-1})-\mathsf{Pr}(X>x_i)=S(x_{i-1})-S(x_i)$ +4252,$\mathsf{E}[WX] \le \rho(X)$ +4253,$r_U \Delta A - \Delta P$ +4254,"$\bar Q_{0,0}$" +4255,$s_0=0$ +4256,$g(S_{\mathsf{j}(a)})=0.5$ +4257,$-g''(s)=\alpha(\alpha-1)s^{\alpha-2}$ +4258,$\bar Q(a) =a-\bar P_g(a)$ +4259,$\exp(a)$ +4260,$s\mapsto g(s)$ +4261,$\alpha X$ +4262,$\mathsf{E}[XM]$ +4263,$c(S)\le c(T)$ +4264,$(1-\lambda)(1+\gamma)$ +4265,$\mathsf{E}[X] = \displaystyle\int_\Omega X(\omega)\mathsf{Pr}(d\omega)$ +4266,$1-\beta_i(t)g(S(t))$ +4267,$\mathsf{Pr}(X>x)$ +4268,$\mathsf{E}[X\mid X>2000]-2000=\mathsf{TVaR}_{F(2000)}(X)-2000=624$ +4269,"$(p, \mathsf{E}[X_i\mid X=q(p)])$" +4270,$L_a^{a+da}$ +4271,"$a_{0,t}' := a_{0,t-1}-X_{0,t}$" +4272,$-Y$ +4273,$P = \mathsf{E}[X] + \pi \mathsf{Var}(X)$ +4274,$\rho(X)=\mathsf{E}_\mathsf{Q}[X]-\alpha(\mathsf Q)$ +4275,$W_{t}$ +4276,$2^n$ +4277,$(1 - \nu F(a))$ +4278,$<$ +4279,$g'\left (S_X(X)\right )$ +4280,"$X_1=(0,0,0,0,0,0,2,4,8,0)$" +4281,$R_L=R_f + \beta_L(R_M-R_f)$ +4282,$cv=0.137$ +4283,$\mathsf{E}[(X-x)^+]$ +4284,$\mathsf{Pr}(X>a)$ +4285,"$(2,2)$" +4286,$1/x$ +4287,$A(1_{X_1>x_1}+1_{X_2>x_2}) \le A(1_{X_1>x_1}) + A(1_{X_2>x_2})$ +4288,$\Delta=\Phi(d^*)$ +4289,$F_g(x) = 1- g(S_X(x))$ +4290,$Z(1000)=(1-0)/(0.1-0)=10$ +4291,$\tilde X_1=X_1 + \mathsf{E}[X_2\mid X_1]$ +4292,$\tau=1$ +4293,$\rho(X_1) \ge D\rho_X(X_1)$ +4294,"$\mathcal Q =\{ \mathsf Q \mid \mathsf Q\ll \mathsf P,\ \alpha(\mathsf Q)=0 \}$" +4295,"$X_j=\sum_i X_{i,j}$" +4296,$\mathsf{SD}$ +4297,$n>1$ +4298,$-\phi(d^*)<0$ +4299,$\rho(\tilde X+X)=\rho(\tilde X)+\rho(X)$ +4300,$a(\mathbf{v})$ +4301,$P(dx)$ +4302,$\mathsf Q(X>a)/P_X(X>a)=g(S(a))/S(a)$ +4303,$\rho(X-P)=\rho(X)-P$ +4304,$a+da$ +4305,$r_pq$ +4306,"$m\ge 1, n\ge 0$" +4307,$(1-g)$ +4308,$x=2$ +4309,$T_k$ +4310,$X\le c$ +4311,$X \succeq Y$ +4312,$t$ +4313,$x=a$ +4314,"$\mathsf{biTVaR}_{p_0,p_1}^w(X)=\bar P$" +4315,$\log(X)$ +4316,$\mathsf{E}$ +4317,$\mathsf{P}(X=X_j)$ +4318,"$[0, 1-p)$" +4319,$\mathsf{VaR}_{0.95}(X)$ +4320,$S_t=a_0 + (1+c)\mu t - X_t$ +4321,$1/(1+r) = 0.893$ +4322,$D^n\rho_X(X_i)$ +4323,$A\subset \mathbb{R}$ +4324,$\mathsf{E}[X_1]=\mathsf{E}[Y_{0}]$ +4325,$\bar P_t = \rho(Y_{t})$ +4326,$W_1$ +4327,$s/(1-s)$ +4328,$E_1$ +4329,"$f:[0,1]\to[0,1]$" +4330,$A\cap B$ +4331,"$(p,q(1-g^{-1}(1-p)))=(1-g(S(x)),x)$" +4332,$X_1=0$ +4333,"$\beta = \mathsf{cov}[r,r_M]/ \sigma^2_{r_M}$" +4334,"$f(x, \cdot)\in L_p(\Omega, \mathcal{F}, \mathcal{P})$" +4335,$\bar P(a)=\rho_g(L_0^a(X))$ +4336,$S_t=\exp(\mu t + \sigma W_t)$ +4337,$P_i(x)=\beta_i(x)g(S(x))$ +4338,$1-L/P = (P-L)/P$ +4339,"$x_{1,2}$" +4340,$\mathsf{Q}\in\mathcal Q$ +4341,$w/(1-w)$ +4342,$\sup_\Omega |X_n - X| \to 0$ +4343,$\beta_i(a) g(S(a))$ +4344,$\prec_n^*$ +4345,$2^{-t}$ +4346,$\mathsf{Pr}(p(\omega)=0)=0$ +4347,$X_2/X$ +4348,$\Delta X=80-11=69$ +4349,$g(S(x))=\exp(-\alpha H(x))$ +4350,$\rho(\lambda X + (1-\lambda)\rho(X))$ +4351,$X\le Y+\Vert X-Y\Vert$ +4352,$(1-p)/(p\nu(p)^2)$ +4353,$\mathsf{E}[X_2\mid X_1]$ +4354,$I(F(x) < p)=\begin{cases} 1 & F(x)< p \\ 0 & F(x)\ge p\end{cases}$ +4355,$g(s)=1$ +4356,$\mathsf{Pr}(X = q(p)) > 0$ +4357,$x < x^\ast$ +4358,$X\wedge a(X)\le Y\wedge a(Y)$ +4359,$t\uparrow 0$ +4360,"$\eta_{p,\alpha_1}(X) < \eta_{p,\alpha_2}(X)$" +4361,$\mathsf{Pr}(\{\omega \mid X_n(\omega)\to X(\omega) \})=1$ +4362,$a_{\min}$ +4363,$\phi(t)=\int_0^t (1-p)^{-1}\mu(dp)$ +4364,$\mathsf{E}_\mathsf{Q_r}[X_j]$ +4365,$\mathsf{EPD}_s(X)$ +4366,$\mathsf{E} X + c{X-\mathsf{E} X}_p$ +4367,$x\ge a$ +4368,$N=r_a$ +4369,$\int S(x)dx = \int xdF(x)$ +4370,$\mathsf{VaR}_{0.7}(X)=$ +4371,$\mathsf P(X=\sup(X))>0$ +4372,$L(X)=(1-p)^{-1}1_{X\ge x_p}(X)$ +4373,$\mu = w \delta_{\alpha_1} + (1-w) \delta_{\alpha_2}$ +4374,$A(-X)$ +4375,$\mathsf{j}(90)=6$ +4376,$a - P$ +4377,$q^-_X(0.95)$ +4378,$1100 \le x \le 1250$ +4379,$\sigma\sqrt{t}$ +4380,$(L-A)^+$ +4381,"$\int Zd\mathsf P = \int d\mathsf Q/d\mathsf P\, d\mathsf P = \int d\mathsf Q =1$" +4382,$\mathcal A$ +4383,$d\mathsf Q/d\mathsf P$ +4384,$\rho(X) = \mathsf{E}_{\mathsf{Q}}[X] = \mathsf{E}_{\mathsf{Q}}[X\wedge a + (X-a)^+] = \mathsf{E}_{\mathsf{Q}}[X\wedge a] + \mathsf{E}_{\mathsf{Q}}[(X-a)^+] \le \rho(X\wedge a) + \rho((X-a)^+) = \rho(X)$ +4385,$t_f$ +4386,$\hat q(p)$ +4387,$\mathbf {a_{2}}'$ +4388,$p<1$ +4389,$r < n$ +4390,$\mathcal S(X)=\mathsf{VaR}_p(X)$ +4391,"$(\omega',\omega'')$" +4392,"$u\in[0, 1-p]$" +4393,$k_i=a_i/v_i$ +4394,"$(s_j, g_j)$" +4395,"$(3,3)$" +4396,$T(p)$ +4397,$D/C$ +4398,"$s\in[0, 1-p]$" +4399,$p=1$ +4400,$a_{d}' = a_{d-1}-X_{d}$ +4401,$g'\left (S(X)\right )$ +4402,$p=0.75$ +4403,$\mathbf {X_{1}/X}$ +4404,$\bar M(a) = \bar P(a) - \mathsf{E}[X\wedge a]$ +4405,$\{\omega\mid X(\omega) = x_1\}$ +4406,$\tau a_i$ +4407,$d\downarrow 0$ +4408,$p\ge 1$ +4409,"$g(s)=\min(s/(1-p),1)$" +4410,$\mathit{ROE}(s) = r_f + Ck(s)$ +4411,$\phi'(p)\ge 0$ +4412,$q(p)\phi(p)$ +4413,$\mu_0=\mu_1$ +4414,$w_l=1-c\gamma$ +4415,$0.7 \le p < 0.8$ +4416,$\int_\Omega X(\omega)\mathsf \mathsf{Pr}(d\omega)$ +4417,"$j=1,\dots, n$" +4418,$x=q_X(1-s)=\mathsf{VaR}_{1-s}(X)$ +4419,$\{p \ge p_-\}$ +4420,$(1-p)^{-1}$ +4421,$\alpha_i(x) = \mathsf{E}[X_i /X \mid X> x]\not=\mathsf{E}[X_i\mid X> x]/\mathsf{E}[X\mid X>x]$ +4422,$\omega<1/n$ +4423,$\tilde X = (x_{ij})$ +4424,$\mathbf {\mathsf{VaR}_p(X_1+X_2)}$ +4425,$q(p)=c$ +4426,$a(X_i;X)\le \rho(X_i)$ +4427,$\rho(0) \ge 0$ +4428,$g'(s-)\ge 0$ +4429,$\mathbf {\max a}$ +4430,$\exists$ +4431,$^1$ +4432,$x^{-\alpha}$ +4433,$k>1$ +4434,$D\rho_X(X_2)=45.1801$ +4435,$\mathsf{E}_\mathsf{Q_2}[X_j]$ +4436,$\mathsf{E}[X_i (X\wedge a)/X]$ +4437,$V(2)$ +4438,"$\rho_g(X)=\int_0^\infty g(S(x))\,dx$" +4439,$c(1)$ +4440,$\mathsf{E}[(X-x_l)^+]$ +4441,$X=X_{-1}+X_{0}$ +4442,$\mathsf{E}[Z_A]=1$ +4443,$xf(x)dx$ +4444,$t=0.06405$ +4445,$Y_{0}=\sum_{d>0} X_{d}$ +4446,$a=Q+P$ +4447,$Y\preceq Z$ +4448,"$a_{0,0}:=a(Y_{0,0})$" +4449,$X_1+X_2\sim 2X$ +4450,$l$ +4451,$r_h=r+\pi$ +4452,$\Delta Q_{ro}(a)$ +4453,$\bar Q_{0}$ +4454,$=1-\nu F(a)$ +4455,$X \le 0$ +4456,$X^{-1}(A)\in\mathcal F$ +4457,$\sup(X\wedge a)=a$ +4458,$\mathbf{P_i} \in \mathbb{R}^2$ +4459,$\mathsf{E}[\kappa_i(X)g'(S(X))]$ +4460,$\mathsf{E}[X_i(x)]$ +4461,$\bar\nu=1/(1+\bar\iota)$ +4462,$\rho(X/n)=\rho(n(X/n))/n=\rho(X)/n$ +4463,$1_{U_X\ge p}=0$ +4464,$\mathsf x\mathsf{TVaR}$ +4465,$F(X) - F_X(X-)=0$ +4466,$\tilde X$ +4467,$m'(0) = (m_1-m_0)/s_1$ +4468,$B \in\mathcal B_p$ +4469,$1/16$ +4470,$\tilde X_1 = X_1 + \mathsf{E}[X_2]$ +4471,$\bar\iota(a)=\bar\iota$ +4472,$\mathsf{TVaR}_p(X)-\mathsf{E}[X]$ +4473,$\mathsf P(X=X(\omega_0))>0$ +4474,$X=X_i + (X-X_i)$ +4475,$X(\omega)\mathsf{Pr}(\omega)$ +4476,$\rho(X_1+X_2) \le \rho(X_1)+\rho(X_2)$ +4477,$(0)$ +4478,$\mu_i$ +4479,$\mathsf{E}[kX]$ +4480,$\mathsf{VaR}_1=\esssup$ +4481,$#4$ +4482,$v=x$ +4483,$\phi(p)=g'(1-p)\ge 0$ +4484,"$(0,0)$" +4485,$s_2$ +4486,$\mathbf {a_1'}$ +4487,$F(x) < p \iff q^-(p) > x$ +4488,"$\mathbf {g(S)\,\Delta X}$" +4489,"$g(S)\,\Delta X$" +4490,$E'$ +4491,$\delta+\nu$ +4492,$\mathsf P(X\ge x_p)=1-p$ +4493,$\mu=7.8044$ +4494,$a_{2}'$ +4495,$p>0$ +4496,$z$ +4497,$\mathsf{j}(91)=7$ +4498,$\zeta_s = 8$ +4499,$ag(0+)$ +4500,$\rho-\iota g>0$ +4501,$R = P-L$ +4502,$\mathrm{sgn}(z)|z|^{1/(q-1)}/\|z\|_p^{q/p}$ +4503,$o(dt)$ +4504,$q^-$ +4505,$A_4 = [0; \epsilon_1 + \epsilon_2]$ +4506,$\mathsf{E}[q(U_X)1_{U_X\ge p}]$ +4507,$\mathsf{Var}(Y) \ge \mathsf{Var}(X)$ +4508,$0\le \omega\le 1$ +4509,$q(p)=e^{\mu+z_p\sigma}$ +4510,"$[f'_-(x_0), f'_+(x_0)]$" +4511,$p/\mathsf{E}[p]=p(1+r_f)$ +4512,$a(\cdot)$ +4513,$\mathsf{E}[X_i \mid X]$ +4514,$L_p$ +4515,"$X\ge 0,(\tilde X-X)\ge 0$" +4516,$\rho(\lambda X)$ +4517,$\mathbf {j}$ +4518,$Pr(X_{-1} > a)$ +4519,$X\wedge a / X$ +4520,$\tau=-1$ +4521,$\mathsf{TVaR}_{p^*}$ +4522,$X\ge x_p$ +4523,$A(1_{U>0.95})=A(1_{U\le 0.05})=g(0.05)=0.3017$ +4524,$\mathsf{TVaR}_{0.9}$ +4525,$2.576\sigma_d$ +4526,$\mathbf {Z_7}$ +4527,$\mathsf{TVaR}_1$ +4528,$\mathsf{E}[X\mid X>x]/\mathsf{Pr}(X>x)$ +4529,$X_1=\mathsf{E}[X\mid \mathcal F_1]$ +4530,$\bar P_n$ +4531,$\mathit{MV}_{gc}(a_{gc})=a_{gc}-\rho(X\wedge a_{gc})={{mv_gc}}$ +4532,$\mathsf{Pr}(X_n\in A)=1$ +4533,$1\wedge \cdot$ +4534,"$g(s)= \displaystyle\int_0^s \phi(1-p)dp = \min(s/(1-\alpha), 1)$" +4535,$\ll$ +4536,$0\le \alpha \le 1$ +4537,$\bar P=\mathsf{E}[W]+\lambda\sigma(W)$ +4538,$j=8$ +4539,$\rho(X)=\mathsf{E}[Xe^{kX}]/\mathsf{E}[e^{kX}]$ +4540,$\int_{\mathsf{E}[X]}^\infty (x-\mathsf{E}[X])^2 f(x)dx$ +4541,$S_0=1-p_0$ +4542,$\mathsf{E}[\cdot]$ +4543,$\mathbf {Z_\mathit{lift}}$ +4544,$g_{ROC}$ +4545,$\rho_1(X)$ +4546,$f(s) \ge s$ +4547,$Q(a)=h(F(a))$ +4548,$P = \mathsf{E}[X] + \pi \mathsf{E}[((X-\mathsf{E}[X])^+)^p]^{1/p}$ +4549,$\mathbf {a=1}$ +4550,"$\bar P_i(v_1, v_2, a) / v_i$" +4551,$q_C\le q_A$ +4552,$. Thus $ +4553,$k\mapsto k\rho(-X)$ +4554,$\mathsf{TVaR}_1=\sup$ +4555,$\lambda = \dfrac{E( r_{M} ) - r_{f}}{\sigma_{rM}}$ +4556,$g'(1)<1$ +4557,$u'''' \le 0$ +4558,$\mathbf {g_3(s)=s^{0.7}}$ +4559,$-X_i$ +4560,$ROE=(g-s)/(1-g)=m/(1-s-m)$ +4561,$X > a$ +4562,"$f(0,0)=0$" +4563,$\mathsf{Var}$ +4564,$l(kX)\le\rho(kX)$ +4565,$\lambda \ge 0$ +4566,"$0, 1/p$" +4567,$X\ge m$ +4568,$E(X_{0}(a))$ +4569,"$(0,1)$" +4570,$i=1\dots N$ +4571,$-(1-s)g''(1-s) + g(0+)\delta_1 + \sum_s s\Delta_s \delta_{1-s} + g'(1)\delta_0$ +4572,"$\mathsf{E}[X_{t,d}\mid \mathcal F_0]=\mathsf{E}[X_{t_d}]$" +4573,$\phi(0)=\mu(\{0\})$ +4574,$X_1=X_2=10$ +4575,$80=9.56 + 70.44$ +4576,"$\kappa_{i}(x) = \dfrac{\sum_{j:X_{j} = x} X_{i,j} p_j}{\sum_{j:X_{j} = x}p_j}$" +4577,$S(p)=1-p$ +4578,$x=q(\hat p)$ +4579,$g(s)\le 1$ +4580,$N\times d$ +4581,$X=a$ +4582,$P_{g}$ +4583,$x=q_{\mathbf{v}}(s)$ +4584,$dW_t$ +4585,$a_x=2$ +4586,$f(x)=\exp(-x/\mu)/\mu$ +4587,$\bar M_i(a)$ +4588,$Z\in \mathcal Q$ +4589,$U=4$ +4590,$f(x)=e^x$ +4591,$X_{-1}=C_1 + \cdots + C_N$ +4592,$M_i(x)+Q_i(x)$ +4593,$V=1_{X\le x^\ast}$ +4594,$\bar Q_{2}$ +4595,$\bar P_g(a)=\rho(X\wedge a)$ +4596,$g''(s)=0$ +4597,$\mathsf x\mathsf{VaR}_p(X):=\mathsf{VaR}_p(X)-\mathsf{E}[X]$ +4598,$K=3$ +4599,$a\mathsf{E}_{\mathsf{Q}}[...]$ +4600,$g(s)=\sqrt s$ +4601,"$\bar P_{0,0}$" +4602,"$(x,-x)$" +4603,$n=9$ +4604,$\hat q(p)=q(1-g^{-1}(1-p))$ +4605,$A(0)=0$ +4606,$\rho(X)\le\liminf \rho(X_n)$ +4607,$c$ +4608,$p^*=48.25/71=0.6796$ +4609,$\mathsf{E}[X]+k\mathsf{Var}(X)$ +4610,$d\tilde p=g'(1-p)dp=\phi(p)dp$ +4611,$BC$ +4612,$1 in a layer with loss probability $ +4613,$d=1$ +4614,$s/g(s)\le 1$ +4615,$1/\lambda$ +4616,$1-\alpha_i(x)S(x)$ +4617,$Z=Z_X$ +4618,$E[Z]$ +4619,$\rho(\tilde X_1)=\rho(X_1)+\rho(\mathsf{E}[X_2\mid X_1])$ +4620,$\sum_i X_i(a)=X\wedge a$ +4621,$\mathsf{P}(\{X\in A\})$ +4622,$M(x)/Q(x)$ +4623,$d\omega$ +4624,$\mathcal{Q}$ +4625,"$(x_B, g(S(x_B-))$" +4626,$X=q(U)$ +4627,$q_A(p) = \sup A$ +4628,$\lambda > 1$ +4629,$a \in \mathbb{A}$ +4630,$y\le q_C(p)$ +4631,$\rho(kX)$ +4632,$u=ug(1)=ug(1)+(1-u)g(0) \le g(u)$ +4633,$\Delta Q_{ro}(a) = a-a_{ro}$ +4634,$x_A=\partial x/\partial A$ +4635,$\mathsf{TVaR}_0$ +4636,$\lambda=0.73$ +4637,$Q^* > S$ +4638,$c\le 1$ +4639,$\omega=1$ +4640,$\tau=0.03$ +4641,"$\mathbf {S\,\Delta X}$" +4642,$p<0.9$ +4643,"$\beta, \kappa$" +4644,$a=a_0+(1+c)\mu$ +4645,$f_{\mathbf{v}}$ +4646,$(d\mathsf{Q}/dP)(x) = (1-p)^{-1}1_{x >\mathsf{VaR}_p(X)}$ +4647,$\frac{1}{1-p}\int_{1-p}^q \mathsf{VaR}_s(X)ds$ +4648,"$\bar L, \bar P, \bar M$" +4649,$\mathsf{E}(X)=$ +4650,$\nu$ +4651,$\tau$ +4652,$x_l < x=\mathsf{VaR}$ +4653,$0\le p < 1$ +4654,$Z\mid X$ +4655,"$X:\Omega\to[0,\infty)\subset \mathbb R$" +4656,"$[a, a+da]$" +4657,$f>0$ +4658,$S(x-)=0.1$ +4659,$\rho(X)\le b$ +4660,$s=0.45$ +4661,$(1-p)$ +4662,$Z(X(\omega))$ +4663,$\mathsf{E}[X_i]$ +4664,$\mathit{MV}_{ro}(a) = a-P(X_{-1}\wedge a)$ +4665,$9+1=10+0=10$ +4666,$\mathbf {\mathsf{VaR}_p(X_1)}$ +4667,$g \circ S$ +4668,$1+2c(1-\mathsf{Pr}(Z>\mathsf{E} Z))$ +4669,$\mathsf{P}(\{\omega_i\})=1/4$ +4670,"$\bar S_i(\mathbf{v}, a) := \mathsf{E}[X_i(\mathbf{v}, a)]$" +4671,$r_f>0$ +4672,$\sum_\omega Z(\omega)\mathsf{P}(\omega)=\mathsf{E}[Z]$ +4673,$X\wedge a$ +4674,$NT$ +4675,$p\ge p_0$ +4676,$-\rho(-H)=\rho(H)$ +4677,$\mathbf {X'\Delta S}$ +4678,$L_X(X)=\rho(X)$ +4679,"$\{(s_j, g_j)\} \cup \{(0,0), (1,1)\}$" +4680,$\displaystyle\int_0^\infty xdF(x)$ +4681,$g = s^{0.4}$ +4682,"$0.06 \times (64,861 - 7,500)=3,442$" +4683,$a_1'=a_0-X_{1}$ +4684,$N(t)$ +4685,$\rho(X)=\mathsf{E}_\mathsf{Q}[X]=\mathsf{E}[XZ]$ +4686,$v=S$ +4687,$\mathsf{VaR}_p(X)$ +4688,$\mathsf{E}_{\mathsf Q}[Y] = \mathsf{E}[Yg'(S_X(X))]$ +4689,$u^{iv}<0$ +4690,$\lambda_1$ +4691,$X_1=c_1-Y/2$ +4692,$\alpha < 1$ +4693,$Y+W$ +4694,$\bar q(s)=q(1-s)$ +4695,$P=\mathsf{E}[X]$ +4696,$L-f(L)$ +4697,$X=MX_2$ +4698,$a=a(X)$ +4699,$\alpha_i(a)$ +4700,$\bar\iota : 1$ +4701,$a < kP$ +4702,$\rho(X)=\sum_n X(n)\mathsf{P}(n)$ +4703,$X_i/X$ +4704,$\partial a/\partial v_i$ +4705,$U(-X)\ge U(-Y)$ +4706,$\rho(X)\le \lim\rho(X_n)$ +4707,$wq_Y(p)+(1-w)q_Z(p)$ +4708,$\mathsf{var}(\sum C'_i)=v_{res}^2 \sum c_i^2$ +4709,$p^-$ +4710,$h(0)=0$ +4711,$0\le p^\ast\le 1$ +4712,$\alpha\ge A(n)=\sum_s n_s(1-g(s))$ +4713,$af=1$ +4714,$N=n$ +4715,$q=1-p$ +4716,"$\{x_1,\dots,x_N\}$" +4717,"$(0,0,0,0,0,0,0,0,5,5)$" +4718,$X_n(\omega)=0$ +4719,$1-s_j$ +4720,$c(1)-c(\mathsf{var}nothing)=c(1)$ +4721,$\mathsf{TVaR}_p(X)-\mathsf{VaR}_p(X)=\sigma(\phi(\Phi^{-1}(p))/(1-p) - \Phi^{-1}(p))\to 1$ +4722,$\alpha_j'(x)<0$ +4723,$\mathsf{E}[X_1h(X)]$ +4724,$P=D$ +4725,$f(w) = \exp(-w)$ +4726,$1+r^*=(1+r)(1+\tau)$ +4727,$a=P+Q$ +4728,$X\wedge 10$ +4729,$\mathsf{E}_{\mathsf{Q}}[Y\mid X]\mathsf{E}[Z\mid X] = \mathsf{E}[YZ \mid X]$ +4730,"$u\in[0,1]$" +4731,$L_0^l(X)$ +4732,$j=1$ +4733,$g(s)=\mathsf{TVaR}_{.99}$ +4734,$m+1$ +4735,$\rho_h(X):=\mathsf{E}[X_h]$ +4736,$S(x):=\mathsf{P}(X>x)$ +4737,$9.67$ +4738,$\|\cdot \|_\rho=\rho(|\cdot |)$ +4739,$L^*$ +4740,"$(x_{2,1}, x_{2,2})$" +4741,"$(x,y)$" +4742,$p>1$ +4743,$\mathsf{VaR}_1$ +4744,$p=\Phi^{-1}(4)=3.17\times 10^{-5}$ +4745,$g(s)=s^a$ +4746,$X_i\Delta g(S)$ +4747,$x'$ +4748,$\mathsf{E}[g'(S(X))]=\int_0^\infty g'(S(x))dF(x)=\int_0^\infty -\frac{d}{dx}g(S(x))dx=g(S(0))-g(S(\infty))=g(1)-g(0)=1$ +4749,$\rho_g(X)=51.156$ +4750,"$(s,g(s))=(0.2, 0.36)$" +4751,$\delta^{\star}$ +4752,$\mathsf Q^t\cdot X$ +4753,$\mathsf{Pr}(\Omega)=1$ +4754,$s(0)=s_0=0$ +4755,$dS=-f(x)dx$ +4756,$1_{\{X>x\}}$ +4757,$\ge x$ +4758,$g'(1-p)$ +4759,$Z(x)$ +4760,$0.495(r-i)$ +4761,$\tau(a-\bar P_\tau(a))$ +4762,"$\bar Q_{0,2}$" +4763,"$u'>0, u''>0$" +4764,$Y(\omega)=0$ +4765,$g(S(x))=s$ +4766,$P/S$ +4767,"$p\in[0,1]$" +4768,$X=F^{-1}(U)$ +4769,$>1$ +4770,$r\times m$ +4771,$\mathsf{E}_{\mathsf{Q}}[(X-a)^+] \le \rho((X-a)^+)$ +4772,$\mathbf {D^n\rho_{X\wedge 30}(X_1)}$ +4773,$s=0$ +4774,$\hat q(p)=x$ +4775,$\mathscr{O}(f)$ +4776,"$1/2,1/4,1/4$" +4777,$n-5$ +4778,$q(1-g^{-1}(1-p))/q(p)$ +4779,$Z-X$ +4780,$s>0$ +4781,$\mathsf{E}_\mathsf{Q}[X_i \mid X=x]=\mathsf{E}[X_i g'(S(X))1_{\{X=x\}}] / \mathsf{E}[g'(S(X))1_{\{X=x\}}] = \mathsf{E}[X_i1_{\{X=x\}}]/\mathsf{E}[1_{\{X=x\}}]=\mathsf{E}[X_i\mid X=x]$ +4782,"$\pmb{j, p, S, \kappa_1, \Delta X, \Delta(X\wedge a)}$" +4783,$S(x)$ +4784,$0\le x < 1/6$ +4785,$K = \mathsf{E}[\exp (\lambda x)]^{-1}$ +4786,$Y=\mathsf{E}[Z\mid\mathcal G]$ +4787,"$\omega=0,1,\dots, 99$" +4788,${}^2S(t)=\mathsf{E}[(X-t)_+]$ +4789,"$0,0,1,2,3,6,10,18,36,52$" +4790,$\mathbf {X_{n}}$ +4791,$t=0$ +4792,$p=0.791$ +4793,$\ge \mathsf{E}[X]$ +4794,$f(x+)$ +4795,$X_{2c}$ +4796,$\mathsf{E}_{QQ'}[X_i(a)] \ne \mathsf{E}_{QQ}[X_i(a)]$ +4797,$\mathcal S$ +4798,$\mathbf {M_{2}}$ +4799,$q_{\mathbf{v}}(p)=\mathsf{VaR}_p(X(\mathbf{v}))$ +4800,$p(\omega)$ +4801,$0\le p^*\le 1$ +4802,$r_N$ +4803,$\rho(X_g)-\rho(X_n)=51.1560-49.8986=1.2574$ +4804,$\sum_i x_i\mathsf{Pr}(X=x_i)$ +4805,"$\mathsf{Q}_2(A)=2\mathsf{P}(A\cap (0.5, 1])$" +4806,$S(x)=0$ +4807,$5/6$ +4808,$\bar S(a)=\mathsf{E}[X\wedge a]$ +4809,$s^\ast=1/2$ +4810,"$a_{0,t}:=a(Y_{0,t})$" +4811,$0\le p_0 \le p_1\le 1$ +4812,$0.2$ +4813,"$X_1,X$" +4814,$(1-r_0)\delta_1$ +4815,"$[0,1-p)$" +4816,$\mathcal Q_2$ +4817,$\mathsf{E}[X\mid \mathcal F_{t+1}]$ +4818,$\mathsf{Pr}(\mathsf{var}nothing)=0$ +4819,$\rho\mapsto a^\rho(\ \cdot\ ;\ \cdot\ )$ +4820,$\lambda\rho(X)$ +4821,$\mathcal D(X)=c\mathsf{Var}(X)$ +4822,$\le$ +4823,$u''' \ge 0$ +4824,$\rho(X\wedge a)$ +4825,$Y_2$ +4826,$X=X_1+...+X_n$ +4827,$\mathsf{E}[X\mid A]$ +4828,$g_j$ +4829,$\mathsf{E}[X] + \pi \mathsf{SD}(X)$ +4830,$1 < \alpha < 2$ +4831,$\Delta \mathit{MV}_{ro}(a)$ +4832,$\phi(s)$ +4833,$p\cdot X$ +4834,$\mathsf{E}[U]=\mathsf{E}[X]$ +4835,$\mathbf {Z_\mathit{lin}}$ +4836,$\beta_i(x)=\mathsf{E}_\mathsf{Q}\left[ \dfrac{X_i}{X}\mid X > x\right]$ +4837,$p_0 \le p^\ast \le p_1$ +4838,$D\rho(\cdot)$ +4839,$\lambda=0.25$ +4840,$u'''>0$ +4841,${}^nS^{-1}_X(q)\le {}^nS_Y(q)$ +4842,$S_X$ +4843,"$(\Omega, \mathcal{F})$" +4844,$S\subset\Omega$ +4845,$\hat q(p) > q(p)$ +4846,$\mathsf j(a)$ +4847,$X+c$ +4848,$S(y_j-)-S(y_j)$ +4849,$\mathbf {Z_3}$ +4850,$\Phi^{-1}(0.995)=2.576$ +4851,$g(p)/p$ +4852,$\alpha_2(99)=0.9$ +4853,$\alpha_iS\Delta X$ +4854,$L_0^a(X)=X\wedge a$ +4855,$\mathsf{E}[v^T] \ge v^{\mathsf{E}[T]}$ +4856,$g(s)=s^{0.7}$ +4857,${}^2S^{-1}(t)=q\mathsf{TVaR}_q(X)$ +4858,$\bar P_{0}=\rho(Y_{0})$ +4859,$\mathsf{E}_\mathsf{Q}[\mathsf{E}[X_i \mid X]]$ +4860,$g'(s)=\phi(1-s)\ge 0$ +4861,$\mathsf{MONO}$ +4862,$\bar M_i(a)>0$ +4863,$g(S(0-))=1$ +4864,$\rho(0)=\rho(0+0)=\rho(0)+\rho(0)$ +4865,$\rho(X_0)$ +4866,$\delta_p/\nu_p = \iota_p$ +4867,$\mathcal{Q}=\mathcal{M}$ +4868,$\rho \ge \mathsf{E}[X]$ +4869,"$d=1,\dots,N$" +4870,$x=0$ +4871,$\mathsf{j}$ +4872,$\mathsf{E}[X] + c\mathsf{E}[(X-\mathsf{E} X)_+^2]$ +4873,"$E_1,\dots,E_N$" +4874,$\mathsf{E}[Z\mid X]=0$ +4875,$(1-p)x_0$ +4876,$U\le p$ +4877,"$(x_1-\epsilon,x_1]$" +4878,$\sigma=0.15$ +4879,$pl(p)$ +4880,$g'(0)$ +4881,$P = \mathsf{VaR}_\pi(X)$ +4882,$C_i$ +4883,$x\mapsto (x-a)^+$ +4884,$\beta_L$ +4885,$D\rho_X(X_i)$ +4886,"$\alpha_1,\alpha_2$" +4887,$\{X>x\}$ +4888,"$x_{2,2}$" +4889,$w=1$ +4890,$\mathbf n$ +4891,$\mathsf{E}[Z\mid X]$ +4892,$F_2\prec_2 F_1$ +4893,$Z_8$ +4894,$T^{-1}(A)$ +4895,$\mathsf{TVaR}_{0.95}(Y)=0.8\mathsf{E}[X]=2000$ +4896,$g'(s) = rs^{r-1}$ +4897,$\mathsf{Q}\in\mathscr{M}$ +4898,$\mathsf{P}_X(A) :=\mathsf{Pr}(X\in A)$ +4899,$\rho(X)/2$ +4900,$ro$ +4901,$\alpha(\mathsf Q) < \infty$ +4902,$x_p$ +4903,$X\Delta S$ +4904,$S=e^{\mu t}$ +4905,$\Delta gS$ +4906,$s^{th}$ +4907,$\mathsf{E}[(X-\mu)^n]$ +4908,$X=X_0+Y$ +4909,$20$ +4910,$\mathsf{TI}$ +4911,$b-a$ +4912,$\sigma(Z)=\sqrt{\mathsf{var}(Z)}$ +4913,"$\tau(a-\rho_{a,\tau}(X))$" +4914,$\rho(X) < \infty$ +4915,$Y=c$ +4916,$\rho(W_0\wedge a_0)$ +4917,$g(0.01)=0.1$ +4918,$f(x)=(x-d)^+1_{\{x \le m \}}$ +4919,$X\circ T$ +4920,$\mathsf{E}[X\wedge 0]=0$ +4921,"$(\x*.75, -2)$" +4922,"$\mu=8.7, \sigma=2.5$" +4923,$p=0.05$ +4924,$\mathit{RV}$ +4925,$n=7$ +4926,$X_4=X_5=10$ +4927,$\mathsf{E}_{\mathsf Q}[.]$ +4928,$n\to \infty$ +4929,$i\not=j$ +4930,$H(x)$ +4931,$\mathsf{E}_{\mathsf Q}[X_i \mid X]$ +4932,$a\le \rho(X)\le b$ +4933,$EL$ +4934,$\alpha_i'(x)<0$ +4935,$q_Z$ +4936,$dp=f(x)dx$ +4937,$v_{res}\sqrt{(1+v^2)/n}\approx v_{res}v/\sqrt{n}$ +4938,$\rho(X_i)\le 0$ +4939,$s=s_1+s_2$ +4940,$\displaystyle\int_0^1 X(1-g^{-1}(1-\tilde p))d\tilde p$ +4941,$F(x_0)= p_+>p_0$ +4942,$g''(s)<0$ +4943,"$(s,m)$" +4944,$U = A = 8.149$ +4945,$P(a)da$ +4946,$B(p)$ +4947,$Q=a-P$ +4948,"$2^1, 2^3, ...$" +4949,$c(S)= \rho\left( \sum_{i\in S} X_i \right)$ +4950,$\partial f_{\bar x}/\partial x_i$ +4951,$\log(x)$ +4952,$L_d^{d+l}$ +4953,$\alpha(\mathsf Q)\not=0$ +4954,$X\le a$ +4955,$\kappa_i(x)/x$ +4956,$\bar P_{d}=\rho(Y_{d})$ +4957,$D^n\rho_X(X_2)=45.1838$ +4958,$f(x)=1$ +4959,$X_0+\epsilon Y$ +4960,$g_\tau$ +4961,$\phi'(s)ds$ +4962,$S\approx \mathsf{E}[X]$ +4963,$g'(1)>0$ +4964,$A=8.13$ +4965,$\mathbf {\kappa_1}$ +4966,$X_n$ +4967,$a=P+Q=EL+M+Q$ +4968,$\mathit{MV}_{ro}(a_{ro})$ +4969,$g(0-)f(\esssup(X))$ +4970,$S(x)=d/dx(\mathsf{E}[X \wedge x])$ +4971,$g_1$ +4972,$g'(1-p)=\nu$ +4973,$\mathbf {\rho(X\wedge a)}$ +4974,$X_1+X_2=X$ +4975,$|t|$ +4976,$\prec_n$ +4977,$P(X\wedge a)=\bar P(a)$ +4978,$\bar x$ +4979,$x_h>x=\mathsf{VaR}$ +4980,$1+\gamma$ +4981,$S/P$ +4982,$X_0$ +4983,$b_h$ +4984,$\mathsf P(X>a)>0$ +4985,$(1+\gamma)^{t-x}$ +4986,$n > 2$ +4987,$=\displaystyle\int_0^\infty x \P(\{X \in dx \})$ +4988,$\mathbf {a_2'}$ +4989,$\phi'(p)=f(p)/(1-p)\ge 0$ +4990,$\mathsf{VaR}_{0.98}$ +4991,$\sup X$ +4992,$h_f$ +4993,$\lambda>0$ +4994,${10\choose 5} = 252$ +4995,$T$ +4996,"$i,v$" +4997,$a_i':=\sum \alpha_i(1-S)\Delta (X\wedge a)$ +4998,$F:\mathbb{R}^n \to \mathcal{X}$ +4999,$u_i$ +5000,$N=40$ +5001,$Pr(X > a)$ +5002,$X_i(a')$ +5003,$t\mapsto s(t)$ +5004,$a_{1}$ +5005,$\int_0^1 f(p)dp = 1 - \alpha < 1$ +5006,$X=q(U_X)$ +5007,$t=w$ +5008,$E[X_2 | X]$ +5009,$B=\Omega$ +5010,"$1 million auto accident, a $" +5011,$3^{20}$ +5012,"$(-\x*0.75, -2)$" +5013,$\bar S_i(x)$ +5014,$dX$ +5015,$D\rho_X(X_1)$ +5016,"$\int_0^\infty z(x)\,dF(x)=1$" +5017,"$X_{t+1,1}$" +5018,$\log$ +5019,$(1-g(s))q$ +5020,"$(0,0,\dots,0,10)$" +5021,"$\iota, \iota(p)$" +5022,$\mathsf{Var}^+(X) = \int_{\mathsf{E}[X]}^\infty (x-\mathsf{E}[X])^2 f(x)dx$ +5023,$\mathsf{TVaR}_{0.5}(X_2)=45.5$ +5024,$t-2$ +5025,$Z_2$ +5026,$\prec_2$ +5027,$0\le x < X_1$ +5028,$\mathsf{E}_{\mathsf{Q}}[X] \le \rho(X)$ +5029,$a=\sum_i a_i$ +5030,$s<0.1$ +5031,"$a(x_1,x_2)=\sqrt{3x_1^2 + 4x_2^2}$" +5032,$\mathsf{E}[XZ_j] = (5)(1/10)(8)+(5)(1/10)(9)=8.5=\mathsf{TVaR}_{0.8}(X)$ +5033,$E[u_j(W_j - X_j + Y_j - H[Y_j])]$ +5034,$0\le \mathsf{Pr}(E)\le 1$ +5035,$1-p=0.9$ +5036,$h(s)=1-g(1-s)$ +5037,$(P-L)/A$ +5038,$X_1(10)$ +5039,$AR\succ BY$ +5040,$w_0$ +5041,$q_X\le q_Y$ +5042,$0 < \alpha\le 1$ +5043,"$\mathsf{biTVaR}_{0,1}^w$" +5044,"$a_i=\rho(X_i, p^*)$" +5045,$\mathsf{E}[1_{U_X\ge p}]=\mathsf{E}[B]$ +5046,$\phi(p)=g'(1-p)$ +5047,$1/(1+r)$ +5048,$\dfrac{1}{1+\iota} p$ +5049,$p(1-p)$ +5050,$\rho(X) = \int_0^\infty g(S(x))dx$ +5051,$\sum S\Delta(X\wedge a)$ +5052,$V^*$ +5053,$\partial a/\partial v_1$ +5054,"$A_1=[-k,-k]$" +5055,$p=0.25$ +5056,$a^{\star}(X)$ +5057,$0.8 \ge p < 0.9$ +5058,$\mathcal{G}$ +5059,$g'(s-)$ +5060,$k$ +5061,$\rho(X_n) \downarrow \rho(X)$ +5062,$q_X(U)$ +5063,$wq_X(p)+(1-w)q_Z(p)$ +5064,"$p\in [0,1]$" +5065,$g(s) \approx m_0+(1+m'(0))s$ +5066,"$Y_{0,0}:=\sum_{d>0} X_{0,d}$" +5067,"$Y_m=\max(X_1,\dots,X_m)$" +5068,$\mathsf{VaR}_{0.99}(X_1)=150$ +5069,$0.01$ +5070,"$t^\star \in [0,1]$" +5071,"$\{1,2,\dots, n\}$" +5072,$a < \max(X)$ +5073,$\mathcal N_X(X_i)$ +5074,$x^{\ast}:=\min(x)$ +5075,$0.5L_{250}^{500}(x)+0.75L_{500}^{750}+L_{750}^{1000}$ +5076,"$x_0, x_1, x_2$" +5077,$\sum (1-S)\Delta (X\wedge a)$ +5078,"$[0,\infty)\subset\mathbb{R}$" +5079,$\mathsf{Pr}(X=\mathsf{VaR}_p(X))=0$ +5080,$\mathsf{E}[(X-a)^+]/\mathsf{E}[X]$ +5081,$\bar Z = F(\bar x)$ +5082,$^2$ +5083,$q_{X}(p)=\sqrt{2}\Phi^{-1}(p)$ +5084,$a = a(\mathbf{v}) = a(X(\mathbf{v}))$ +5085,$s=1$ +5086,$S\cdot dX$ +5087,$s$ +5088,$\mathsf{E}[X\mid \mathcal F']$ +5089,$S(x)=u$ +5090,$\sup_{\omega\in\Omega} (f(\omega)+g(\omega)) \le \sup_{\omega\in\Omega} f(\omega) + \sup_{\omega\in\Omega} g(\omega)$ +5091,$\mathsf{E}_\mathsf{Q}[0]=0$ +5092,"$0,1,1,1,2,3, 4,8, 12, 25$" +5093,$\triangleright$ +5094,$\mathsf{TVaR}_p(X)=51.156$ +5095,"$A\subset [0, \infty)$" +5096,"$\Delta\,g(S)$" +5097,$f(x)\le f(y)$ +5098,$\rho(X) = \mathsf{E}[X] + \lambda \mathsf{E}[(X-\mathsf{E}[X])^+]$ +5099,$da$ +5100,$\mathsf{E}[X]=0.6$ +5101,$S=1$ +5102,$L_{250}^{\infty}$ +5103,$\mathcal S(X)=\mathsf{E}[X]$ +5104,$0 = x_0< x_1<\cdots < x_n < \cdots$ +5105,"$\nu p\,da=\nu F(a)\,da$" +5106,$X=\mathsf{E}[Y \mid \mathcal F']$ +5107,$\hat p:=1-g^{-1}(1-p)$ +5108,"$X(x_1, x_2)=(x_1+x_2)Y$" +5109,$1-F(x)=1-p$ +5110,$\mathcal F_t$ +5111,$\rho(X)=\rho(X-Y+Y)\le \rho(X-Y) + \rho(Y)$ +5112,$c \le 0$ +5113,$S(x_{(j)})(x_{(j+1)}-x_{(j)})$ +5114,$p=0.9$ +5115,$\rho(X+Y) \le \rho(X) + \rho(Y)$ +5116,$e^{X_t}$ +5117,$n\times r$ +5118,"$f'_\omega (\bar x, h)$" +5119,"$Y_{t,d+1}$" +5120,$F(b)-F(a)$ +5121,$\rho(X)\ge\rho(X+Y)\ge \rho(X)+\mathsf{E}[YZ]$ +5122,$a_{ro}:=\mathit{VaR}_{p}(X_{-1})=10743.5$ +5123,$\rho(X) - (-\rho(-X))=\rho(X)+\rho(-X)$ +5124,$Z_\epsilon$ +5125,$(\beta_i g(S))'(x)=-\mathsf{E}[X_i\mid X=x]g'(S(x))f(x)/x=-\kappa_i(x)g'(S(x))f(x) / x$ +5126,$\{3\}$ +5127,$\lim_{\epsilon \downarrow 0} (f(x-\epsilon)-f(x))/\epsilon$ +5128,$\max_{\mathsf{Q}} \mathsf{E}_\mathsf{Q}[0] -\alpha(\mathsf Q) =\max_{\mathsf{Q}} -\alpha(\mathsf Q)= -\min_{\mathsf{Q}} \alpha(\mathsf Q) = 0$ +5129,"$1,9,4,4,2,$" +5130,$g(S)$ +5131,$\mathsf{WCE}_p(X) := \sup\ \{ \mathsf{E}[X \mid A] \mid \mathsf{Pr}(A) > 1-p \}$ +5132,$\mathsf{TVaR}_p( X )$ +5133,$\mathsf{MON}'$ +5134,$\mathsf{TVaR}_{p_1}(X)$ +5135,$1_{X < q(1-s)}-(1-g)$ +5136,$g(x)=e^{2\pi i x\theta}$ +5137,"$\mathsf{E}[Y_{0,0}]+\lambda\sigma(Y_{0,0})=58.129$" +5138,$f=f(s)$ +5139,$l=a$ +5140,"$H(A, L, t)$" +5141,$\mathsf{TVaR}_{0.75}=4\left( \frac{90}{8}+\frac{98}{16}+\frac{100}{16}\right)=94.5$ +5142,$\mathit{NPV}$ +5143,$E_k$ +5144,$g(s)=s^\rho$ +5145,$X\ge 0$ +5146,$1.2\times 10^9$ +5147,$p=F(x)=\mathsf{Pr}(X\le x)$ +5148,$\mathsf{Pr}(X > \mathsf{VaR}_p(X))$ +5149,$f'(a)$ +5150,$y\in A$ +5151,$0 < \lambda \le 1$ +5152,"$\mathsf{cov}(X_i,X)/\sigma_X$" +5153,$s=S(x)=\mathsf{Pr}(X>x)$ +5154,$t_1$ +5155,$\lambda>1$ +5156,$g(S(x))=g(0)=0$ +5157,$D^n\rho_{X\wedge a}(X_i)$ +5158,$\mathsf{E}[X\mid \mathcal F_t](\omega)$ +5159,$\tau < t+d$ +5160,$s_2=1$ +5161,$\mathsf{E}[X_i\mid \{X=X(\omega)\}]$ +5162,$\mathsf j(a)=\max \{ j:X_j < a \}$ +5163,$g'(S(x))\ge 1$ +5164,$1-\tilde p=g(S(x))$ +5165,$F_m\succ_m F_0$ +5166,"$X_{t,d+1}$" +5167,$A(-X)=-B(X)\not=-A(X)$ +5168,$g=1$ +5169,$0.99$ +5170,$f_t$ +5171,$\mathsf{Var}^+(X)$ +5172,$E[YZ]$ +5173,$1-r_0$ +5174,$\lambda=0$ +5175,$\mathsf{E}[X_i\mid X]$ +5176,$\beta_2g-\alpha_2S$ +5177,$\rho(X)=\mathsf{E}[Xg'(S(X))]=\mathsf{E}[\sum_i X_i g'(S(X)))]=\sum_i \mathsf{E}[X_ig'(S(X))]$ +5178,$\mathsf{E}[X_i\wedge a_i]$ +5179,$x^*$ +5180,$\lambda t$ +5181,$\{X > \mathsf{VaR}_p(X)\}$ +5182,$r_f = 0.02$ +5183,$x=1$ +5184,"$[s_0, s_1]$" +5185,$(\beta g(S))'(x)=-\kappa_i(x)g'(S(x))f(x)/x$ +5186,"$a_{0,1}$" +5187,$X_{d}$ +5188,$q(p)=\inf\{x \mid F(x)\ge p \}$ +5189,"$([0,1], \mathcal B, \mathsf P)$" +5190,$\alpha(\mathsf{Q})=\infty$ +5191,$\rho_a(0) = \rho(0 \wedge a(0)) = \rho(0 \wedge 0) = \rho(0) = 0$ +5192,$\rho(X)\le \rho(\lambda X)/\lambda$ +5193,$c(\sum_{i\in S} X_i)$ +5194,$g(0)=0$ +5195,$\alpha_{1}$ +5196,$0 < b \le 1$ +5197,$pX + (1-p)Z$ +5198,$\pi(X)$ +5199,${}^nS^{-1}(q)$ +5200,$\sup X=\inf$ +5201,$Q^*$ +5202,$v-\nu^{\star}=(\iota^{\star}-i)/v\nu^{\star}$ +5203,$_{ro}$ +5204,$\iota=\delta/\nu$ +5205,$m'(1) = -m_2/(1-s_2)$ +5206,$D^n\rho_{X\wedge a}(\cdot)$ +5207,$P_i=\mathsf{E}_\mathsf{Q}[X_i]$ +5208,$(M-N)\times d$ +5209,$S(x_0)=1$ +5210,$\mathsf{E}_\mathsf{Q}[X1_A] / \mathsf{E}_\mathsf{Q}[1_A]$ +5211,$10/11$ +5212,$f(L)=(L-a)^+$ +5213,$\tilde Z_X:=\mathsf{E}[Z\mid X]$ +5214,$\mathsf{j}(a) = \max\{ j:X_j < a \}$ +5215,"$3.807=\lambda \sigma(W_{0,0})$" +5216,$h(x)=\sqrt x$ +5217,$ for $ +5218,$S(x-)=1$ +5219,$\{ Z\not=0 \}$ +5220,$\iota=(g(s)-s)/(1-g(s))$ +5221,$\tau=0$ +5222,$\mathsf{E}[X_iZ]$ +5223,$(r-i)Q_t$ +5224,$\delta p$ +5225,$\mathsf{TVaR}_p = q(p)$ +5226,$\mathsf{E}[Z]=g(1)-g(0)=1$ +5227,$\sigma=0.1980$ +5228,$X_1> x_1$ +5229,$\mathsf{E}[X_1\mid X=20]= 14$ +5230,$1/(1-p)>1$ +5231,$\mathbf {\mathcal Q}$ +5232,$\lambda\mathsf{E}[X]$ +5233,$q_V(p)=0$ +5234,$(1-s)^{-1/2}/4$ +5235,$g(0-)$ +5236,$(s+\iota) / (1+\iota)$ +5237,$k = 1.4 + 1.8s$ +5238,$\Psi(x)=1-\exp(-e^x)$ +5239,$=\displaystyle\int_0^\infty S(x)dx$ +5240,$dp$ +5241,$da\to 0$ +5242,"$(lee.east |- lee.north)+(0.25,0.25)$" +5243,$G$ +5244,$X'=0$ +5245,$\rho_g$ +5246,$\mathsf{E}[X_i/X\mid X>x]$ +5247,$s > 0.5$ +5248,$\rho=0.12$ +5249,$\beta_1g(S)dx$ +5250,$X(x)=x$ +5251,$g(S(x)) = S(x) + \delta(F(x))F(x)$ +5252,$L_X \in \mathcal L_\rho$ +5253,$g-S$ +5254,$x_0$ +5255,$0=\rho(0)$ +5256,$Xm1=X_{-1}$ +5257,$1-g^{-1}(1-p')$ +5258,$B(1_{U>0.95})=B(1_{U\le 0.05})=h(0.05)=1-g(1-0.95)=0.0203$ +5259,$\mathcal D(X)=\rho(X)-\mathsf{E}[X]$ +5260,$\phi(p)\ge 0$ +5261,$E(X_{-1}\wedge a)$ +5262,$n=8$ +5263,$R/Q$ +5264,$q < p$ +5265,$x=wy + (1-w)z$ +5266,"$B_3=[-k, \epsilon]$" +5267,$Q = 5.0449$ +5268,$n'=7$ +5269,$g'(t)>0$ +5270,"$j=0,\dots, N-1$" +5271,$0\ < p < 1$ +5272,$(S_t-a)^+$ +5273,$\alpha+\beta = \iota^\ast/(1+\iota^\ast)$ +5274,$\sin(x)$ +5275,$\mathbf{P_i}$ +5276,$a_{gc}:=\mathit{VaR}_{p}(X)={{a_x}}$ +5277,$\mathsf{VaR}_{0.995}$ +5278,$P(X_{-1}(a_{gc}))={{mvp_gc}}$ +5279,$\rho''(x)=-U''(x)>0$ +5280,$\{\omega\mid X(\omega)=x\}$ +5281,$\kappa$ +5282,$e$ +5283,$\omega'=\omega$ +5284,$0.3 < s <0.4$ +5285,$\mathbf {d=2}$ +5286,$g(s)=s^\alpha$ +5287,$X_1-X_2$ +5288,$\mathbf {g(S)\Delta X}$ +5289,$a = \sum_i a_i$ +5290,$\rho(X)=1$ +5291,$H(X)\le H(Y)$ +5292,$Y=X$ +5293,$\{\omega\in \Omega \mid (X\wedge a)=a \}$ +5294,$X\ge x_0$ +5295,$r=1$ +5296,"$\bar Q_{0,1}$" +5297,$Y\preceq_2 X$ +5298,$\rho(X)=k\mathsf{Var}(X)$ +5299,$\delta = \iota\nu$ +5300,$g'(1-s)=\phi(s)$ +5301,$q(U_X) < m$ +5302,$\alpha_1$ +5303,$A(X+Y)\le A(X)+A(Y)$ +5304,"$a_{0,t}' = a_{0,t}$" +5305,"$j=5,6$" +5306,$\mathsf Q_k$ +5307,$\lambda < 1$ +5308,$\mathcal E:=\{Y \circ T \mid T \text{ PPT} \}$ +5309,$Xp$ +5310,$F(x)=\mathsf{P}(\{X\le x\})$ +5311,"$(lee.east |- lee.south)+(0.375,-0.25)$" +5312,$p_j=\mathsf{P}(X=X_j)$ +5313,$dF=-dS=$ +5314,$m(s) := (1-s)\wedge m(s)$ +5315,$\mu_{rU} = M/K = 0.133$ +5316,$y \wedge (x-a)^+$ +5317,$\mathcal A=\{X\mid \rho(X)\le 0 \}$ +5318,"$Y_{0,0}$" +5319,$\bar P_{1}$ +5320,$\alpha_1+\alpha_2=\beta_1+\beta_2=1$ +5321,$a_l>b_l$ +5322,$X_0=0$ +5323,$\Delta Q_{gc}(a)$ +5324,$P_j=\sum_{i=0}^j p_i$ +5325,$\{y_j\}$ +5326,$X=3$ +5327,$\mathsf{Pr}(q^-(F(X))\not=X)=0$ +5328,$\rho(X)=\bar P$ +5329,$\alpha(\mathsf Q)\ge 0$ +5330,$a_l$ +5331,$A$ +5332,$v(AB) + v(ABCD) = 3/2 > v(ABC) + v(BCD) = 4/3$ +5333,$\sum p_jX_j$ +5334,$0.5+U/4$ +5335,$n=3$ +5336,$\bar\nu$ +5337,$p^*=1$ +5338,$r_K = \exp (\lambda) - 1$ +5339,$v(\mathsf{var}nothing) =0$ +5340,$n\mathsf{Pr}(Y > y_c)$ +5341,$x<1$ +5342,$a(X)=a(\sum_i X_i) = \sum_i a_i$ +5343,$P(X_{-1}(a))=\bar P^a_0$ +5344,$\kappa_{1}$ +5345,$\{\omega\in\Omega \mid X(\omega) \le x\}\in\mathcal F$ +5346,$\mathsf{TVaR}_{0.6975}$ +5347,$F(q^-(p))=p$ +5348,$\mathsf{E}[X]+\mathsf{var}(X)/\mathsf{E}[X]$ +5349,$B_2 \succ A_2$ +5350,$\hat{s}$ +5351,$\rho(X+\rho(X))=\rho(X)-\rho(X)=0$ +5352,$\mathsf{NORM}$ +5353,$Y\succeq X$ +5354,$\lim_{x\to\infty} xg(S(x))=0$ +5355,$\int xdF$ +5356,$\mathbf {M_1\Delta X}$ +5357,$t > 2/3$ +5358,$\mathsf{E}_\mathsf{Q}[X]=\mathsf{E}[XZ]$ +5359,$p=1-s_j$ +5360,$\mathsf{E}[\mathsf{E}[X_iZ\mid X]]\not=\mathsf{E}[\mathsf{E}[X_i\mid X]\mathsf{E}[Z\mid X]]$ +5361,"$d,v\ge 0$" +5362,$X_1\le X_2$ +5363,$r_D$ +5364,$x=\max(X)$ +5365,$c=0$ +5366,$1/\lambda = \sum_j 1/\lambda_j$ +5367,$>0$ +5368,$\rho_a(X)>2\rho_a(X_1)$ +5369,$Z(200)=0$ +5370,$A=\{X>x\}$ +5371,$n\ge 0$ +5372,$\bar P(a)\le a$ +5373,$\mathsf{Pr}(X < x)=p=\mathsf{Pr}(X\le x)$ +5374,$\displaystyle\int_0^\infty g(S(x))dx$ +5375,$M(x)$ +5376,$\mathbf {M\Delta X}$ +5377,$\rho(\tilde X)=\mathsf{E}_{\mathsf{Q}}[\tilde X]$ +5378,$\int_0^1 F^{-1}(p)dp$ +5379,$e_x=\sum_t {}_tp_{x}$ +5380,$g'\left (S_{X\wedge a}(X\wedge a)\right )$ +5381,$0 < g' \le 1$ +5382,$\mathit{NPV}_1$ +5383,$w(Z)/\mathsf{E}[w(Z)]$ +5384,$0.75+U/4$ +5385,$g_2$ +5386,$r_D=0$ +5387,$\displaystyle\int_\Omega X(\omega)\P(\omega)$ +5388,$p:=1-s$ +5389,$\bar\delta=\bar\iota\bar\nu$ +5390,$\rho(aX)=a\rho(X)$ +5391,$f(x-)$ +5392,$\mathsf{E}_\mathsf{Q}[X_i(a)]$ +5393,$A_i\cup A_i^c$ +5394,"$(s_0,g(s_0))$" +5395,$Q_0=0.25$ +5396,$3$ +5397,$X=\sum_t B_t/2^i$ +5398,$\iota(s)=(1-s)/(1-1)=\infty$ +5399,$Z_A=(1-p)^{-1}1_A$ +5400,$Q\circ T\in\mathcal{Q}$ +5401,$\mathsf{Pr}(B\le t) = 1/2 + 1_{t>1/2}(1/2)$ +5402,$\mathcal Q$ +5403,$\Delta X_j=X_{j+1}-X_j$ +5404,$w$ +5405,$t>\tau$ +5406,$1-g(S(t))$ +5407,$ to be the set of all sample points where the insurance event $ +5408,$1-1_{X>a}=1_{X\le a}$ +5409,$s=1-p$ +5410,$f(x)=x$ +5411,$s \approx 0$ +5412,$j=9$ +5413,$\mathsf{E}[Z(X)]=1$ +5414,$k\le m$ +5415,$\{\mathsf{E}_{\mathsf Q}[X_i] \mid \mathsf Q\in\mathcal Q(X)\}$ +5416,$\mathsf{E}[|X|]<\infty$ +5417,$\epsilon$ +5418,$\mathsf{E}[X_i (X\wedge a)/X \mid X=x] = \mathsf{E}[X_i\mid X=x] (x\wedge a)/x$ +5419,$\bar Q(a)=a-\bar P(a)$ +5420,$#2$ +5421,$\rho(X) = \mathcal{N}_{\tilde X}(X)$ +5422,$p$ +5423,$\mathbf {a=0.93}$ +5424,$3/4 \pm 1/4$ +5425,$10^{-2}$ +5426,$\mathsf{E}[X\wedge a] + d(a - \mathsf{E}[X\wedge a])$ +5427,$\mathcal B$ +5428,"$(\Omega,\mathcal F, \mathsf{P})$" +5429,$\epsilon>0$ +5430,"$g(s) = \nu s + \delta, s>0$" +5431,$X(\omega)\ge a'$ +5432,$\mathsf{E}[Xe^{hX}]/\mathsf{E}[e^{hX}]$ +5433,$r=0.025$ +5434,$\mathsf{E}[1_{X>a}]=\mathsf{P}(1_{X>a}=1)$ +5435,$\{X=q_X(p)\}$ +5436,$m$ +5437,$\mathcal F_0$ +5438,$L_0$ +5439,$m\le 4$ +5440,$\mathsf{TVaR}_1(X)=\sup(X)$ +5441,$\mathbf {d=0}$ +5442,$q(p)=\mathsf{VaR}_{p}(X)$ +5443,$\rho(X-Y)\le 0$ +5444,$P_{i}(a)$ +5445,$\rho(X)=\mathsf{TVaR}_p(X)$ +5446,"$\mathbf{v}=(v_1,v_2)$" +5447,$\kappa_i(t)=E[X_i \mid X=t]$ +5448,"$(s, g(s))$" +5449,"$(-1,1)$" +5450,$n\times 1$ +5451,$g'(S(x))<1$ +5452,$X_{1}$ +5453,$\rho(X)\le\lim \rho(X_n)$ +5454,$q^+(p) := \sup\ \{x \mid F(x) \le p \} = \inf\ \{ x \mid F(x) > p \}$ +5455,$M-N$ +5456,"$i=2,3,4,5$" +5457,$\mathsf{Pr}(X_n=Y)=\mathsf{Pr}(X=Y)=0$ +5458,$X_i(v_i)=v_iX_i(1)$ +5459,$X\le Y$ +5460,$S\Delta X'$ +5461,$t\mapsto \rho(X) + t\mathsf{E}_{\mathsf Q_X}[Y]$ +5462,$\rho(X\wedge a)=0.909$ +5463,$(1+\gamma)F_0$ +5464,$\sigma=\sqrt{s(1-s)/N}$ +5465,$\iota(s)$ +5466,$a-\bar P(a)$ +5467,$\mathbf {\mathsf{P}(X)=\Delta S}$ +5468,$F^{-1}$ +5469,$\rho(X) = \max_{\mathsf Q\in \mathcal Q} \ \mathsf{E}_\mathsf{Q}[X]$ +5470,$\mathbf {X'p}$ +5471,$\kappa_2(X)$ +5472,$U$ +5473,"$Y_{t,1}$" +5474,"$k=1,2,\dots,n-1$" +5475,$g(S(x-))=1$ +5476,$X_0 + \epsilon Y$ +5477,"$\displaystyle\int_0^a \kappa_i(x)g'(S(x))f(x)\,dx + a\beta_i(a)g(S(a))$" +5478,$m(s)$ +5479,$x_0 \ge q^-(p)$ +5480,$X(\mathbf{v}) = \sum_i X_i(v_i)$ +5481,$a=9532.0$ +5482,$L_{250}^{1000}(x)$ +5483,"$\sigma=13,108$" +5484,$\mathsf{E}[r] = \mu_r = M/K = 0.132$ +5485,$T_2 := ((n+1)-pN)x_n$ +5486,$\{ X>x \}$ +5487,$\rho(X)=\mathsf{E}[X] + c\sigma(X)$ +5488,$\iota = \dfrac{g(s)-s}{1-g(s)}$ +5489,$\mathsf{E}[|X_1|]<\infty$ +5490,$S_{\mathbf{v}}(t)=\text{Pr}(X({\mathbf{v}})>t)$ +5491,$g(s) = s^r$ +5492,$\Delta X$ +5493,$=$ +5494,$R^2$ +5495,$\mathsf{E}[X(1_{U_X\ge p}-B)]=\mathsf{E}[(X-m)(1_{U_X\ge p}-B)]\ge 0$ +5496,$S(x_4)$ +5497,$\kappa\ge K(n)=\sum_s n_s(1-g(s))k(s)$ +5498,$X+100$ +5499,"$\Omega=\{0,1,2,\dots \}$" +5500,$S_X(x) \ge S_{X_1}(x)$ +5501,$R(X)$ +5502,$g(S_6)\Delta X'_6$ +5503,$\rho(X-\rho(X))=0$ +5504,$\alpha_i(x) = \mathsf{E}[X_i /X \mid X> t]\not=\mathsf{E}[X_i\mid X> t]/\mathsf{E}[X\mid X>t]$ +5505,$P = \mathsf{E}[X] + \pi \mathsf{E}[(X-\mathsf{E}[X])^+]$ +5506,$g(0+) > 0$ +5507,"$X_i,X$" +5508,$p=0$ +5509,$r_h=\mu_L=0$ +5510,$g(0^+)>0$ +5511,$\mathrm{Pr}_{rn}\{P_{act}>P\}$ +5512,"$(I, \mathcal B, \mathsf P)$" +5513,$\mathbb{R}^3$ +5514,$ is not continuous and $ +5515,$\mathsf{Pr}(X=1)=s$ +5516,$E'=\Omega\setminus E\in\mathcal F$ +5517,$a_x$ +5518,"$\{1,2,\dots,10000\}$" +5519,$\Pi$ +5520,$ipl(p)$ +5521,$a'(x)=a(1)$ +5522,$-g''(t) = w \delta_{\alpha_1}/\alpha_1 + (1-w) \delta_{\alpha_2}/\alpha_2$ +5523,$a=\infty$ +5524,$\bar P_g$ +5525,$\sum_i a_i=\sum_i a(X_i;X)=\rho(X)$ +5526,$a^\rho$ +5527,"$(\Omega,\mathcal F,\mathsf{P})$" +5528,$p_{\mathit{cl}}$ +5529,$h(1)=1$ +5530,$\Delta_{1}$ +5531,$p^+=\mathsf P(X\le q_X(p))$ +5532,$\rho=\dfrac{M}{l} = \dfrac{1-\lambda}{\lambda}$ +5533,$\rho_1$ +5534,$S_{\mathbf{v}}(a)$ +5535,$^\circledR$ +5536,"$g(s)=\min(g_1(s), g_2(s))$" +5537,$a(X)=\mu+4\sigma$ +5538,$\mathsf{Pr}(X < x) \le 0.4 \le \mathsf{Pr}(X\le x)$ +5539,$pq$ +5540,$\rho(X+Y)=\rho(X) + \rho(Y)$ +5541,$g'=2/3$ +5542,$\mathsf{VaR}_p(X) = \mathsf{E}[X] + \pi(X)\mathsf{SD}(X)$ +5543,$X\wedge a\Delta g$ +5544,$P=80$ +5545,$\rho(-H)=\rho(C)-1=-0.05$ +5546,$i$ +5547,$S_i(x)=\alpha_i(x)S(x)$ +5548,$X'(\omega) \le Y'(\omega)$ +5549,$B_t(\omega)=\omega_t$ +5550,$S(x)/P(x)$ +5551,"$\int_0^s g'(t)\,dt=\nu s$" +5552,$L_X(v)=l(v)$ +5553,$\mu=\log(\theta)$ +5554,$\mathsf{E}[X] + \pi\mathsf{Var}^+(X)$ +5555,$T_{(1)}=W$ +5556,$t\in\mathbb{R}$ +5557,"$(x_{1,i}, x_{2,k(i)})$" +5558,$\rho_g(X\wedge a)=\bar P(a)$ +5559,$g'>0$ +5560,$X\wedge a = \sum_i X_i(a)$ +5561,$t=-\log(1-p)$ +5562,$\mathsf{E}[X]$ +5563,$S_X(y)$ +5564,$n=2^m+k$ +5565,$\mu t$ +5566,"$1/2, 1/4$" +5567,$\mathsf{CX}$ +5568,$\sigma^2 = \sum \sigma_i^2$ +5569,$\iota=M/Q$ +5570,$\sup_\mathsf{Q} (\mathsf{E}_\mathsf{Q}[X] - \alpha(Q))$ +5571,$AB$ +5572,"$\displaystyle\int_0^a \beta_i(x)g(S(x))\,dx$" +5573,$\bullet$ +5574,$366.4$ +5575,"$\tilde X:[0,\infty)\to[0,\infty)$" +5576,$1-\alpha_i(t)S(t)$ +5577,$F_1$ +5578,$a=\mathsf{VaR}_p$ +5579,$(a'-X)^+$ +5580,$\mathbf {X_3}$ +5581,$\mathsf{FSD}$ +5582,$a={{a_x}}$ +5583,"$(0.2, 0.304)$" +5584,$e^{\mu_A}-1$ +5585,$\mathsf{E}[(X-\mathsf{E} X)^+]={(X-\mathsf{E} X)^+}_1$ +5586,$-\rho(X-Y)\le \rho(Y)-\rho(X)$ +5587,$\mathsf{Pr}(X > a) \le \epsilon$ +5588,$\mathsf{E}[Z_A\mid X]$ +5589,$B^c_k$ +5590,$-$ +5591,$d+l$ +5592,$0.1005$ +5593,$\mathsf{E}[X\wedge a]$ +5594,$r_i$ +5595,$=v_f \mathsf{E}_\mathsf{Q}\left[\dfrac{X_i}{X}(X\wedge a)\right]$ +5596,$\bar\delta a$ +5597,$c > 1/2$ +5598,"$\mathsf{PML}_{n, \lambda}(X)=\mathsf{PML}_{n, \lambda}$" +5599,$f(x)$ +5600,$h(1-p)=1-g(p)=1-\sqrt{0.9}=0.051$ +5601,$\mathsf{E}[X_i\mid X](x)$ +5602,$\mathsf{P}(B)=0.5$ +5603,$Gn$ +5604,$\mathcal F$ +5605,$g_2(s)=\sqrt{s}$ +5606,$v_f=1/(1+r_f)$ +5607,$B\subset \Omega$ +5608,$\bar S(x)$ +5609,"$s_j,g_j\in[0,1]$" +5610,$\mu=21.315$ +5611,$a_{gc}=P(X_{-1}(a_{gc}))+P(X_{0}(a_{gc}))+\mathit{MV}_{gc}(a_{gc})$ +5612,$X0=X_{0}$ +5613,$X=(X\wedge a) + (X-a)^+$ +5614,$(\mathsf{TVaR}_p - q(p))/(1-p)$ +5615,$X \preceq_n Y$ +5616,$\lambda_i$ +5617,$\mathsf{Pr}(X_n>\epsilon)\to 0$ +5618,$\mathsf{VaR}_{0.95}(X)=3395$ +5619,"$W_2=\sum_{t+d=2} Y_{t,d}$" +5620,$\mathsf{E}[X\mid X\ge \mathsf{VaR}_p(X)]$ +5621,$a\ge \sup(X)$ +5622,$\mathsf{E}[\log(X)]$ +5623,$a=Q+R$ +5624,$p/q-1=(p-q)/q>0$ +5625,$\alpha_1\ge \beta_1$ +5626,"$c_1=(c(1) + c(1,2)-c(2))/2$" +5627,$Z\in D\rho(X_0)$ +5628,$\cdots$ +5629,$\mathsf{E}[Xw(X)]/\mathsf{E}[w(X)]$ +5630,$d\bar S(a)/da$ +5631,$\mathsf{P}(X=X_j)=S_{j-1}-S_j$ +5632,$\omega'=0$ +5633,$\rho(Y)=g(pq)$ +5634,"$\phi(s) = (1-p)^{-1}1_{[p, 1]}(s)$" +5635,$S(x_i-)-S(x_i) =\mathsf{Pr}(X=x_i)$ +5636,$dg/ds$ +5637,$T_1 := X_{n+1} + \cdots + X_{N-1}$ +5638,$\displaystyle\int_0^\infty xd(g\circ F)(x)$ +5639,$\mathsf{E}[X \mid X \ge q^+(p)]$ +5640,$\mathsf{POS\ LOAD}$ +5641,$\mathsf{E}[X_iZ]=500$ +5642,$R_x$ +5643,$t\mapsto W_t$ +5644,$\mu+\lambda\sigma$ +5645,$\rho(X)\le\rho(0)=0$ +5646,$\kappa_2$ +5647,$k(i)$ +5648,$\mathsf{E}[X^k]\le \mathsf{E}[Y^k]$ +5649,$\chi( s ) = p - \log(s)$ +5650,$C$ +5651,$0\le x\le 1000$ +5652,"$\Omega=(0,1)$" +5653,$D(t)$ +5654,$\mathsf{Pr}[X > a]$ +5655,$w(x)=x$ +5656,$\mathsf{E}_{\mathsf Q}[\kappa_i(X)]$ +5657,$Z(X)$ +5658,$1 < x < 2$ +5659,$P/A$ +5660,$\mathsf{TVaR}_{p^*}(X_1)+\mathsf{TVaR}_{p^*}(X_2)=80$ +5661,$g(S(x))$ +5662,$s<0.20$ +5663,$M_i = \beta_ig-\alpha_iS$ +5664,"$[0,1,\dots,n]$" +5665,$X\Delta g(S)$ +5666,$\mathsf{E}[X^k] \le \mathsf{E}[Y^k]$ +5667,$\mathbf {M}$ +5668,$\mathsf Q\not\ll \mathsf P$ +5669,$q(p')=q(p)$ +5670,$100G$ +5671,$g(x)$ +5672,$c-1$ +5673,$\lambda$ +5674,$S(x)=\mathsf{Pr}(X>x)=1-F(x)$ +5675,$C^1$ +5676,$q^-(F(x))\le x$ +5677,$\mathsf{E} X + c\mathsf{E}[\vert X-\mathsf{E} X \vert^p]^{1/p}$ +5678,$\alpha_i(a)S(a)=\mathsf{E}[(X_i/X)1_{X>a}]$ +5679,$h(p)p$ +5682,$a=f=1$ +5683,$R_L=(L-P)/P$ +5684,$\omega\mapsto \psi=F(X(\omega))$ +5685,$\tilde M_i(a) = \bar P_i(a) - \mathsf{E}[X_i(a)]$ +5686,$r-i$ +5687,$\sigma=0.4$ +5688,$y$ +5689,$d>0$ +5690,$\mathsf{TVaR}_p(X)= \sum_i X_iZ_i / 10$ +5691,$F_0=2$ +5692,$\rho(X+c) = \rho(X)+c$ +5693,$X\ge X+Y$ +5694,$X > x$ +5695,$c(X(\mathbf v))$ +5696,$\beta-\alpha$ +5697,"$(1+t)(1), (1+t)(2),\dots,(1+t)(10)$" +5698,$q = 1-p$ +5699,$\rho_g(X)=g(s)$ +5700,$\kappa_1$ +5701,$\Delta_d=a_{d}'-a_{d}$ +5702,$_{gc}$ +5703,$\mathbf {p}$ +5704,$q(p')$ +5705,$f_i(x+y)=f_i(x)+f_i(y)$ +5706,$=\mathrm{MV}(T(X))$ +5707,$F(a-)=\lim_{x\uparrow a} F(x)$ +5708,$g(S(x))>S(x)$ +5709,$s_0/2^{n}$ +5710,$\alpha f/(1-g)$ +5711,"$a_i=a(X_i, p^*)$" +5712,$\mathsf{E}[X_1]=\mu$ +5713,$\Delta X=X_1$ +5714,$V(U)$ +5715,"$f(x)=\int_0^1 f'(tx)\,dt$" +5716,$9$ +5717,$\rho(X_0)\ge \mathsf{E}[X_0 Z_\epsilon]$ +5718,$S_{X_{-1}}(a)$ +5719,$g(S_4)=0.5$ +5720,$S(x)>0$ +5721,$\mathsf{E}[YZ\mid X]=Z\mathsf{E}[Y\mid X]$ +5722,$q(1)$ +5723,$x_{max}$ +5724,$a \ge 0$ +5725,$E[s|t]=0.08353$ +5726,$ag(S_{\mathsf{j}(a)})=(80)(0.5)=40$ +5727,$\rho(\tilde X\wedge a)\le a$ +5728,$\mathsf{E}[X](1+\pi)$ +5729,$\preceq$ +5730,$X'$ +5731,"$\mathsf{NORM,TI}$" +5732,"$X^+=\max(X,0)$" +5733,$h(s) < s$ +5734,$g(s)>s$ +5735,$1_{U a)=0$ +5750,$f(p)=(1-p)\phi'(p)$ +5751,$gc$ +5752,$\mathcal F_1$ +5753,$\kappa/x$ +5754,$r_0$ +5755,$=\exp(8.7103 + \Phi^{-1}(0.995)\times 1)$ +5756,$\rho(X)=\mathsf{E}_{\mathsf Q}[X]$ +5757,$(1-f)$ +5758,$1+V^{\ast}(1) > V(2)$ +5759,$\mathsf{E} X + c{(X-\tau)^+}_p$ +5760,$D = L^* - L$ +5761,$Z_\mathit{lift}$ +5762,$\pi_1$ +5763,$p<0.01$ +5764,$f(s)$ +5765,$\lambda X_1$ +5766,$\mathbf {X_{2}/X}$ +5767,$h(X)$ +5768,$\mathsf{E}[X\wedge a] = \dfrac{k}{\beta-1}F(a)-\dfrac{a}{\beta-1}S(a)$ +5769,"$\int |X_n(\omega) - X(\omega)|^p\, \mathsf{P}(d\omega)\to 0$" +5770,$\rho_2(X_i)=0.5$ +5771,$W0$ +5800,$\mu(\{\alpha \})=1$ +5801,$\mathsf{Var}(U)>\mathsf{Var}(X)$ +5802,"$Y_{t,d=0}$" +5803,$(l-X)^+$ +5804,"$\rho(X)=\max(\rho_1(X), \rho_2(X))$" +5805,$9/6$ +5806,$j=2$ +5807,$\rho_1(X_i)=1$ +5808,$D^n\rho(\cdot)$ +5809,$\mathsf{FATOU}$ +5810,$p_0$ +5811,$\bar P=\bar P_1+\bar P_2$ +5812,$\mathsf{CTE}_p(X)=(12+25)/2=18.5$ +5813,$f=1$ +5814,$U_X = F(X-) + V(F(X) - F(X-))$ +5815,$ROL = EL + \lambda (\mathit{EL} (1 - \mathit{EL})/w)^{1/2}$ +5816,$q$ +5817,$v_{res}$ +5818,"$\{1,\dots,n \}$" +5819,$\mathit{MV}_{gc}(a_{gc})=a_{gc}-P(X\wedge a_{gc})=5583.9$ +5820,$q_Y(U)$ +5821,$x^{\ast}$ +5822,$P = \mathsf{E}[X] + \pi \max(X)$ +5823,$g''(s)=-s^{-3/2}/4$ +5824,$d\tilde p=g'(S(x))f(x)dx$ +5825,$\mathsf{E}[(-Y)Z]\ge 0$ +5826,$N\times 1$ +5827,$F_X$ +5828,$\mathcal{G}\subset\mathcal{F}$ +5829,$\preceq_n$ +5830,$s \to 0$ +5831,$A\subseteq \Omega$ +5832,$r =$ +5833,$t=1$ +5834,"$(s_i,m_i)$" +5835,$F_X(x)\ge F_Y(x)$ +5836,$g'''>0$ +5837,$T=1$ +5838,$\mathsf{E}[X_ih(X)]=\mathsf{E}[\mathsf{E}[X_ih(X)\mid X]]=\mathsf{E}[\mathsf{E}[X_i\mid X]h(X)]=\mathsf{E}[\kappa_i(X)h(X)]$ +5839,$\mathsf x\mathsf{VaR}$ +5840,$\mathcal F_{\tau}$ +5841,$\mathsf{E}[X]=\int_0^\infty S(x)dx$ +5842,"$\mathbf X = (X_1, \dots, X_n)$" +5843,$\bar P_{act} = \bar P + F_0 > \bar P$ +5844,$x=\sum_i \mathsf{E}[X_i\mid X=x]$ +5845,$(f)$ +5846,$y^2 - 2\sigma y=(y -\sigma)^2 -\sigma^2$ +5847,"$[0,t]$" diff --git a/greater_tables/data/notes.csv b/greater_tables/data/notes.csv new file mode 100644 index 0000000..079c41e --- /dev/null +++ b/greater_tables/data/notes.csv @@ -0,0 +1,3592 @@ +,expr +0,$\sigma=0.075$ +1,"$(\s, 4-\s)$" +2,$\le 1/(1-p)$ +3,$x\mapsto \mathsf{E}(X_i\wedge x)$ +4,$a(f)=\dfrac{gs_g}{1-f-fgs_g}$ +5,$\int_0^1 \phi(p)dp=1$ +6,$\ge 5000$ +7,$E_2\not=0$ +8,$\rho(-1_{A^c}) = \rho(-1_{B_l} - 1_{B_r}) = \rho(-1_{B_l}) + \rho(-1_{B_r})$ +9,$X:\{\text{Explicit Events}\}\to\mathbb{R}$ +10,$\tilde p0$ +67,$v=(1+i)^{-1}$ +68,$E_{\Bbb{Q}}[X] := E[Xg'(S(X))]$ +69,$1-S(a)=F(a)=(\nu + \delta)F(a)$ +70,"$(Alice)+(0,-2.5)$" +71,$\lambda / p$ +72,$R_2=C_2$ +73,$\tilde p=1-g(1-p)$ +74,$R_i > C_i$ +75,$-\log(1-\Phi(x))$ +76,$g(s) = \dfrac{r_o+s(1+r_K)}{1+r_o+r_Ks}$ +77,$\alpha_\epsilon=\alpha$ +78,"$\mathscr{P} =\{1+\lambda(\zeta-\mathsf{E}\zeta) \mid \zeta\ge 0, \|\zeta\|_q\le 1 \}$" +79,$r_c\le r_i$ +80,"$(A.north east)+(0.1, -0.05)$" +81,$\hat\rho(A_0)\ge \rho(A_0)$ +82,$1=P(x) + Q(x)$ +83,$G>c(x)$ +84,"$(X-a)^+=\max(X-a, 0)$" +85,"$(Alice)+(0,-3.5)$" +86,$\phi\equiv 1$ +87,$xy^4 / (x^2 + y^8)$ +88,$\beta_i(t)/\alpha_i(t)$ +89,$X=X(\bar x)=G\circ F(\bar x)=GF(\bar x)$ +90,$X(\omega)=q(T(\omega))$ +91,$\mathsf{E}(X_i/X ; X > a)$ +92,$g=2\nu^4/(1-f)+3c+1$ +93,$Y=c\in \mathbb R$ +94,$R(x)=pd_i+(v-\nu^*)\sqrt{pq}$ +95,$st=k$ +96,$X\ge X+Y$ +97,"$L_{p,p+\delta}$" +98,$s^*=1-p^*\le 1$ +99,$\rho(X) = sup_Q \mathsf{E}_Q(X)$ +100,$\bar P=\bar P_1+\bar P_2$ +101,$1 for each $ +102,$S_g = g\circ S$ +103,$\pi(x)$ +104,$\int S$ +105,$\Delta\tilde p > \Delta p$ +106,"$(0,1)$" +107,$\rho_{(g)}(X)=\int xg'(S(x))f(x)dx$ +108,$\tpx=e^{-1}$ +109,$\{ r_i \}$ +110,"$p \in [1,\infty]$" +111,$\approx\sqrt{2Np}$ +112,$\rho(X)=\int_0^1 q(p) \phi(p) dp$ +113,$g_0$ +114,$qq$ +115,$q_{X+Y}=q_X+q_Y$ +116,"$(fun2.north west)+(-\spcer, \spcer)$" +117,$p(1-\nu(p)-il(p))$ +118,$0.725$ +119,$D$ +120,$=P=\mathrm{MV}(X\wedge a)$ +121,$\Delta \tilde p\times T$ +122,$L_0$ +123,$\int_{1-p}^1 \phi(t)dt =\int_0^p \phi(1-t)dt=g(p)$ +124,$x$ +125,$\mathsf{E}[X_1g'(S(X))]$ +126,$\mathsf{E}(X\wedge a)$ +127,$0\le\beta<1$ +128,$\rho_{t+1}(X)$ +129,$X_i(X\wedge a)/X$ +130,$P=\nu(\bar S + \iota a)$ +131,$\rho_i(X_i)$ +132,$\downarrow$ +133,$\nabla_x f= \nabla_xq_\alpha -\nabla_x G$ +134,$\eta\gg\zeta$ +135,$v-\nu^*=\delta^*-d$ +136,$\sup_n \| X_n \|< \infty$ +137,$A=P+Q$ +138,$B = g^{b} \pmod{p}$ +139,$\alpha$ +140,$X=C(\bar x)+N(\bar x)=$ +141,$X_1$ +142,$\mathrm{PQ}$ +143,$v-\nu^*=(\iota^*-i)/v\nu^*$ +144,$\mu_X\le\mu_X$ +145,$\lambda X$ +146,$g(x)\ge x$ +147,$\rho(A_k)\le\hat\rho(A_0) + k\rho(N)=\hat\rho(A_k)$ +148,$\rho_t$ +149,$Z=\mathsf{E} Z$ +150,$\beta$ +151,"$(A.north east)+(0.2, -0.05)$" +152,$\tilde\rho_T=\rho_T$ +153,$>$ +154,$c_h>c=\mathsf{VaR}$ +155,$a\ge c$ +156,$F(x)=\mathsf{Pr}(X\le x)$ +157,$X_i(x_i)$ +158,$P + \rho_i(F_i) < \rho_i(X_i) \iff P < \rho_i(X_i) - \rho_i(F_i)$ +159,$\tilde \rho$ +160,"$L^\infty(\Omega, \mathsf{P})$" +161,$0\le Y\le 1$ +162,$R(a)=\delta N(a)$ +163,$\bar P$ +164,$F_Y$ +165,"$(fun3a.south -| fun3a.south east)+(\smlspc,-\smlspc)$" +166,$\sigma(1-t)=g'(t)$ +167,$g'(1-p) dp$ +168,$\mathsf{E}(X) = \int_0^1 q(p)dp$ +169,$(3) \rightarrow (9 = 9) \rightarrow (27 = 4) \rightarrow (12 = 12) \rightarrow (36 = 13) \rightarrow (39 = 16) \rightarrow (48 = 2) \rightarrow (6 = 6) \rightarrow (18 = 18) \rightarrow (54 = 8) \rightarrow (24 = 1)$ +170,$t=T_xa)=1-\exp(-\lambda S(x))$ +210,$YL$ +211,$X\le 0\implies\rho(X)\le 0$ +212,"$u_1,\dots, u_n$" +213,$t_2-\epsilon/2$ +214,$F_t$ +215,$p=F(\mathsf{E}(X))$ +216,$1 \times 10^{24}$ +217,$\nabla (\zeta NF) = \zeta\nabla NF$ +218,$p=p_a$ +219,$\iff$ +220,"$L,P,M,Q,a,LR,PQ,COC$" +221,$\approx$ +222,$\mathsf{E}(X_i\mid X)$ +223,"$\eta\gg \zeta:[0,1]\to\mathbb{R}$" +224,$\phi:=\rho\circ F$ +225,$i=0$ +226,$\iota^*$ +227,$\partial a/\partial x_1$ +228,$\mathsf E[X_i]$ +229,"$\rho(Z)=\sup_{\zeta\in\mathcal{A}} \langle \zeta, Z \rangle$" +230,"$\Omega=[0,1]$" +231,"$s\in[0,1]$" +232,$\bar\nu=1/(1+\bar\iota)$ +233,$\rho(X+m)=\rho(X)-m$ +234,$K = A^{k}=g^{ak} \pmod{p}$ +235,$\rho E/(1-\tau) - rA$ +236,$=E(X_i / X)$ +237,$\mathscr{O}(\eta)$ +238,"$\mathbf{x}=(x_1,\dots,x_n)$" +239,$t_10 \}$ +256,$1-g(S(x))=\tilde F(x)$ +257,"$a, b$" +258,$\xtext$ +259,$\bar h$ +260,$g'(S(x))dF(x)$ +261,$1=S(a) + \delta F(a) + \nu F(a)$ +262,$\Omega=\mathbb{R}$ +263,$\mathsf E[XY]\not=\mathsf E[X]\mathsf E[Y]$ +264,$\sum \alpha_i=1$ +265,$Z_p^\times$ +266,$h_\epsilon$ +267,$\rho(X) = \mathsf{E}(X) + \| (X-\mathsf{E} X)_+ \|_p$ +268,$\mathbb{R}_+=[0\infty)$ +269,$\delta_p+\nu_p=1$ +270,$Z=g'(S(X))$ +271,$\rho=0.6$ +272,$\rho(L) = q(p)>q(p)$ +273,$\mathsf{E}(X\mid X > a)$ +274,$L^\infty$ +275,$p(\nu_p-l_p)$ +276,$\rho(B(s_u)) - \rho(B(s_l))$ +277,$\rho(X)= (1+r_f)^{-1}\mathsf{E}_Q(X)$ +278,$(1-{}_b\bar V)$ +279,$a\theta^2=c$ +280,$(1-\nu_p-il_p)/(\nu_p-l_p)=\iota_{1/2}$ +281,$p=23$ +282,$\nu=1/(1+\iota)=1-\delta$ +283,$X=X_1+X_2+X_3$ +284,$\rho(-k_i 1_{A_i}) \le c < 0$ +285,$\bar a_x$ +286,$a=1/c$ +287,$\rho(-1_{A^c})=0$ +288,$c=\bar A^{1}_{x:\lcroof{1}}/\bar a_{x:\lcroof{1}}$ +289,$\mathcal F^G$ +290,$\bar a_{x:\lcroof{1}}$ +291,$g^ag^k=g^{a+k}$ +292,$\pi_X(t)\le \pi_Y(t)$ +293,$Y=\sum_i X_iY_i$ +294,"$(Alice)+(0,-3)$" +295,$\beta_i(a)/\alpha_i(a) < 1$ +296,$(3\times 6 + 2\times 2)/ 8 = 11/4$ +297,$g\ge 0$ +298,$X(u)$ +299,$\displaystyle\int_0^1\phi(s)ds=\displaystyle\int_0^1\displaystyle\int_{1-s}^1\dfrac{\mu(dt)}{t}ds = \displaystyle\int_0^1\displaystyle\int_{1-t}^1ds\dfrac{\mu(dt)}{t}=\displaystyle\int_0^1\mu(dt)=1$ +300,$\rho(A)>\hat\rho(A)$ +301,$= \rho(B(s_l)) (1 - s) + \rho(B(s_u)) s$ +302,$\int_0^1 μ(dt) = 1 - α < 1$ +303,$P_idx_i$ +304,"$\omega_1,\omega_2\in\Omega$" +305,$X\wedge a$ +306,$C_2(0) = \mathsf{E}[X_2]$ +307,$X(\mathbf{x})=\sum_i x_i X_i$ +308,$\rho(X)=50=:r$ +309,$|Z|$ +310,$\rho(X)=\int_0^1 q(1-g^{-1}(1-t))dt$ +311,$N(1-p)$ +312,$1+2c(1-\mathsf{Pr}(Z>\mathsf{E} Z)$ +313,$r$ +314,$\bar P^a_i$ +315,$E_2$ +316,$m_j / r_j$ +317,$\int_0^x (x-y)^{n-1}dG(y)$ +318,$P =\{ Q \mid dQ/dP \le k \}$ +319,$\pi = \mathsf E[PR]$ +320,$A_{x+b}$ +321,$\rho(-X_n)\downarrow 0$ +322,$q(\epsilon)\approx q + \epsilon\mathsf{E}_q(X_i)$ +323,$\mathsf{E}(Y(a))=\mathsf{E}(Y\wedge a)=\int_0^a S_Y(t)dt$ +324,$g'(t)=1-r_0$ +325,"$\langle \zeta_{\bar x}, N(\bar x) \rangle$" +326,$\sum_i h^i= 0$ +327,$g(S(x))=1$ +328,"$(A.north east) + (-0.07mm,0)$" +329,$E(XZ \mid \mathcal{G})=ZE(X \mid \mathcal{G})$ +330,"$(\nodespc/2, -\nodespc/2)$" +331,$\{ v_i \}$ +332,$\int_0^q = \int_0^{\mathsf{E}_q(X_2)} + \int_{\mathsf{E}_q(X_2)}^q$ +333,$q(\epsilon)=q+\epsilon\mathsf{E}_q(X_1)$ +334,$1-U$ +335,$\log_{10}(N(m))) \propto -bm$ +336,$\not=$ +337,"$[a, a+da]$" +338,$1_Af_t(X)=1_Af_t(1_AX)$ +339,$\mathbf{X}\times\mathbb{R}$ +340,$X_i\ge 0$ +341,$a>a(f)$ +342,$p(a)=\nu S(a) + \delta = S(a) + \delta F(a) = 1-\nu F(a)$ +343,$\sigma=0.125$ +344,"$D_n,D_n^*$" +345,$X+Y$ +346,$X_n \downarrow 0$ +347,$\rho(X)=\int_0^1 q(s)g'(1-s)ds$ +348,$\mathsf{E}(X) = \displaystyle\int_0^\infty xf(x)dx = \displaystyle\int_0^1 q(p)dp$ +349,$\rho(X^{\oplus n}) \ge \rho(X^{\oplus n-1}) + \mathsf E[X] > \rho(X^{\oplus n-1})$ +350,$\pi_\sigma(L)$ +351,$X_1\wedge a$ +352,$\rho_p$ +353,"$p=0,1$" +354,$\hat \rho$ +355,$X_p=^d Y_p$ +356,$\mathbb{R}^2$ +357,$B(b)\approx -b\mu_x$ +358,$L(a)=$ +359,$1 = m(x) + \nu F(x) = S(x)+\delta F(x) + \nu F(x)$ +360,"$I_i\in\{0,1\}$" +361,$Y\ge X$ +362,$X(\mathbf{1})$ +363,$\bar P'(x)=P(x)$ +364,$(34.05-23.81) / (100-34.05)=15.5$ +365,$\rho_w$ +366,"$(-\x, 2)$" +367,$a=F^{-1}(1-\delta)$ +368,"$\sigma=0.5, 1.0$" +369,$Y\wedge a$ +370,"$[0,1]$" +371,$\mathcal G$ +372,$2/3$ +373,$\bar Q'(x)=Q(x)$ +374,$G(\bar x)$ +375,$F(a)$ +376,$p\delta_p$ +377,$R_1(t)<\mathsf{E}[X_1]$ +378,$5.14\times 10^{19}$ +379,$\mathsf{TVaR}_p(X) \le r$ +380,$0\le\alpha\le K$ +381,$= 10^{1+6+12}=10^{19}$ +382,$g^k$ +383,$\phi(0)=0$ +384,"$g, g^2, \dots,g^{q-1}, g^q\equiv 1$" +385,$||\cdot ||$ +386,"$g, g', g''$" +387,"$\Omega=\{1,2,3 \}$" +388,$\sum x_iX_i$ +389,$X_1=s$ +390,$\mathbb{R}^n\to\mathbb{R}$ +391,$X\wedge a\not\in \mathbf{X}$ +392,$T_i\circ T$ +393,$\delta+\nu=1$ +394,$X(\cdot)$ +395,$q+p\delta_p$ +396,$q_Z(U)$ +397,$1_A = 1 - 1_{A^c}$ +398,$P=\rho(X\wedge a)$ +399,$1-p \ge g^{-1}(1-p) \implies 1-g^{-1}(1-p) \ge p \implies q(1-g^{-1}(1-p))>q(p)$ +400,$G=X_1+X_2$ +401,$=\rho(B(\mathrm{current\ best\ estimate\ of\ } s)) = \rho(B(s))$ +402,"$\zeta, \zeta_t\ge 0$" +403,"$p=0.98,0.99$" +404,"$(3,6-4.724)$" +405,"$=\mathsf{E}(X_{i,2}(a))$" +406,$\square\rho_i$ +407,$602.6 billion and converted to net premium based on $ +408,"$(r,s)$" +409,$F^{\times}_{23}$ +410,$\mu_\sigma$ +411,$E_{\Bbb{Q}}[Y]=E[Yg'(S(X))]$ +412,$n\times 1$ +413,$\phi(s)=\displaystyle\int_{1-s}^1\dfrac{\mu(dp)}{p}=\int_0^s\dfrac{\mu(dp)}{1-p}$ +414,"$(Alice)+(0,-1.75)$" +415,$\rho(1_A) \le \rho(1)=1$ +416,$1 \times 10^{20}$ +417,$\rho(X)<\rho(Y)$ +418,$1-U^2$ +419,$G=N+C$ +420,$\alpha_i$ +421,$0 1$ +454,$g'(s)=1$ +455,$1-p=g(1-\hat p)$ +456,$r_i=\rho(X_i)$ +457,"$\langle \zeta, G \rangle$" +458,$g(s)=O(d)$ +459,$g'(p)=\phi(1-p)$ +460,$ be the compound of $ +461,$\rho_t = \rho_t(-\rho_{t+1})$ +462,$\circ$ +463,$\mathsf{TVaR}_\beta\mathbin{\square}\mathsf{TVaR}_\gamma = \mathsf{TVaR}_\gamma$ +464,$\rho=\rho_\phi$ +465,"$(rep.east) + (1.5, 0.5)$" +466,$\mathsf{E}[h_\epsilon Y]\to\mathsf{E}[h Y]$ +467,$\rho =$ +468,$\phi(1-t)$ +469,$A:=g^a \pmod{p}$ +470,$a-L_0^a$ +471,$1=v+d$ +472,$\nu(p)=p$ +473,$\rho(A+B)64.5>63.5=\rho(A)+\rho(B)$ +474,$p<\infty$ +475,$\alpha_i(X_u)= \text{E}[u_iX_i \mid X_u > F_u^{-1}(p)] = u_i \partial T/\partial u_i$ +476,$\alpha_\epsilon-\alpha$ +477,$F^{(2)}=[F^{(-2)}]^*$ +478,$\mathrm{P}$ +479,"$q\in[1, \infty]$" +480,$\uparrow$ +481,"$(0.5,1.5)$" +482,$\omega\in\Omega$ +483,$\phi(p)=1$ +484,$0a$ +521,$\mathsf{E}[g]\le 1$ +522,$\mathsf{Pr}(I=1)=s$ +523,$-norm less than $ +524,$\bar M_i(a)$ +525,$\mathbf{r}\ge 0$ +526,$u_i\partial\pi / \partial u_i$ +527,$\rho_m(X)=\rho_m(X\wedge k) + \rho_m((X-k)_+)$ +528,$\bar P(x) = \bar S(x) + \bar R(x)$ +529,$\mathsf{TVaR}_{p^*}$ +530,$\rho(1_A) = \rho(1) = 1$ +531,"$s,t$" +532,$ρ$ +533,$[xf(x)] \times dx$ +534,$p<1$ +535,"$c\in[0,1]$" +536,$R(x)=pd+(\delta^*-d)\sqrt{pq}$ +537,$g'(1-p)=\phi(p)$ +538,$\nu=\nu_p$ +539,$q=11$ +540,$\bullet$ +541,$\iff \mathcal A_{t+1}\subseteq \mathcal A_t$ +542,$\sigma=0.5$ +543,"$n=1,2,\dots$" +544,$\mathcal{A} = \{ X \mid \rho(X)\le 0 \}$ +545,$age^2$ +546,"$\phi_{\bar x}(Z)=\langle Z,\zeta_{\bar x} \rangle$" +547,$3.2 \times 10^{15}$ +548,$P=L + \delta (a-L)$ +549,$\mathsf{E}[X\cdot Z\circ T]=\mathsf{E}[X\cdot Z\circ T_B\circ T_A ]=\mathsf{E}[X \cdot Z\circ T_A]=\mathsf{E}[X\circ T_A^{-1}]=\mathsf{E}[X Z]$ +550,"$(X,Y)$" +551,$\partial \zeta_{\bar x}/\partial x_i$ +552,$\delta = \delta(p) = 1-\nu(p)$ +553,$\lim_n \mathsf{E}_{\mathsf{Q}_n}(X)=\rho(X)$ +554,$g^a\equiv n\pmod{p}$ +555,$P(a) = S(a) + \delta F(a)$ +556,$\hat\rho(A_k)$ +557,$g'=0$ +558,$X=X(I)$ +559,$g(s)=\dfrac{r_{occ}+s(1+r_{use})}{1+r_{occ}+r_{use}s}$ +560,$\mathsf{Q}_n\in\mathscr{P}$ +561,$g^-1(p)$ +562,$S=1-F$ +563,${}^nS^{-1}_X(t)\le {}^nS^{-1}_Y(t)$ +564,$\iota\alpha(X)=\iota a$ +565,$\mathsf{E}[Y\mid X] = X$ +566,$f^*_i$ +567,"$(fun3.north west)+(-\smlspc,\smlspc)$" +568,"$[0, 1]$" +569,"$(fun1a.south east)+(\smlspc,-\smlspc)$" +570,"$(rep.south) + (0.5, -1.0)$" +571,$\rho^*=\rho(0.5)$ +572,$g(S(x))\approx S(x)\approx 1$ +573,$=(1-\alpha)\mathsf{TVaR}_\alpha(X)$ +574,$B\cup B_t = (B\cap B_t) \cup C_t$ +575,"$X^n_t=1_{[1+T_n, \infty)}$" +576,$(p-\nu-il)/(v-l)$ +577,"$G(x+th, \omega+d\omega) = c_k(x+th)$" +578,$(k+1)\times 1$ +579,$X_n\le 1$ +580,"$\langle NF(x), Th_i \rangle+\langle \partial NF/\partial x_i, \zeta_{GF(x)} \rangle$" +581,$\partial a/ \partial x_i$ +582,$\rho(I)\rho(X)=g(s)\rho(X)$ +583,$\mathsf{Var}(\Pi)$ +584,$\nu(p)=1/(1+\rho(p))$ +585,"$g:[0,1]\to[0,1]$" +586,$g(S(a))-S(a)$ +587,$\delta N(a)$ +588,$n$ +589,$H(x)$ +590,$\rho(X)=\mathsf{E}_Q(X)=\mathsf{E}_Q(Y)+\mathsf{E}_Q(Z)$ +591,$\rho(X)\le \rho(Y)$ +592,$P_Q = \mathsf{P}[(X-a)V(a)]$ +593,$q_j$ +594,"$(fun1.north west)+(-\medspc,\medspc)$" +595,$\zeta NF$ +596,$a = q_X(0.995)$ +597,$S(x)=1=F(x)$ +598,$d+v=1$ +599,$K=g^k$ +600,$b-a$ +601,"$(Bob)+(0,-2)$" +602,$X^{\oplus N}$ +603,"$(a-X)^+:=\max(a-X, 0)$" +604,$\rho_{1/2}$ +605,"$s,t \in[0,1]$" +606,$10^{17}$ +607,$M_0$ +608,$\int_0^1 Z=1$ +609,$\rho(X)=\mathsf{TVaR}_1=\esssup$ +610,$\nu < 1$ +611,"$\pi : X\mapsto (X, \alpha(X))\mapsto E_g(X\wedge \alpha(X))$" +612,$\rho_m(X) = \mathsf{E}(X) + (\rho_m(X)-\mathsf{E}(X))$ +613,$(x-y)^n$ +614,"$u\in D_n=\{ u \mid u^{(k)} \ge 0, k=1,\dots,n-1, u^{(n-1)}\text{ nondecreasing} \}$" +615,$\mathsf{E}(XZ \mid \mathcal{G})$ +616,$\bar a_{\lcroof{n}}$ +617,$(S(x) + \delta(F(x))F(x)) dx$ +618,$m + ra = ks$ +619,$Q$ +620,$n-1$ +621,$-1$ +622,"$(1-g(S(x)),x)$" +623,$k_0>\ge 2$ +624,$\Pi$ +625,$\rho_i$ +626,$\bar a_x = \bar a_{x:\lcroof{b}} + v^b{}_bp_x\bar a_{x+b}$ +627,$v-l$ +628,$\delta_p/\nu_p = \rho_p$ +629,$\rho(-X+a)=\rho(-X) + a \le 0$ +630,$r = g^k$ +631,"$(0,1) < 1$" +632,$\mathcal F_1=\sigma(N)$ +633,$dt$ +634,$Z_1=Z\circ T_A$ +635,"$(fun4a.south -| fun3a.west)+(-\medspc,-\medspc)$" +636,$dx=x_{i+1}-x_i$ +637,"$x=1.5, M=1.5,\sigma=0.75, K=6$" +638,$ from policyholder as premium and capital $ +639,$\hat X_i=\hat x_i$ +640,$\nu(dx)$ +641,$0.5$ +642,$\liminf \rho(X_n) \ge \rho(X)$ +643,$M(a)=\mathsf{E}(X\wedge a) + \delta N(a)$ +644,$S(x) + \delta F(x)$ +645,"$(ckey1.north west)+(-\boundpad,\boundpad)$" +646,$10 million I **must care more** about a loss of $ +647,$1.5\times 10^{37}$ +648,$Q\in \mathscr{P}$ +649,"$a,b$" +650,$\zeta>0$ +651,$\mathsf{E}(X_i \mid X \le a)$ +652,$X=C+G$ +653,$\rho(X)\le\liminf_{n\to\infty} \rho(X_n)$ +654,$M_X(k)\le M_Y(k)$ +655,$t=t_2$ +656,$T_t$ +657,$H:\mathcal X\to\mathbb R$ +658,$\phi_i = \mathsf{E}(X_i)/\mathsf{E}(Y)$ +659,$\mathsf{E}_Q(Y\mid X)\mathsf{E}(Z\mid X) = \mathsf{E}(YZ \mid X)$ +660,$g(s) = t_{df}(t_{df}^{-1}(s)+\lambda)$ +661,$\rho=\rho(p)$ +662,"$2*(1,1)$" +663,$\lambda > 0$ +664,$\rho=0.12$ +665,${}_nE_x$ +666,$\rho(T)\ge T$ +667,$p\mathsf{E}[X\mid XC$ +671,$\delta=\log(1+i)$ +672,$a=1$ +673,$\approx (920+961)/2=940.5$ +674,"$(Alice) + (0,-2)$" +675,$v\mathsf E[X] + d\max(X)=\rho(X)$ +676,$\rho_{(g)}=\max\{\mathsf{E}(ZX) \mid Z\in \mathcal{A}\}$ +677,"$G(x,\omega)=c_k(x)$" +678,$P=\displaystyle\sum_i P_i$ +679,$t \le g(t) = \displaystyle\frac{t}{1-p}$ +680,$\lambda_{t}$ +681,$P + \rho_i(F_i)$ +682,$X\circ T=X$ +683,$\sigma_\mu(\alpha) = \int_0^\alpha \frac{1}{1-p}\mu(dp)$ +684,$X_i < cx/a$ +685,$age$ +686,$\zeta=\Omega$ +687,$X = X_0 + M + A$ +688,$l(p)= \nu(p)-\sqrt{(1-p)/p}$ +689,$Y \Leftrightarrow \rho(X)\le \rho(Y)$ +690,$\beta_i(t)<\alpha_i(t)$ +691,$\sqrt{FS}\gg S$ +692,$dx_i$ +693,$\rho^*(\mu)=\infty$ +694,$\mathsf{E}[hY]$ +695,$U$ +696,$\mathsf{TVaR}_{1}$ +697,"$\mathsf{E}(X_{i,2}(a))$" +698,$p_a$ +699,$4/3$ +700,$\infty$ +701,$12.318 / 260.81 = 4.7\%$ +702,$1-\tilde p=g(S(x))$ +703,$c_x-c_{\text{Nov 1}}$ +704,$k-\rho_m(X)$ +705,"$P(X_1+X_2)=M(X_1+X_2, \psi(X_1+X_2))=$" +706,$\rho(\cdot\mid\mathcal F_1)$ +707,$h\in \nabla\rho(X)$ +708,$\bar x$ +709,$(v-\nu^*)\int_0^a \sqrt{F(x)S(x)}dx$ +710,$Z\circ T_B=Z$ +711,$\displaystyle\int_0^\infty xd(g\circ F)(x)$ +712,$\rho:L_p\to\bar\mathbb{R}$ +713,$x=z$ +714,$A_k=A_0 + kN$ +715,$ is the total return on invested assets and $ +716,$b\approx 0.95$ +717,$t=1-g(1)=0$ +718,$\le_{\mathrm{cx}}$ +719,$\nu(F(x))F(x) = \nu(p)p$ +720,$q=1-p=S(x)$ +721,"$r_o,r_K$" +722,$q(u_i)$ +723,$\bar P^a(t)=\bar P^a_1(t)+\bar P^a_2(t)$ +724,${}_b\bar V=1-\bar a_{x+b}/\bar a_x$ +725,$\mathsf{E}(X) = \displaystyle\int_0^\infty xf(x)dx = -xS(x)\Big\vert_0^\infty + \displaystyle\int_0^\infty S(x)dx = \displaystyle\int_0^\infty S(x)dx$ +726,$\theta(p)=q(1-g^{-1}(1-p))/q(p)$ +727,$v-\nu^*$ +728,$t\to 1$ +729,$p(a) = \nu S(a) + \delta = \nu (S(a) + \rho)$ +730,$1 \times 10^{19}$ +731,$10^{20}$ +732,$i=0.025$ +733,"$(1,1)$" +734,$\nu$ +735,$\rho_k\to\infty$ +736,$\mathcal F_1=\sigma(I)$ +737,$f(s) = -g''(1-s)(1-s)$ +738,$r_O$ +739,"$50 of the amount allowed on each claim in the classes under subsections 3, 4, 4-B, 5 and 6 must be deducted from the claim and included in the class under subsection 8. Claims may not be cumulated by assignment to avoid application on the $" +740,$k\mathsf B(s)$ +741,$r_{qp}=\sqrt{pq}$ +742,$\phi$ +743,$\mathsf{MON}$ +744,$g^{ak}=(g^a)^k$ +745,$F_{\mathbf{x}}(t)=s$ +746,$\rho(X)=\sum_i \mathsf{E}_\mathsf{Q}(X_i)$ +747,$Y\le X=0$ +748,$k=\mathsf E[X]$ +749,$g\in\mathscr{P}$ +750,$p(1-p)$ +751,$x\not= y$ +752,"$\rho(X)= \sup_\zeta \langle \zeta, X \rangle$" +753,"$h_{i,\epsilon}$" +754,"$(X_1,\dots,X_n)$" +755,$R$ +756,$A=X_1 + \cdots + X_N$ +757,$g\in\mathscr P$ +758,$=F^{-1}(p)=$ +759,$g^{-1}$ +760,$q(1-g^{-1}(1-p))$ +761,$Y=\log(X)$ +762,$r = \nabla r$ +763,$\bar S_i(\mathbf{x}; a) := \mathsf{E}[X_i(\mathbf{x}; a)]$ +764,$N(m)$ +765,$a_i = \mathsf E[X_i \mid X \ge a]$ +766,$X\in L^\infty$ +767,$-\int xdS=\int Sdx$ +768,$p=\sigma^{-2}$ +769,$X=\displaystyle\sum_i X_i$ +770,$\partial \rho(X)$ +771,$da > 0$ +772,$s$ +773,$a_1\not=a_2$ +774,$H_k(X)=H_k(Y)$ +775,$\theta(p)\equiv 1$ +776,$\mathsf{E}_\mathsf{Q}(\cdot)$ +777,$g(1-F(x))=1-\tilde p$ +778,"$\mu_t:=\lambda_t / \int_0^1\lambda_s \,ds$" +779,$d\mathsf{Q}=g'(1-p)dp$ +780,$\ge a$ +781,$\mathcal{M}$ +782,"$k \in_{R} \{2,\dots,p-2\}$" +783,$\int_0^x$ +784,$F:\mathbb{R}^n\to \mathcal{X}^n$ +785,$N$ +786,$\bar M(a)$ +787,$C_i=\partial \bar P^a/\partial x_i$ +788,$\mathcal F_1\subseteq \mathcal F$ +789,$2.6 \times 10^{12}$ +790,$\| \sigma \|_p \le c$ +791,$\|Z\| = \mathsf{E}(| Z|^p)^{1/p}$ +792,"$700 million. Enstar, which owns 9.1% of Watford’s common shares, at the same time agreed to abandon its quest to buy the insurer. In May 2020, activist investor Capital Returns Management LLC called for Watford to be sold or put into runoff, complaining about “consistently poor operating and stock performance” in comparison with its peers in the industry. When an initial offer of $" +793,$t=0$ +794,$0.125$ +795,$=\mathsf{E}(X\mid X > a)$ +796,"$t\in[t, t+dt]$" +797,$\rho_\phi(X)=\displaystyle\int_0^1 q(p)\phi(p)dp=\displaystyle\int_0^1 q(p)g'(1-p)dp=\displaystyle\int_0^\infty xg'(1-F(x))f(x)dx$ +798,$\rho(X)=\mathsf{E}(q(U)\phi(U))=\mathsf{E}_Q(q(U))$ +799,$-\log(1-\alpha)$ +800,$ds=g'(1-t)dt$ +801,$Z\ge 0$ +802,$M_r$ +803,$g^mA^r == r^s$ +804,$b\mu_x v^b$ +805,$\mathsf{E}_\mathsf{Q}(X_i) = \mathsf{E}_\mathsf{Q}(\mathsf{E}_\mathsf{Q}(X_i \mid X)) = \mathsf{E}_\mathsf{Q}(\mathsf{E}(X_i \mid X))$ +806,$. Then $ +807,$\iota$ +808,$\epsilon >0$ +809,$\hat\rho(X)<\rho(X)$ +810,$\sum_i I_i=1$ +811,"$M(X_1, a_1)+M(X_1, a_2)=M(X_1, a_1+a_2)$" +812,"$=\mathsf{E}(\min(X,a))=\mathsf{E}(X\wedge a)$" +813,$\pi'(s) = \displaystyle\frac{d}{ds}(g(s)g(k/s))$ +814,"$(A=g^a,a)$" +815,$X \lt a$ +816,"$x=2, M=1.5,\sigma=0.75, K=6$" +817,$h=H(A)$ +818,"$X\sim\text{Lognormal}(\text{mean}=5000, cv=3)$" +819,$\rho(p)=\rho(F(x))$ +820,$ of paying and $ +821,$\mathsf{TVaR}(p)=(1-p)^{-1}\int_{p}^1 q(s)ds$ +822,$p=\infty$ +823,$x+dx$ +824,$d\tilde p =g'(1-p)dp$ +825,$X(x) = \sum_i x_iX_i$ +826,$G>q_\alpha$ +827,$M(a)=g(S(a)) - S(a)$ +828,$x \times [f(x)dx]$ +829,$S_Y(a)$ +830,$\bar a_{x:\lcroof{n}}$ +831,$\rho_\phi(X)=\displaystyle\int_0^1 q(p)\phi(p)dp=\displaystyle\int_0^\infty g(S(x))dx=\rho_{(g)}(X)$ +832,$\delta\bar a_{x:\lcroof{n}}$ +833,$\bar A^{1}_{x:\lcroof{1}}$ +834,$\rho^*(\zeta-1)$ +835,$(v-\nu^*)\sqrt{F(x)S(x)}$ +836,$\theta=c=\nu^2$ +837,$ because $ +838,$\tilde F$ +839,$(\partial \alpha/\partial x_i)q_X(\alpha)q_\zeta(1-\alpha)$ +840,$A$ +841,$-$ +842,$\tilde W$ +843,$\tilde p/p$ +844,"$\bar P_i(\mathbf{x},a):=\mathsf{E}_g[X_i(\mathbf{x}; a)]$" +845,$(x)$ +846,$\mathsf{TI}$ +847,$t_1<\cdots a)S(a)$ +860,$\text{E}(G^3)=g$ +861,$X^{\oplus n}=X_1 + \cdots + X_n$ +862,$Ann+V$ +863,$Z\in L_1$ +864,$F(t)=p$ +865,$-k_i 1_{A_i}$ +866,"$31.5 million. Nine of Argonaut’s 11 top officers were fired, and Singleton began running the operations from headquarters in Los Angeles. Argonaut, one of the last large companies in the malpractice market, discontinued underwriting individual policies for the 20,000 physicians it covered. It continued to offer coverage to the 25 percent of the nation’s hospitals it covered, but at higher rates and covering fewer risks. In the meantime, the company collected $" +867,$a=\inf$ +868,$k_1 >0$ +869,$X_n\downarrow 0$ +870,$\rho=\text{AVaR}$ +871,"$R_1(t),R_2(t)$" +872,$E_Q(N_i) = E_Q(\nabla \rho) + E_2$ +873,$a\le X\le b$ +874,$t<0.12$ +875,$\text{E}(G^r)=\theta^r\Gamma(a+r)/\Gamma(a)$ +876,"$10 monthly premium and pay out as much as, say, $" +877,"$(fun4.north west)+(-\smlspc,\smlspc)$" +878,$\mathsf E[A] \le \mathsf E[\rho(X^{\oplus N})] \le \rho(A)$ +879,$a=\max X$ +880,$s_l = f / (n+1)$ +881,$M^{\tau_n}_t = M_{t \wedge \tau_n}$ +882,$\rho(\cdot\mid \mathcal F_1)$ +883,$0\le \alpha<1$ +884,$n=2^2$ +885,$H_g(X) \le H_g(Y)$ +886,$\nu(p) = v-(v-\nu^*)\sqrt{(1-p)/p}$ +887,$\mathsf{E}_Q$ +888,$\nabla\partial\rho(Z)$ +889,"$\sigma=2.0,3.0$" +890,$w \ge 0$ +891,$Z=\frac{X-\mathsf{E}[X]}{\sigma(X)}$ +892,$k= \mathsf{E}(X\wedge k) + (\rho_m(X\wedge k) - \mathsf{E}(X\wedge k)) + (k-\rho_m(X\wedge k))$ +893,$=q(p)$ +894,$\delta^2 p +\nu^2q-(p-\nu)^2=\delta^2 p -p\nu^2 -p^2+2p\nu =p(\delta^2 -\nu^2) -p^2+2p\nu =p(\delta -\nu) -p^2+2p\nu =p\delta -p^2 + p\nu = p-p^2$ +895,$N\mid G$ +896,$\mathsf{E}(L) = q(p)\delta$ +897,$\rho(X)=\mathsf{E}[gX]$ +898,$u_l>0$ +899,$\alpha_i(t) = \mathsf{E}[X_i /X \mid X> t]\not=\mathsf{E}[X_i\mid X> t]/\mathsf{E}[X\mid X>t]$ +900,$=Q=\mathrm{MV}(a-X)^+$ +901,$\zeta=0$ +902,$\mathsf{Var}(X_i)>0$ +903,$\phi(0)$ +904,$2^2\rightarrow 3^3-1=2\times 3^2 + 2\times 3 + 2 = 26$ +905,$\hat\rho(Y)$ +906,${}_tV$ +907,$\tilde\rho$ +908,$a=0$ +909,$\square^\square-1$ +910,$\rho_t(X) = \rho_t(-\rho_{t+1}(X))$ +911,"$\alpha_p = 1- (\| (X-\eta_{p,\alpha})_+\|_{p-1} / \| (X-\eta_{p,\alpha})_- \|_{p})^{p-1}$" +912,$t$ +913,$c\ge 1$ +914,$g'(0)\le 1$ +915,$\mathsf{E}(\theta)=1$ +916,$\mathsf{TVaR}_{0.99}(X)=119.8=\mathsf{E}(W+Q\mid X\ge 100)=\mathsf{E}(W\mid X\ge 100) + \mathsf{E}(Q\mid X\ge 100)=19.8+100$ +917,$\mathsf{E}(T)=74.25$ +918,"$X\wedge a:=\min(X,a)$" +919,$P_Q$ +920,$\bar\delta$ +921,$\bar a_{40}=17.95$ +922,$Y= IX$ +923,$L^p$ +924,$\mathsf E[A_0\mid N=n]=\mathsf E[X_0^{\oplus n}]=0$ +925,"$(asecret.east) + (0,-0.5)$" +926,$\rho(X_n)\downarrow 0$ +927,$\tilde p=\tilde p(p)$ +928,$dp=$ +929,$t=0.25$ +930,$\zeta_\epsilon$ +931,$s_s < s < s_f$ +932,$\exp(n(e^\zeta-1))$ +933,$M$ +934,$0C_i(t)$ +937,$\rho(X)=\sum_i \mathsf{E}_\mathbb{Q}(X_i)$ +938,$se(\hat\beta)$ +939,$1 - g(s)$ +940,"$j = 1, 2$" +941,$\text{E}(G)=a\theta$ +942,$\rho(-k_1 1_{A_1}) = k_1 \rho(-1_{A_1}) < c$ +943,$H=G_0-F$ +944,$-g''$ +945,$\alpha=d$ +946,$Y=h(Z)$ +947,$\alpha(X)=a$ +948,"$(fun1a.south -| fun4a.south east)+(\smlspc,-\smlspc)$" +949,$m=K^{-1}Km$ +950,"$\langle \cdot,\cdot\rangle:\mathcal{X}\times\mathcal{M}\to \mathbb{R}$" +951,"$p\in[1,\infty]$" +952,$\mathsf{P}$ +953,$q_Y$ +954,$\bar P^a$ +955,$\bar Q$ +956,$\{X\le a\}$ +957,$E_\mathsf{Q}(X_i) = E_\mathsf{Q}(E_\mathsf{Q}(X_i \mid X))$ +958,"$\phi:[0,1]\to [0,\infty)$" +959,$q(p)=\mathsf{VaR}(p)$ +960,$\rho(X)-a$ +961,$m(p)$ +962,$v^b{}_bq_x\bar a_{x+b} /\bar a_x=v^b{}_bq_x(1-{}_b\bar V)$ +963,$\epsilon > 0$ +964,$\mathsf{E}(Q/X | X\ge x)$ +965,$v\mathsf E[X_i]$ +966,$\tau>0$ +967,$\Longleftrightarrow$ +968,$\rho(X+Y)=\rho(X)+\rho(Y)$ +969,$\lambda=0.1525$ +970,$\mathsf E[X_i\mid X=x]$ +971,$=a$ +972,$P_{x+b}-P_x > 0$ +973,$0<\alpha<2$ +974,$p(\delta_p-il_p)$ +975,$1 - \mathsf{Pr}(Z>\mathsf{E} Z)$ +976,"$[a,b]$" +977,"$(valu\x.south east)+(\boundpad,-\boundpad)$" +978,"$\rho(X) = \sup_{\zeta\in A} \langle \zeta, X \rangle$" +979,$P(a) = \nu S(a) + \delta = \nu (S(a) + \rho)$ +980,"$(X,a_2)$" +981,$\mathsf{E}_\mathsf{Q}(Y\mid X)\mathsf{E}(Z\mid X) = \mathsf{E}(YZ \mid X)$ +982,$r=0.045$ +983,$a$ +984,$F(x):=\mathsf{Pr}(X\le x)$ +985,"$C_{1,\cdot}$" +986,$\mathsf{E}_Q(\cdot)$ +987,$g(s)=(s/1-p)^\alpha\wedge 1$ +988,$\omega$ +989,$ = a bond with probability $ +990,$p=0.1$ +991,$26 \rightarrow 2\times 4^2 + 2\times 4 + 1=41 \rightarrow 60 \rightarrow 83 \rightarrow 109\rightarrow\dots$ +992,$x=3$ +993,$p\delta_p/p\nu_p=\iota_p$ +994,$t=0.5$ +995,$c\ge 1/2$ +996,$\mathbb{R}^n$ +997,$\phi(t) = g'(1-t)$ +998,$k0.5$ +1010,$X+\epsilon Y$ +1011,$1-\Phi(x)=\Phi(-x)$ +1012,$>q(p)$ +1013,$k\ge n$ +1014,$\alpha(X_u) = \text{E}[X\mid X > F_u^{-1}(p)]$ +1015,$E(u(X)) \le E(u(Y))$ +1016,$p_n=\mathsf{Pr}(N=n)$ +1017,$\zeta-\zeta_\epsilon$ +1018,$\mathsf{E}_Q(X) =\mathsf{E}(\theta X /\mathsf{E}(\theta))$ +1019,$g(s)g(t)=O(d^2)< g(s)$ +1020,$Y\le a$ +1021,$\zeta\in\partial(X)$ +1022,$\rho(T)$ +1023,$13809$ +1024,$n+2$ +1025,$P(a) = L(a) + \iota (a-P(a)) = \nu L(a) + \delta a$ +1026,$x_1$ +1027,$\sum_j \mathsf{TVaR}_{p_j}(X)m_j$ +1028,$\mathscr{O}(\zeta)=\{\zeta T \mid T\in MPT\}$ +1029,$\rho(1_A) = 1$ +1030,$g'(x)=0$ +1031,$\{X>a\}$ +1032,$\alpha(\cdot)$ +1033,"$h(t)=\int_0^t F_Z^{-1}(1-u)\,du$" +1034,$g''(p)=-\phi'(1-p)\le 0$ +1035,$x = 0$ +1036,$(\bar a_x - \bar a_{\lcroof{b}})/\bar a_x$ +1037,$t=0=1$ +1038,$P=\rho_{PH}(X)$ +1039,$\mathbf{x}'$ +1040,$\mathrm{L}$ +1041,$\mathsf{E}(X) = \mathsf{E}(X\mid X \le a)F(a) + \mathsf{E}(X\mid X > a)S(a)$ +1042,$(1-t)/t$ +1043,$c=1.124$ +1044,"$(Alice) + (0,-4)$" +1045,"$\mathsf{cov}(h^i, Y(\mathbf{X})) = \mathsf{E}_P[h^iY(X)]$" +1046,$0<\alpha_1<\alpha_2<1$ +1047,"$\rho(X,a)=\int_0^a S(x) + \delta(F(x))F(x)dx$" +1048,$l(p)= \nu-\sqrt{p(1-p)}$ +1049,$p-1=22$ +1050,$q_{\cdot}(\mathbf{x})$ +1051,$\cdot$ +1052,$\nabla\rho(X)=\{h\}$ +1053,"$i=0,\dots,n-1$" +1054,"$ is time cheap. Indeed, the condition implies the denominator is $" +1055,$g(\sqrt{st})^2$ +1056,$g(s)g(t)-g(st)$ +1057,$0.475$ +1058,"$(ckey2.north west)+(-\boundpad,\boundpad)$" +1059,$\rho_t(X) = \displaystyle{1}{\beta} \log \mathsf E[e^{-\beta X}\mid \mathscr F_t]$ +1060,$\bar A_{x+b}$ +1061,$\rho(X+\epsilon Y)-\rho(X)$ +1062,$\prec_3$ +1063,$\rho(T)=76.11$ +1064,$\bar R(a)$ +1065,"$4.7\times 10^{21} / 10^{19} = 470 \text{\,seconds} \approx 8\text{mins}$" +1066,$X^{\oplus n}$ +1067,$\sup \{ \mathsf{E}(LZ) \mid Z \preceq \sigma \}$ +1068,"$(Alice)+(0,-2)$" +1069,"$g(s) = \max(g_m, g^0(s))$" +1070,$g'(1)=\alpha < 1$ +1071,"$X_-:=\max(-X,0)$" +1072,$g'(1-s)$ +1073,$X\in \mathcal X$ +1074,$\mathsf{E}_Q(N_i) =$ +1075,$\iff P +\rho_i(F_i) < \rho_i(X_i) \iff P < \rho_i(X_i) - \rho_i(F_i)$ +1076,$g'(S(x))=dQ/dP$ +1077,$G=\sum_i N_i(x_i) + C_i(x_i)$ +1078,$F_X$ +1079,$5 \times 10^9$ +1080,$1-\tilde p$ +1081,$\mathsf{E}_Q=\mathsf{E}$ +1082,$1- \nu F(x)$ +1083,$\delta_p=1-\nu_p=\rho_p\nu_p$ +1084,$X^{\oplus n} -\mathsf E[X] = X^{\oplus n-1} + (X'-\mathsf E[X])$ +1085,$\mathsf{E}[Y]=1$ +1086,"$\langle \mu,Y \rangle - \langle \mu,X \rangle = \langle \mu, Y-X \rangle \ge 0$" +1087,$q(p)=c$ +1088,$\mu$ +1089,$\mathsf{E}(X_ig'(S))$ +1090,$x=0$ +1091,$p\delta_p/p\nu_p=\rho_p$ +1092,$g'(t)=αt^{α-1}$ +1093,$s_u = (f+1) / (n+1)$ +1094,$1-t=g^{-1}(1-s)$ +1095,$\rho(X)=\mathsf{E}_\mathsf{Q}[X]$ +1096,$X=q_X(U)$ +1097,$A=\sum_n 1_{N=n}X^{\oplus n}$ +1098,$i\in I$ +1099,$L_p$ +1100,$\mathsf{CoTVaR}(X_i)$ +1101,$g(st) = \displaystyle\frac{st}{1-p} < \displaystyle\frac{s}{1-p}= g(s)g(t)$ +1102,$\rho(X)=\lim_n \rho(X_n)$ +1103,$\sigma=0.45$ +1104,$\tilde F(x)=\mathsf{Pr}(\tilde X-\lambda\le x-\lambda)=\Phi(x-\lambda)$ +1105,$x_iX_i$ +1106,"$(0,0)$" +1107,$\alpha_i(t)$ +1108,$q_X$ +1109,$g(1)=1$ +1110,$g'(1-s)=\phi(s)$ +1111,"$\mathcal{M}\subset\mathscr{P}[0,1]$" +1112,$34.05$ +1113,$\mathsf{Pr}(X>a)>1-\alpha$ +1114,$k>m$ +1115,$m(x) = \nu S(x) + \delta = \nu (S(a) + \rho)$ +1116,$\sqrt{FS}$ +1117,$P_{x+b}-P_x$ +1118,$c_k$ +1119,"$(X, a)$" +1120,$\mathsf{E}(X_i / X)$ +1121,$ is a measure on $ +1122,$k_i(a) = \phi_i(a) k(a)$ +1123,"$\rho(G(\bar x))=\langle \zeta_{\bar x}, G(\bar x) \rangle$" +1124,$(a-X)^+$ +1125,"$\langle \zeta, G \rangle=\int q_G q_\zeta$" +1126,$g(p)\ge p$ +1127,$\rho(m) = \rho(0) - m$ +1128,"$\mathsf{cov}(X_1, N | G = const_j) f_G(const_j)$" +1129,"$f'_\omega (\bar x, h)$" +1130,$g^a$ +1131,$\mathsf{VaR}$ +1132,$\bar P_{x+b}$ +1133,$L_0^{a-Y}$ +1134,$\sigma=0.25$ +1135,$(\rho)$ +1136,"$\bar P^a(t):=\bar P^a(1-t, t)$" +1137,$\mathsf{E}(X)=\int S(x)dx$ +1138,$a=q_p(\mathbf{x})$ +1139,$a(x)$ +1140,$u^{iv}\le 0$ +1141,$\mathsf{E}(X \mid X\ge q_{1/k}(X))$ +1142,$\bar a_x = (1-\bar A_x)/\delta$ +1143,$(g^{k})^a = K$ +1144,$s=1$ +1145,$X=Y+Z$ +1146,$\le$ +1147,"$(-\x*0.75, -2)$" +1148,$\mathsf{E}(YZ\mid X)=Z\mathsf{E}(Y\mid X)$ +1149,"$X_+:=\max(X,0)$" +1150,$N_i=N_i(x_i)$ +1151,"$50) of the amount allowed on each claim in the classes under subsections (3) to (7), inclusive, of this section, shall be deducted from the claim and included in the class under subsection (9) of this section. Claims may not be cumulated by assignment to avoid application of the fifty dollars ($" +1152,$\bar G'(a)=\frac{d\bar G}{da}=G(a)$ +1153,$\nu(p)<1$ +1154,$\mathsf{E}_q(X_1)$ +1155,$\mathsf{E}(L)$ +1156,$X_c$ +1157,$s_u$ +1158,"$T_{x,\delta}$" +1159,$\phi'(s)=\mu(ds)/(1-s)\ge 0$ +1160,$SD(G')=\nu$ +1161,$x+t$ +1162,"$x=0.5, M=1.5,\sigma=0.75, K=6$" +1163,$g'(1)$ +1164,$\nu(p) F(x)$ +1165,$c_l\mathsf{E}[X_2]$ +1215,$M(0)=1$ +1216,$2\nu$ +1217,$c_k-G\le 0$ +1218,$\forall X\in L^p$ +1219,$B_t$ +1220,$\nu_p$ +1221,$Q(a) = (L-a)V(a) = (L-a)^+$ +1222,$p\nu_p$ +1223,$L_{\sigma_1}\subset L_{\sigma_2}$ +1224,$g(x)=x$ +1225,"$(p,q(p))$" +1226,$S(x)dx$ +1227,$\nabla p$ +1228,$Z_1$ +1229,$\mathsf{E}[X_i(1) \mid X(\mathbf{x}) = q_p(\mathbf{x}) ]$ +1230,$g(S(x))=q(\tilde p)\phi(\tilde p)$ +1231,$r_f$ +1232,$\bar P_{40}=6908.82$ +1233,$\phi(p)$ +1234,$D_i-N_i > 0$ +1235,$A=0.00022$ +1236,"$(X,a_1)$" +1237,$\rho(X)=\int g(S(x))dx$ +1238,$X_i(\mathbf{x}; a)$ +1239,$0.5 R_2(0)$ +1254,$\sigma_\mu(\alpha) = \displaystyle\int_0^\alpha\dfrac{1}{1-u}\mu(du)$ +1255,$\rho(X+Y)\le\rho(X) + \rho(Y)$ +1256,$X \prec_n Y$ +1257,$\phi_i(a)\mathsf{E}(Y\wedge a) = \mathsf{E}(X_i(a))$ +1258,$\rho(X+x)=\rho(X)-x$ +1259,$F:\mathbb{R}^n \to \mathcal{X}$ +1260,$S>0$ +1261,$G = C + \sum_i N_i$ +1262,$\sqrt{2Np}=19$ +1263,"$(fun1a.south -| fun3a.south east)+(\smlspc,-\smlspc)$" +1264,$g'$ +1265,$Y-X\le 0$ +1266,$\rho(X-a)=\rho(X)-a$ +1267,$\mathsf E[F_i]$ +1268,"$750,000,000). The deposit shall be made subject to the approval of the commissioner under those rules and regulations that he or she shall promulgate. The deposit shall be maintained at a deposit value specified by the commissioner, but in any event no less than one hundred thousand dollars ($" +1269,$\mathbb{Q}$ +1270,$f(s) = \alpha(1-\alpha)(1-s)^{\alpha-1}$ +1271,"$\nu \in\mathscr{P}[0,1]$" +1272,$a\ll \sum_i a_i$ +1273,$g^{-1}(x)\le s$ +1274,$\mathsf{TVaR}_p(X)=\frac{1}{1-p}\int_p^1 F_X^{-1}(t)dt$ +1275,$A_1$ +1276,$g_n$ +1277,$\bar R$ +1278,$\mathsf{E}_\mathsf{P}$ +1279,$1/(1-\alpha)$ +1280,$u'''>0$ +1281,$Z_a$ +1282,$t = 1$ +1283,$id\times\tau$ +1284,"$[0.37, 0.55]$" +1285,$B(1/2)$ +1286,"$n=2,3$" +1287,$m(p)=q+p\delta_p$ +1288,$\rho(-X)$ +1289,$X=X_c + X_n$ +1290,$\sigma=0.15$ +1291,$\rho(\cdot)$ +1292,"$[a,a+da]$" +1293,"$(s,t)$" +1294,$g'(0)>1$ +1295,$\le 1$ +1296,$q=1-p$ +1297,$\rho(X)\ge -\rho(-X)\ge a$ +1298,$(\mathsf{E}_q(X_1)-s)/\mathsf{E}_q(X_1)$ +1299,"$200 of losses otherwise payable to any claimant under this subsection. All claims under life insurance policies and annuity contracts, whether for death proceeds, annuity proceeds or investment values, must be treated as loss claims. Claims may not be cumulated by assignment to avoid application of the $" +1300,$X_p =F_X^{-1}(p + (1-p)U_X$ +1301,$X_i(\alpha)$ +1302,$=\mathsf{E}(X_i/X \mid X > a)$ +1303,$N=1$ +1304,"$a\wedge b:=\min(a,b)$" +1305,$t_2-\epsilon$ +1306,"$X_1,X_2$" +1307,$q(1)$ +1308,$\theta<1$ +1309,$\sum_i X_i(a) = X\wedge a$ +1310,$X(T(s))=q(s)$ +1311,$\tpx=\exp(-\int_0^t \mu_{x+s}ds)$ +1312,$H$ +1313,$g^{kS}=R^S$ +1314,$a\mapsto n=g^a\pmod{p}$ +1315,"$(x, g(S(x)))$" +1316,$0 \le \rho(0) = \rho(X-X) \le \rho(X) + \rho(-X)$ +1317,$\bar a_{\lcroof{b}}=(1-v^b)/\delta$ +1318,$CV=\nu=\sqrt{a}\theta$ +1319,$\psi$ +1320,$3.2 \times 10^{18}$ +1321,$a_i=\rho_i(\tilde X_i)$ +1322,$\rho(X-\rho(X))=\rho(X)-\rho(X)=0$ +1323,$v$ +1324,$\lambda_{x+t}=\lambda\mu_{x+t}$ +1325,$\rho(X + \rho(X))=0$ +1326,$\lambda=(1-\alpha_p)^{-1}$ +1327,$\backslash$ +1328,$\delta=\iota\nu$ +1329,$\mathsf{E}[X_2]$ +1330,$\rho(xX)=x\rho(X)$ +1331,$R_1(t) = \bar P^a_1(t)/(1-t)$ +1332,$g^{ak}=(g^k)^a$ +1333,$f(0)=0$ +1334,"$(fun5.north east)+(\medspc,\medspc)$" +1335,$p = 1-g^{-1}(1-\bar p)$ +1336,$1-p$ +1337,$C_1$ +1338,$x<\mathsf{VaR}_p(X)$ +1339,$μ = δ_α$ +1340,"$P_c, P_n$" +1341,$g(s) =$ +1342,$\rho_\phi$ +1343,$\rho_\min(L_i)=\rho_i(L_i)$ +1344,$\mathsf{E}(X_i \mid X=x)$ +1345,$g(s)=s^{2/3}$ +1346,$\epsilon(\mathsf{E}_q(X_1)-s)$ +1347,$\sigma\in L_q$ +1348,$a\ge \psi(X)$ +1349,$l_p=\nu_p-\nu_{1/2}\sqrt{\bar p}$ +1350,"$(N,m)$" +1351,$s=0$ +1352,$x^∗$ +1353,$C_t$ +1354,$\mathsf{E}(X_i\mid X=x)$ +1355,$i=1$ +1356,$\tau_n$ +1357,"$200 of losses otherwise payable to any claimant under this subsection other than the federal government. All claims under life insurance and annuity policies, whether for death proceeds, annuity proceeds or investment values, shall be treated as loss claims. Claims may not be cumulated by assignment to avoid application of the $" +1358,$G=f+G'$ +1359,$-\partial g(S(x))/\partial x$ +1360,$\mathcal X^\perp$ +1361,"$\mathsf{E}_P[h_0]=\mathsf{E}_P[h_{i,\epsilon}]=1$" +1362,"$EL_a =\mathsf{Pr}(Y>a) = \mathsf{Pr}(\max(X_1, \dots, X_N)>a)=\mathsf{Pr}(\text{one or more events $" +1363,$\mathsf{E}_\mathbb{Q}$ +1364,$\rho(0) = 0$ +1365,$xf_i(x)$ +1366,$\delta \ge 0$ +1367,$Z'=ZT$ +1368,$X \preceq_{sl} Y$ +1369,$q(p)=F^{-1}(p)=\mathsf{VaR}_p(X)$ +1370,$A=X_1 + \cdots X_N$ +1371,$x\mapsto |x|$ +1372,${}^1S^{-1}=S^{-1}$ +1373,$m$ +1374,$f$ +1375,$g(s)=1$ +1376,$\mathsf{E}[X_1]=\mathsf{E}[X_2]$ +1377,$1-EL$ +1378,$100$ +1379,$C_k$ +1380,$COC = (P-L) / Q$ +1381,$\mathsf{E}_Q(X \mid \mathcal{G})\mathsf{E}(Z \mid \mathcal{G}) = E(XZ \mid \mathcal{G})$ +1382,$c=\sup_{0\le\alpha<1} \dfrac{\int_\alpha^1 \sigma_2}{\int_\alpha^1 \sigma_1}$ +1383,"$\mathcal{M}_{X,r_X}=\{m \in\mathcal{M} \mid \rho_m(X) = r_X \}$" +1384,$\mathsf{E}_\mathbb{Q}(X_i) = \mathsf{E}_\mathbb{Q}(\mathsf{E}_\mathbb{Q}(X_i \mid X)) = \mathsf{E}_\mathbb{Q}(\mathsf{E}(X_i \mid X))$ +1385,"$ ""the standard way to obtain the $" +1386,$\rho(X)=\mathsf{E}[hX]$ +1387,$R(a)$ +1388,"$f(x, \cdot)\in L_p(\Omega, \mathcal{F}, \mathcal{P})$" +1389,$\pi'(\sqrt k)=0$ +1390,$\rho_{m'}(Y) < 89$ +1391,$i>0$ +1392,$(L^t)^+$ +1393,$P(x) = \sum_i P_i(x)$ +1394,$\dots$ +1395,$X=X_+-X_-$ +1396,$\mathsf{Var}(\pi)=\bar p/(\nu_p-l_p)^2$ +1397,$q_X(p)$ +1398,$a=a(f)$ +1399,$(1-\alpha)^{-1} \min_c c(1-\alpha) + \mathsf{E}(X-c)_+$ +1400,$d=i/(1+i)$ +1401,$\nu(p)$ +1402,"$(rep.south) + (0.5, -2.70)$" +1403,$\mathsf{Pr}(Z>\mathsf{E}(Z))$ +1404,$r=50$ +1405,$\inf_\eta \{ \eta + \phi(X_\eta) \}$ +1406,$X+tY$ +1407,"$p_1, \dots, p_N$" +1408,$\text{Var}(G)=a\theta^2$ +1409,$r=3$ +1410,$Var(G) = a\theta^2$ +1411,$\delta F$ +1412,"$P(X) = M(X, \psi(X))$" +1413,$a\ge 0$ +1414,$X(p)=F^{-1}(p)$ +1415,$K = (A)^{b} = g^{ab}$ +1416,$YN$ +1417,$\bar P_{75}=53123.19$ +1418,$x\to\infty$ +1419,$m_1 / r_1 > m_2 / r_2$ +1420,$0.1$ +1421,$\Delta \tilde p< \Delta p$ +1422,$l_p=0$ +1423,$X_i(u_i)$ +1424,$k>0$ +1425,$\mathsf{E}(L) = F^{-1}(p) dp$ +1426,$X_i(a)=(X\wedge a)X_i/X$ +1427,$\rho_t(X)$ +1428,$1-l-(\nu-l)=\delta$ +1429,$Q_\epsilon \to Q$ +1430,$ +1431,$\rho(0X)=\rho(0)=0\rho(X)=0$ +1432,$r_X=\mathsf{TVaR}_p(X)$ +1433,$. If the insurer has a single insured there is no notion of default: the insured has purchased a policy covering losses up to a limit $ +1434,$R_1(t)$ +1435,$X=q=F^{-1}$ +1436,$Q(a)$ +1437,$q_2(t)=t^2$ +1438,$\mathcal{A}$ +1439,$F:\mathbb{R}^n\to\mathcal{X}^n$ +1440,$\eta\ge$ +1441,"$\subset [\essinf X ,\esssup X]$" +1442,$a'=\mathsf{E}[X|A^c]$ +1443,"$1,2,3,\dots$" +1444,$g\circ S$ +1445,$2\square^2 + 2\square + 2$ +1446,$L_p dp$ +1447,"$A_k=X_{k,1} + \cdots + X_{k, N}$" +1448,$X_n\uparrow 0$ +1449,$\mathsf{Pr}(\mathsf B(s)=1)=s$ +1450,$C_1(t)=C_2(t)=\bar P^a(t)$ +1451,$Q(a) = 1 - P(a) = 1 - g(S(a))$ +1452,"$\rho(X)=\sup\{ \mathsf{E}(XZ) \mid Z\ge 0, \mathsf{E}(Z)=1, \mathsf{E}(Z\log(Z))\le\log(1/(1-\alpha)) \}$" +1453,$C_i$ +1454,$\bar Q(a)$ +1455,$\bar P_i$ +1456,$\mathsf{E}(X_i \mid G=q)=:\mathsf{E}_q(X_i)$ +1457,"$\mathsf{E}[XZ] = \mathsf{cov}(X,Z) \le \sigma(X)\sigma(Z)\le \sigma(X)$" +1458,$\nu=1/(1+\rho)$ +1459,$\mathscr{P}=\{ (1-p)^{-1}1_A \mid P(A)\le 1-p \}$ +1460,$\phi(x)=-\int_x^1 (s-x)^{n-1}d\tau(s)$ +1461,$m(x)=S(x)+d_iF(x)+(v-\nu^*)\sqrt{F(x)S(x)}$ +1462,$\partial B$ +1463,$\mathsf B(s)$ +1464,$t^*$ +1465,"$X,Y,X+Y$" +1466,$a=(X\wedge a) + (a-X)^+$ +1467,"$(rep.south) + (0.5, -1.85)$" +1468,$ for $ +1469,$L_a^{a+y}$ +1470,"$(\sqrt{st}, \sqrt{st})$" +1471,$\sum_{n\ge 0} 1_{N>n} X_n$ +1472,"$X\wedge a =\min(X,a)$" +1473,$\mathsf{TVaR}_p(X)$ +1474,"$L_{p,\delta}(\omega)=\begin{cases} q(p) & \omega\in (p,p+\delta] \\ 0 & \omega\not\in (p, p+\delta]\end{cases}$" +1475,$\mathbb{R}\times \mathbb{R}$ +1476,$\beta_i(t)/\alpha_i(t)> 1 > g(S(t)) / S(t)$ +1477,$\ge 5000 / \text{Probability}$ +1478,$\rho(A_k)\ge \mathsf{E}[A_k] = k\mathsf{E}[N]$ +1479,$1 \times 10^{15}$ +1480,$q\phi$ +1481,"$R_i=\alpha p_i + \beta r_{qp,i} + \gamma\, \text{controls}_i$" +1482,$CV(G) = SD(G') = \nu$ +1483,$+$ +1484,$\eta=(1-\alpha)^{-1}1_A$ +1485,$E(X^k)=E(Y^k)$ +1486,$2 \times 10^{14}$ +1487,$a=a(\mathbf{x})$ +1488,$a=a(x)$ +1489,"$g\in D_n^*=\{ g \mid (-1)^{k+1} g^{(k)} \ge 0, k=1,\dots,n-1, (-1)^n g^{(n-1)}\text{ nonincreasing} \}$" +1490,$\log(1-\Phi(x))$ +1491,$S(x_1)-S(x_2)\approx f(x_1)(x_2-x_1)$ +1492,$\zeta_t\to\zeta$ +1493,$R_1(t) C_2(t)$ +1515,$ is $ +1516,$\mathcal A_\rho= \{ X\mid \rho(X)\le 0 \}$ +1517,$X \prec_n^* Y$ +1518,$\nu F(a)$ +1519,$\mathsf{E}(L)=\int_0^\infty S(x)dx$ +1520,$K_Q=19.473$ +1521,$X=X_i + \hat X_i$ +1522,"$500/year HO insurance then I don't really notice it compared to upkeep, mortgage, property tax etc. It is just a sunk cost. But if I pay $" +1523,$ and investor equity $ +1524,$(x-a)_+^\alpha$ +1525,$r_{pq}$ +1526,$\mathsf{E}$ +1527,$c\le a$ +1528,$g(s)g(k/s)$ +1529,$\phi(1-p)=g'(p)$ +1530,$k= \mathsf{E}(X\wedge k) + (\rho_m(X) - \mathsf{E}(X\wedge k)) + (k-\rho_m(X))$ +1531,"$\langle \zeta_{\bar x}, N_i \rangle$" +1532,$Z>\mathsf{E} Z$ +1533,$\int_0^1 dp$ +1534,$\Bbb{Q}$ +1535,$T_A$ +1536,$E_\mathsf{Q}(X_i\mid X)=E(X_i\mid X)$ +1537,$\beta=0$ +1538,$O(dt)$ +1539,$V=m(L(1+e)P+rS) + (eL+\rho S)$ +1540,$01$ +1545,$\displaystyle\int_0^1 \text{AVaR}_\alpha(X)d\alpha$ +1546,$\rho_m$ +1547,$b_i$ +1548,$\mu_{x+t}$ +1549,${}_tp_x=\mathsf{Pr}(T_x > t) =\mathsf{Pr}(T_0 > x+t \mid T_0 > x)$ +1550,$\mathsf{P}(B)=0$ +1551,"$m_j=m([p_{j-1},p_j])$" +1552,"$(0,\dots,0,r_0,\dots, r_k)$" +1553,$\| X_n \|_\infty \le 1$ +1554,$dF=-d(g\circ S)=$ +1555,"$\rho(X+tY)=\langle \zeta_t, X+tY \rangle$" +1556,$\pi'(k)=...$ +1557,$g:\text{thin layer risk}\mapsto\text{price}$ +1558,$(x-\mu_x)^+$ +1559,"$(\mathsf{E}_q(X_1)(1-\epsilon\mathsf{E}_q(X_2)/q), \mathsf{E}_q(X_2)(1+\epsilon \mathsf{E}_q(X_1)/q))$" +1560,$5 \times 10^{14}$ +1561,$\rho(Z)=\int_0^1\eta(\tau)\mathsf{VaR}_\tau(Z)d\tau$ +1562,"$ xx billion, of which California workers compensation deposits account for $" +1563,$-\int xd(g\circ S)=\int g(S(x))dx$ +1564,$2$ +1565,"$(p,q(1-g^{-1}(1-p)))$" +1566,$S(a)da$ +1567,$\partial Y/\partial x_i$ +1568,$\sum_i F_i=F$ +1569,$\mathsf{E}(X) + c\mathsf{E}(| X-\mathsf{E}(X) |^p)^{1/p}$ +1570,"$\mathcal X^\perp = \{X\in\mathcal X\mid \exists U\text{ uniform[0,1] rv independent of } X\}$" +1571,$\alpha(X)$ +1572,$\bar A^{1}_{x:\lcroof{n}}$ +1573,$\mathsf{TVaR}_{p_2}(X)\ge r$ +1574,$\mathsf{TVaR}_p(X)=$ +1575,$g(s)=s^{1/4}$ +1576,"$\rho(X+tY)\ge \rho(X) + \langle \zeta, tY \rangle$" +1577,$X_n\to X$ +1578,$\rho(X - b)=\rho(X)-b\le 0$ +1579,$t=2$ +1580,$Q\in \partial\rho(X)$ +1581,$g=\mathsf{E}(G^3)=\nu^3 skew(G')+3c+1$ +1582,$375-185=190 > 0$ +1583,"$C_1(t) < \bar P^a(1, 0)$" +1584,"$i=1,2$" +1585,$\partial\rho(Z)$ +1586,$\rho(L) = q(1-g{-1}(1-p))\delta > \mathsf{E}(L)$ +1587,$\rho(p)$ +1588,$1-\delta\bar a_{x:\lcroof{n}}-\bar A_{x:\lcroof{n}}=0$ +1589,$\theta=(1-f)/a$ +1590,$\mathsf{Var}(B(p))=p(1-p)$ +1591,"$p\in[0,1]$" +1592,$\mathsf{COH}+\mathsf{FAT}$ +1593,$=E(X_i \mid X \ge a)$ +1594,$\zeta$ +1595,"$\mathcal{M}_{X,r}=\mathsf{var}nothing$" +1596,$\rho(X\mid \mathcal F_1) =\mathsf E[X g'\mathsf{Pr}(X>x\mid \mathcal F_1) ]$ +1597,$\alpha=d_i$ +1598,$\{ \zeta>0 \} = \{ G>c(x) \}$ +1599,$(v-\nu^*)\sqrt{FS}$ +1600,$\mathsf{TVaR}_{p=1}=\esssup$ +1601,$F_i = X_i(1 - (X\wedge a) / X)$ +1602,$t>0.25$ +1603,$X^∗_i = (X − x^∗)I_{A^∗_i} + x^∗ / n$ +1604,$H_k=H_{g_k}$ +1605,$\lambda\mu_t$ +1606,"$(Bob) + (0,-4)$" +1607,$1 assets: $ +1608,$\sum_i P_i(a)=P(a)$ +1609,$\rho GF$ +1610,"$\rho=0.5, x=1.5, M=1.5,\sigma=0.75, K=8$" +1611,$q_Z$ +1612,"$\langle \mu,tX \rangle - \rho(tX) =t(\langle \mu,X \rangle - \rho(X))$" +1613,$^{*}$ +1614,$\hat p$ +1615,$\delta(F(x))=\delta$ +1616,$L_x^{x+dx}=L_0^{x+dx} - L_0^x$ +1617,$M(a)$ +1618,$\alpha < 1$ +1619,$a-X\le 0$ +1620,$>0$ +1621,$\tilde \rho(X)=\mathsf{E}(X) + \inf_t \rho(X-t)$ +1622,$Y\circ T=g(X\circ T)$ +1623,$\mathsf{E}[X_1]$ +1624,$\rho(X)=-U(X)$ +1625,$-\epsilon(\mathsf{E}_q(X_2)-s)$ +1626,$E_2=0$ +1627,$\mu_{x+t}=-\dfrac{d}{dt}\log({}_tp_x)$ +1628,$a\mapsto g^a \pmod{p}$ +1629,"$(fun1a.south -| fun5a.east)+(\smlspc,-\smlspc)$" +1630,$10^{16}$ +1631,$X=X(x_i)=\sum_i X_i(x_i)$ +1632,$t \le 1-p$ +1633,$\rho(X+c)=\rho(X) + c$ +1634,$h\in\mathscr P$ +1635,$il$ +1636,$697.6 billion underlying Table \ref{tab-equity-what-if} this implies $ +1637,$q=S(a)$ +1638,$\rho(0)=0$ +1639,$Q_\epsilon$ +1640,$k_i=\mathsf{E}_Q(X_i)$ +1641,$\rho(X)\ge\rho(X+Y)\ge \rho(X)+\mathsf{E}[gY]$ +1642,$\rho(A)\le \rho(N)\rho(X)$ +1643,$k>\max(N)\max(|X|)$ +1644,"$\bar P^a(1,0)<\bar P^a(0,1)$" +1645,$st \le 1-p < s$ +1646,$X-\sum f_i(X)$ +1647,$\bar P_x = (1/\bar a_x)-\delta$ +1648,$\beta=v-\nu^*$ +1649,$\mathscr{F}$ +1650,"$310 billion in premiums annually in California. Since 2011 the California Department of Insurance received more than 1,000,000 calls from consumers and helped recover over $" +1651,$d_i=iv=i/(1+i)$ +1652,$\sigma=0.35$ +1653,$t=0.37$ +1654,$R_2(t)a$ +1656,$X(t)$ +1657,"$(4-\s, \s)$" +1658,$1 excess attachment $ +1659,$f(\alpha):=\mathsf{E}[X^\alpha-Y^\alpha]$ +1660,$t=1-g(0)=1$ +1661,"$x=0.1, M=1.5,\sigma=0.75, K=6$" +1662,$\partial\rho(X)=\{\zeta\}$ +1663,$t>t_2$ +1664,$x\ge 0$ +1665,$Q(a)=\nu N(a)$ +1666,$(3+2)/2=5/2$ +1667,$\displaystyle\int_0^\infty xg'(1-F(x))f(x)dx = -xg(S(x))\vert_0^\infty + \displaystyle\int_0^\infty g(S(x))dx=\displaystyle\int_0^\infty g(S(x))dx$ +1668,"$(K=g^k, mg^{ak})$" +1669,$kN$ +1670,$\mathsf{E}_Q(X \mid \mathcal{G}) = E(X \mid \mathcal{G})$ +1671,$F(x)$ +1672,"$[l_c, r_c)$" +1673,$\mathsf{Var}(B(p)/p\nu_p)=p(1-p)/(p\nu_p)^2$ +1674,$F(a)=p$ +1675,$\mathsf{E}[x_iX_i\mid X(\mathbf{x}) \le a]F_{\mathbf{x}};a) = \mathsf{E}[x_iX_i 1_{X(\mathbf{x}) \le a}]$ +1676,$(-1)^nf^{(n)}(x)<0$ +1677,"$h^i = \lim_{\epsilon\downarrow 0}(h_{i,\epsilon}-h_0)/\epsilon$" +1678,$Z_1=q_Z(U)$ +1679,"$[1,2]$" +1680,$\approx 10^{-40}$ +1681,$\hat\rho(A_k) =\rho(\rho((X+k)^{\oplus N})) = \rho(\rho(X^{\oplus N})+kN)= \hat\rho(A_0) + k\rho(N)$ +1682,$\tau=0.156$ +1683,$\mathsf{E}_\mathsf{Q}(X)$ +1684,$f_G$ +1685,"$424) for the initial filing of each letter of credit utilized pursuant to subdivision (a). In addition, the commissioner shall require payment, in advance, of a fee of two hundred eighty-three dollars ($" +1686,$\displaystyle\int_0^\infty xdF(x)$ +1687,"$(4.5-\s, \s)$" +1688,$g(s) = t_{df}(\Phi^{-1}(s)+\lambda)$ +1689,$B-p(\nu(p) + il(p))$ +1690,"$R, S$" +1691,$a = b$ +1692,$\nabla \zeta=0$ +1693,"$X\sim\text{Lognormal}(\mu=19.9, \sigma=2.36)$" +1694,$\sqrt{F(x)S(x)}$ +1695,$\rho(X)=35/9$ +1696,$X(p)$ +1697,"$\langle X(\epsilon),\zeta_\epsilon \rangle-\langle X,\zeta \rangle=\langle X(\epsilon)-X,\zeta \rangle$" +1698,$\rho_{t+1}(X)=\rho_{t+1}(Y)\implies \rho_{t}(X)=\rho_{t}(Y)$ +1699,$\bar P_x:=\bar A_x / \bar a_x$ +1700,$p=0.5$ +1701,"$(\Omega, \mathcal{F}, \mathbb{P})$" +1702,$l\ge 1$ +1703,$X(\omega)=$ +1704,$g(st) = 1= g(s)g(t)$ +1705,$\int_x^\infty$ +1706,$p=F(a)=1-q$ +1707,$\bar S$ +1708,"$(ckey\x.north west)+(-\boundpad,\boundpad)$" +1709,$\rho_{t+1}(X) = \rho_{t+1}(Y) \implies \rho_{t}(X) = \rho_{t}(Y)$ +1710,$\rho_\phi=\mathsf{E}$ +1711,$\rho(X)=\int_\Omega X(\omega)\theta(\omega)dP(\omega)$ +1712,$B(0.5)$ +1713,$U\subset\Bbb{R}^n$ +1714,$a(x) = \sum_i x_i a_i = \sum_i x_i v_i a$ +1715,$\phi(p)dp$ +1716,$\gamma$ +1717,"$p\in (0, 1)$" +1718,$ since $ +1719,$p\mapsto q(\hat p)=q(1-g^{-1}(1-p))$ +1720,$S =$ +1721,$p(x)$ +1722,$H(x)=y$ +1723,$x\mapsto \mathsf E[f(X_2)\mid X_1=x]$ +1724,$B(b)>0$ +1725,$\mathsf E[X^{\oplus n}]\le\rho(X^{\oplus n})$ +1726,$g(st) = \displaystyle\frac{st}{1-p} < 1 = g(s)g(t)$ +1727,$\pi_X(t_{2j-1})\le \pi_Y(t_{2j-1})$ +1728,$ϕ$ +1729,"$i=1,\dots, n_r$" +1730,$\mathsf PV$ +1731,$\le 1/(1-\alpha)$ +1732,$A \hat\rho(A)=4.8125$ +1755,$\nabla_y f=-\nabla_y G$ +1756,$\| f^*-f\|_2$ +1757,$\iota(0.5)=\iota^*$ +1758,$\rho_{t+1}(X) \ge \rho_{t+1}(Y) \implies \rho_{t}(X) \ge \rho_{t}(Y)$ +1759,$\rho(X_1\mid \mathcal F_1)\le \rho(X_2\mid \mathcal F_1)$ +1760,$\mathsf{E}(X|X\ge a)$ +1761,$ and $ +1762,$L_0^a$ +1763,$\rho(X)=\int_0^1 \mathsf{TVaR}_p(X)m(dp)$ +1764,$g(S(x))\to d$ +1765,$0.1525$ +1766,$l$ +1767,$U=X$ +1768,$\rho_m(X)=r$ +1769,$=1.75$ +1770,$\rho(X) = \max \{ \rho_\phi(X) \mid \phi\in A \}$ +1771,$\zeta\in\mathscr{P}$ +1772,$\rho$ +1773,$Z_i$ +1774,$x=q(p)$ +1775,$\rho(-1_{A^c}) = c < 0$ +1776,$\delta(p)=1-\nu(p)=d+(\delta^*-d)\sqrt{(1-p)/p}$ +1777,$\mathsf{E}[Z_1]=1$ +1778,"$X_t=1_{[1,\infty)}$" +1779,$N\sim\text{Poisson}(1.74)$ +1780,$M(a)=\mathsf{E}(X\wedge a)+dN(a)+(\delta^*-d)\displaystyle\int_0^a \sqrt{F(x)S(x)}dx$ +1781,$c=\mathsf{VaR}$ +1782,"$L^\infty(a, b)$" +1783,$dp$ +1784,$\tilde p=\tilde F(F^{-1}(p))=1-\tilde S(F^{-1}(p))=1-g(S(F^{-1}(p)))=1-g(1-F(F^{-1}(p)))=1-g(1-p)$ +1785,$D_i-N_i$ +1786,$1-t=g(1-s)$ +1787,$\dfrac{d}{dx}g(S(x))=-g'(S(x))f(x)$ +1788,$\mathsf{Pr}(Y\le a)=\exp(-\lambda (1-F(x)))=\exp(\lambda (\int_0^x f(s)ds -1))$ +1789,$0.06333 / 247.798 = 0.026\%$ +1790,$X_n$ +1791,$dx$ +1792,$_1$ +1793,$S_i$ +1794,$\mathsf{E}(X_i/X)$ +1795,$g(p)$ +1796,$g(s)=\displaystyle\frac{s}{1-p}\wedge 1$ +1797,$\mathsf{E}(X\wedge a)=\int_0^a S(x)dx$ +1798,$\mathscr P$ +1799,$})$ +1800,$\bar\delta=\bar\iota\bar\nu$ +1801,$g'(1-p)$ +1802,$k$ +1803,$J$ +1804,$\hat\rho(A)\ge \rho(A)$ +1805,$t=b$ +1806,"$x=4, M=1.5,\sigma=0.75, K=6$" +1807,$\delta=\rho\nu$ +1808,$E(X_i \mid X=a)$ +1809,$c\ge \mathsf{E}[cg]$ +1810,$ϕ(1-t)=g'(t)$ +1811,$\rho:\mathcal{X}\to \mathbb{R}$ +1812,$q_{Z_k}$ +1813,$\rho=\rho_\gamma$ +1814,$T^{-1}$ +1815,$X(p)=q(p)$ +1816,$\\leftrightarrow$ +1817,$F(x_1)=1-S(x_1)=p$ +1818,$V(c)=0$ +1819,$\bar P_1$ +1820,$X_i$ +1821,$\mathsf{E}(X)=\sum_i x_i$ +1822,$a>1$ +1823,$(\delta^*-d)\sqrt{FS}$ +1824,$\mathsf{ABOVE}$ +1825,$C_i(t^*)=R_i(t^*)$ +1826,$T_n$ +1827,$\text{E}(G)=M_G'(0)=1$ +1828,$pl(p)$ +1829,$P(A)=1-\alpha$ +1830,$\mathsf{E}(L) = F^{-1}(p)dp$ +1831,"$\rho(X) = \sup_{\mu\in \mathcal{A}} \langle \mu, X \rangle$" +1832,$\bar P(x+dx) - \bar P(x)$ +1833,"$a=98,99,\dots,104$" +1834,$F^-1$ +1835,$E_\mathsf{Q}(X_i)= E_\mathsf{Q}(E(X_i \mid X))$ +1836,$\hat\rho_N$ +1837,"$a,b=\pm 1/n$" +1838,$N\times r$ +1839,$U(x)$ +1840,$p=0.99$ +1841,$g(t) = \mathsf E[u(X-\pi(R+tQ) +R+tQ)]$ +1842,"$\mathbf{X}=(X_1,\dots,X_n)$" +1843,$\rho_m(Y)$ +1844,$2\square^2 + 2\square - 1$ +1845,$\bar P^a(t)$ +1846,$q(\hat p)$ +1847,"$g(0)=0,\ g(1)=1$" +1848,$\Leftrightarrow$ +1849,$\delta_p/\nu_p = \iota_p$ +1850,$100\cdot (1-g(s))$ +1851,$\delta=\iota/(1+\iota)$ +1852,$\bar X\ge 0$ +1853,$1-g(s)$ +1854,"$X,Y$" +1855,$(g)$ +1856,$\mathscr{P} = \{P\}$ +1857,$\displaystyle\int_0^\infty xg'(S(x))f(x)dx$ +1858,$P'$ +1859,$\displaystyle\int_0^\infty xf(x)dx$ +1860,$Y\le 0$ +1861,$0\le\beta\le \gamma\le 1$ +1862,$\tilde S(x)=g(S(x))$ +1863,$\rho_{t+1}(X)\le\rho_{t+1}(Y)$ +1864,$N=365$ +1865,$b\le 1$ +1866,$g^a=g^{\log_g(n)}=n$ +1867,"$(2,-\x*0.75)$" +1868,$r_X$ +1869,$\min_{\eta\in \mathbb{R}} \eta + \alpha \mathsf{E}(X-\eta)_+ -\beta\mathsf{E}(X-\eta)_-$ +1870,$\bar P_x$ +1871,$T_s(p) = \mathsf{TVaR}_p(s)$ +1872,$\bar A_{x:\lcroof{n}} = \bar A^{1}_{x:\lcroof{n}} + e^{-\delta n}{}_np_x$ +1873,"$\partial \rho(X)=argmax_{\zeta\in A} \langle \zeta, X \rangle$" +1874,$=\mathsf{E}(X \mid X\le a)$ +1875,$p_i(a)=\phi_i(a)p(a)$ +1876,$\mathsf{E}(X_i(a))$ +1877,$Y$ +1878,"$f_x(x_i, \hat x_i) = f(x_i, \hat x_i) / f_X(x)$" +1879,"$\mathbf{x}=(1-t, t)$" +1880,$\mu\in \mathscr{P}$ +1881,$0 \le f'(z) \le 1$ +1882,$p=0. $ +1883,$\bar Z = F(\bar x)$ +1884,"$[0,1]\to [0,1]\times [0,1]$" +1885,$2.592 \times 10^{16}$ +1886,$u_i$ +1887,$\zeta_t$ +1888,$\rho = AVaR$ +1889,$X(u)=X_1(u_1) + X_2(u_2)$ +1890,$E2$ +1891,$g'(0)$ +1892,$ at $ +1893,$1/(1+r_f)$ +1894,$\le a$ +1895,$f(x)dx = dp$ +1896,$\mathsf{E}(X)=$ +1897,$X_3$ +1898,$g'(S(x))$ +1899,"$(Alice) + (0,-3.75)$" +1900,$x=q(1-g^{-1}(1-\tilde p))$ +1901,$d=iv=i/(1+i)$ +1902,$m =$ +1903,$\tau_\sigma(\alpha) = \int_\alpha^1 \sigma$ +1904,$\rho(-1_{B_l}) \le \rho(-1_{B_r})$ +1905,$(g(s)-s)/(1-g(s))$ +1906,"$p\in [0,1]$" +1907,$\rho_{(g)}$ +1908,$X^{\oplus 2}$ +1909,"$(\Omega, \mathcal{F}, \mathsf{P})$" +1910,"$[l_i, r_i)$" +1911,$(1-X)^+$ +1912,$A=\sum_i I_iX_i$ +1913,$\sup\{ \mathsf{E}[Y\sigma(U)] \mid U\text{\ uniform} \}$ +1914,$X>F_u^{-1}(p)$ +1915,$R_2(t)= \bar P^a_2(t)/t$ +1916,$d\tilde p/dp = g'(1-p)=\tilde f(F^{-1}(p))/f(F^{-1}(p))$ +1917,$\rho(X) = \mathsf{E}(X) + c\mathsf{E}( |X-\mathsf{E}(X)|^p)^{1/p}$ +1918,"$30,000 per accident up to $" +1919,$\sigma(X)$ +1920,$A^c\supset A_1\supset A_2\supset \dots$ +1921,$C > cx/a$ +1922,$\omega < 1/n$ +1923,$\phi_W(a)=\mathsf{E}(W/Y \mid Y>a)$ +1924,$\mathsf{E}(X_i/X \mid X > a)$ +1925,$q_L(\tau_\sigma^{-1}(U)$ +1926,$4.7\times 10^{21} / 10^{19} \approx 8\text{mins}$ +1927,"$\mathsf{E}(\min(X_i,a))=\mathsf{E}(X_i\wedge a)$" +1928,$v=1/(1+i)$ +1929,$\tau_\sigma(p)=\int_0^p\sigma(u)du$ +1930,"$50 of the amount allowed on each claim in the classes under paragraphs II, V, and VI except claims of the guaranty associations as defined in RSA 404-B, 404-H, 404-D, and 408-B shall be deducted from the claim. Claims may not be cumulated by assignment to avoid application of the $" +1931,$(p-\nu)/\nu$ +1932,"$50.00) of the amount allowed on each property, casualty or fidelity claim in the classes under Subsections B through F of this section, shall be deducted from the claim and included in the class under Subsection I of this section. Claims may not be cumulated by assignment to avoid application of the fifty dollar ($" +1933,$r=0.038$ +1934,"$X_1(x_1), \dots, X_n(x_n)$" +1935,$ into aggregate premiums $ +1936,"$u_0,u_1,\dots,u_k$" +1937,"$S(1-t,t;x)$" +1938,$\mathsf{E}[gY]\le 0$ +1939,$\mathsf{TVaR}_0(\cdot)=\mathsf{E}[\cdot]$ +1940,$\mathsf{E}(X-c_l)_+$ +1941,$P(a)=\mathsf{E}(Y\wedge a)+\rho K(a)$ +1942,$\iota(p)$ +1943,${}_b\bar V$ +1944,$X_i=q(p_i)$ +1945,$x_1q}$ +1952,$Z=d\mathbb{Q}/d\mathbb{P}$ +1953,$Z^* = \sum_i \alpha_i Z\circ T_i$ +1954,$X(t):=X(\mathbf{x})=(1-t)X_1 + tX_2$ +1955,"$(ccc.south |- mcc.south)+(0,-0.5)$" +1956,$\sum t_i=\infty$ +1957,"$(fun1a.south -| fun2a.east)+(\smlspc,-\smlspc)$" +1958,$1_D$ +1959,$\rho(X)=\mathsf{E}_\mathsf{Q}(X)$ +1960,"$T_{700,100}$" +1961,$< 1$ +1962,$t=q-s$ +1963,$0$ +1964,$M_X(k)=M_Y(k)$ +1965,$\{ X=a \}$ +1966,$a = M(a)+Q(a)= \mathsf{E}(X\wedge a) + \delta N(a) + \nu N(a)$ +1967,"$[p, p+dp]$" +1968,$(v-\nu^*)\sqrt{pq}=$ +1969,$(X−x^∗)I_{B_i}$ +1970,$r=g^k$ +1971,$n=g^a\pmod{p} \mapsto a=\log_g(a)$ +1972,$10^{13}$ +1973,$\gamma = 2/\sqrt(a) = 2\nu$ +1974,$\sigma=1$ +1975,$0\le \tau\le 1$ +1976,"$(fun2.north west)+(-\smlspc,\smlspc)$" +1977,$\rho_g$ +1978,$\mathsf{E}(X) = E(X_i \mid X\le a)F(a) + E(X_i \mid X > a)S(a)$ +1979,$\alpha>1$ +1980,"$b \in_{R} \{2,\dots,p-2\}$" +1981,$N\times 1$ +1982,$g$ +1983,"$(Bob)+(0,-2.5)$" +1984,$\alpha=\text{E}[X \mid X > F_u^{-1}(p)]$ +1985,"$(B.north east) + (-0.07mm,0)$" +1986,$\mathsf{E}[Y]$ +1987,$\mathsf{E}[X^k]=\mathsf{E}[Y^k]$ +1988,$\mathsf E[X]\rho(N) \le \rho(A)$ +1989,$\sigma$ +1990,$C_2$ +1991,$S(a)=1-p$ +1992,$\nu=\nu(F(a))=\nu(p)$ +1993,$\tau_\sigma(p)=\int_0^p \sigma$ +1994,$100\cdot g(s)$ +1995,$\phi(1)\le 1$ +1996,$\mathsf{E}(X)=0$ +1997,$\mathsf{E}(X_i\mid X=x)f_X(x)/x$ +1998,$\mathbf{T}^+\mathbf{r}$ +1999,$\mathsf{E}[Y\tilde W] = n^{-1}\sum_T \mathsf{E}[Y \cdot W\circ T] = n^{-1}\sum \mathsf{E}[Y\circ T^{-1} \cdot W] = \mathsf{E}[YW]$ +2000,$\mathsf{Q}_1$ +2001,$D_i$ +2002,"$(Bob) + (0,-1)$" +2003,$-1\le X_n\le 0$ +2004,$[F(x)](\cdot)$ +2005,$g_k(s) = 1-(1-s)^k$ +2006,$10^{15}$ +2007,$P_i(a)=\phi_i(a)P(a)$ +2008,$F^{-1}(1-g^{-1}(1-p))$ +2009,"$(Alice) + (0,-1)$" +2010,$\mathsf{Pr}(X>a)=S(a)$ +2011,$b\le a$ +2012,$\tilde\rho(X) = \mathsf{E}_Q(Y(\mathbf{X}))=\mathsf{E}_Q(Y)$ +2013,$\mathsf{E}_\mathbb{Q}(Z \mid X)=\mathsf{E}(Z \mid X)$ +2014,$\bar P_i(a)$ +2015,$L_\infty$ +2016,"$k=1,\dots,K$" +2017,$\delta(p) F(x)=dF(x) + (\delta^*-d)\sqrt{FS}$ +2018,$k<\sup X$ +2019,"$t=0,1$" +2020,$M_i\not=C_i$ +2021,$S(x)\to 0$ +2022,$\mathsf{E}[X_2 Z_1] = \mathsf{E}[X_2]\mathsf{E}[Z_1] =\mathsf{E}[X_2]$ +2023,$P(a)= S(a) + \bar\delta F(a)$ +2024,$\rho(L) = F^{-1}(p)g'(1-p)dp$ +2025,$\rho_t(X)=\rho_t(-\rho_{t+1}(X))$ +2026,$\bar p=1$ +2027,$\mathsf{E}(L_\sigma)= \int_0^1 q_L(s)\sigma(s)ds =:\pi_\sigma(L)$ +2028,$\rho(A)\le\rho(A_0) +\mathsf E[X]\rho(N)$ +2029,$\ge\mathsf{VaR}_p$ +2030,$T_x$ +2031,$\mathbf{m}=(m_j)$ +2032,$0\lt p \lt 1$ +2033,$B^2$ +2034,$\mathsf{E}(Q|X\ge a)$ +2035,$X=0$ +2036,$e^* \in E^*$ +2037,$-Y\ge 0$ +2038,$F^{-1}(U)$ +2039,$\kappa_i(x)$ +2040,$C0$ +2068,$\mathsf E[Q\mid \mathcal F_1]$ +2069,"$n=1,2,3,\dots$" +2070,$p(a) = S(a) + \rho k(a)$ +2071,$n\ge 1$ +2072,"$\rho(X) = \sup_{\zeta\in\mathcal{A}} \langle \zeta,X \rangle$" +2073,$O(mn\times n\log(n))$ +2074,"$x\mapsto (f(x), g(x))$" +2075,$\sigma=2.70$ +2076,$w$ +2077,$\Phi$ +2078,$aq(\alpha)$ +2083,$1-\tilde p=g(1-p)=g(S(x))$ +2084,$0\le a-L_0^a\le a$ +2085,$F^{\times}_{359}$ +2086,$S\not=xf$ +2087,$q(\hat p)=q(1-g^{-1}(1-p))$ +2088,$a \le b$ +2089,$\sum_i \phi_i(a) = 1$ +2090,$N=N(\bar x)$ +2091,"$C_{2,\cdot}$" +2092,$T_0$ +2093,$r_0$ +2094,"$1,2,\dots, m$" +2095,$dQ/dP$ +2096,$n \ll p$ +2097,$1-2c\mathsf{Pr}(Z>\mathsf{E} Z)$ +2098,$1 \times 10^{16}$ +2099,$f_X$ +2100,$(\mathsf{E}(X_i)-\mathsf{E}(X\wedge a))/\mathsf{E}(X_i)$ +2101,$p(a)$ +2102,$c=$ +2103,$dv$ +2104,$\mu_{t+1}=\mu_t$ +2105,$\epsilon$ +2106,$X'$ +2107,$\rho(A_k) \le \rho(A_0) + k\rho(N)$ +2108,$Y_n$ +2109,$\delta(s)=g(s)g(k/s)-g(k)$ +2110,$Y>a$ +2111,$R(x)$ +2112,$X_u=X=u_1X_1 + u_2X_2$ +2113,$=\int_0^c S(x)dx = \int_0^c xf(x)dx + cS(c)$ +2114,$Q \sim P$ +2115,$=L/(1+r)$ +2116,"$[x,x+dx)$" +2117,"$, $" +2118,$f(x)<\infty$ +2119,$1-S(a)=F(a)$ +2120,$=\dfrac{g(s)-s}{1-s}$ +2121,"$\langle \zeta_{\bar x}, X_i \rangle$" +2122,$i$ +2123,$\lambda S(a)$ +2124,$a \ge a'$ +2125,$g'(S(X))$ +2126,$\bar P = \bar S + \bar R$ +2127,$a<1$ +2128,$p+dp$ +2129,$L_1$ +2130,$1-\hat p=g^{-1}(1-p)$ +2131,$\mathsf{E}_q(X_2)$ +2132,$\mathsf{E}(X_i(a)) = E(X_i \mid X\le a)F(a) + aE(X_i/X \mid X> a)S(a)$ +2133,"$\eta_{p,\alpha_1}(X) < \eta_{p,\alpha_2}(X)$" +2134,$μ = t ν$ +2135,$1-S(x)=F(x)$ +2136,$\delta=1-\nu=\rho\nu$ +2137,$\bar A_{x:\lcroof{n}}$ +2138,$\mathscr{O}(\zeta)$ +2139,$X=\mathsf E[Y\mid X]$ +2140,$\rho_{t+1}(-\rho_{t+1}(X))=\rho_{t+1}(X)$ +2141,$\rho(n^{-1}\sum X\circ T) = n^{-1}\sum \rho(X\circ T)$ +2142,$=\int_0^\infty xf(x)dx = \int_0^\infty S(x)dx = \int_0^1 q(p)dp$ +2143,$\notiff$ +2144,$\hat\rho$ +2145,$\lambda=0.045$ +2146,"$[x, x+dx)$" +2147,$C$ +2148,$\mathsf{E}(B)=p$ +2149,$O(mn\log(n))$ +2150,$\mathcal F^{NS}$ +2151,$P(\alpha(X))$ +2152,$F(a)/\nu F(a)=1/\nu=1+\rho$ +2153,$da$ +2154,$(\partial P_i / \partial x_i)dx_i$ +2155,$\tilde p=g(p)$ +2156,"$\min(X,a)=X \wedge a$" +2157,$1=S(x)+F(x)$ +2158,"$(valu2.south east)+(\boundpad,-\boundpad)$" +2159,$\theta > 1$ +2160,"$[0,1]\to[0,1]$" +2161,$\lambda_t=\lambda \mu_t$ +2162,$\ge 5$ +2163,"$A = \{ \zeta \mid \|\zeta\|_q\le c, \zeta\ge 0 \}$" +2164,$U(X) a) = (\mathsf{E}(X)-\mathsf{E}(X\mid X \le a)F(a))/S(a)$ +2177,$D(x)$ +2178,$\mathsf{E}(X_i(a)) = \mathsf{E}(X_i \mid X \le a)F(a) + a\mathsf{E}(X_i/X \mid X > a)S(a)$ +2179,$nG$ +2180,$y\ge x$ +2181,$d=iv$ +2182,$\mathsf E[T_s T_t] \ge \mathsf E[T_s] \mathsf E[T_t]=g(s)g(t)$ +2183,$\rho(X) = \mathsf{E}[gX]$ +2184,"$(Bob)+(0,-3.5)$" +2185,$\mathsf{Pr}(X=\mathsf{E}(X))=0$ +2186,$u\mapsto \mathsf{E}[X_i/u\mid X(t)=u]$ +2187,$X_2$ +2188,"$\displaystyle\int g(S_X) = \sup\{ E_Q(X) \mid Q(A)\le g(P(A)), \forall A\in \mathcal{F}) \}$" +2189,$(LL^t)^{-1}L^t$ +2190,$g\leftrightarrow \rho$ +2191,$g(s)$ +2192,$a=P+Q$ +2193,$n=2$ +2194,$Z=d\mathsf{Q}/d\mathsf{P}$ +2195,$n=3$ +2196,$W$ +2197,$g(t)=O(d)$ +2198,$\sqrt{F(x)S(x)}\approx \sqrt{S(x)}$ +2199,"$\ge 50,000$" +2200,$g=3$ +2201,$10^{19}$ +2202,$L_\infty\subset L_p \subset L_\sigma\subset L_1$ +2203,$^{**}$ +2204,$s\mapsto g(s)$ +2205,$X=\sum_{i=1}^n X_i$ +2206,$\tilde F^{-1}(\tilde p)=F^{-1}(p)$ +2207,"$[p, d+dp]$" +2208,$\rho_g(X)=\int xg'(S(x))f(x)dx$ +2209,$\mathsf{E}_q(X_1)/q$ +2210,"$\delta(s,t)\ge 0$" +2211,$\delta F(x)$ +2212,"$\lambda=0.045, 0.0625, 0.085, 0.125,$" +2213,$\bar \zeta$ +2214,$\Delta p\times T$ +2215,$1+2c(Z-\tau)$ +2216,$s_l$ +2217,$\mathbf{T}^+$ +2218,"$\alpha\in [0,1]$" +2219,$\mathsf{E}_\mathsf{Q}[Y \mid X] = \mathsf{E}[Y \mid X]$ +2220,$\epsilon\mathsf{E}_q(X_1)$ +2221,$0\le (-X_n) \le 1$ +2222,"$\rho(X+tY)-\rho(X) = \langle \zeta_t, X+tY \rangle -\rho(X) \le \langle \zeta_t, X+tY \rangle - \langle \zeta_t, X \rangle = \langle \zeta_t, tY \rangle$" +2223,$=18\times 4 = 72$ +2224,$1/N$ +2225,$(\delta^*-d)\int_0^a \sqrt{F(x)S(x)}dx$ +2226,$t_1$ +2227,$\rho(1_A)=1$ +2228,$g(s)=s$ +2229,$(x+b)$ +2230,$\mathsf{E}(U(Z))=\mathsf{E}(U(Z) \mid A) = \mathsf{E}(U(X))p + \mathsf{E}(U(Y))(1-p)$ +2231,$\phi_i(a)=\mathsf{E}(X_i/Y \mid Y>a)$ +2232,$s\le 1-p < t$ +2233,$B(b)$ +2234,$\rho(X_n)=1$ +2235,$\rho(X)=\max_{Q\in\mathsf{Q}} \mathsf{E}_Q(X)$ +2236,$x\in\mathbb{R}^n$ +2237,$1 \times 10^{23}$ +2238,$N=4$ +2239,$H(X) > -H(-Y)$ +2240,$1=\nu+\delta$ +2241,$t=0.55$ +2242,$t = 0$ +2243,$=E(X_i \mid X=a)$ +2244,$\Delta p$ +2245,$p+q=1=\nu+\delta$ +2246,$(\delta^*-d)\sqrt{pq}=$ +2247,"$X_i, Y$" +2248,$A=X+Y$ +2249,$\rho(X)<\infty$ +2250,$b^2 \mu_x /2$ +2251,$\displaystyle\int_0^\infty S(x)dx$ +2252,$\Phi_i(y)=\mathsf{E}(X_i \mid Y = y)$ +2253,$-g''(t)=α(α-1)t^{α-2}$ +2254,$g^mA^R=g^m(g^a)^R=g^{m+Ra}$ +2255,$l_p>0$ +2256,$\sigma(X_1)$ +2257,$\mathsf{E}_\mathsf{Q}(Y \mid X) = \mathsf{E}(Y \mid X)$ +2258,$B$ +2259,$f=1$ +2260,$p(1-p)/p^2(\nu_p-l_p)^2$ +2261,$\rho(X)=r$ +2262,$Z(t\mathbf{X})=tZ(\mathbf{X})$ +2263,$\mu_x = -d\log(\tpx)/dt = \lim_{t\downarrow 0} {}_tq_x/t$ +2264,$g(s)=s^{1/\rho}$ +2265,$(k+1)\times n$ +2266,$f_{\hat i}$ +2267,$2^{20}$ +2268,$\beta_i/\alpha_i$ +2269,$EL$ +2270,$B_i$ +2271,$\phi F$ +2272,$du$ +2273,"$a,0\le a\le\infty$" +2274,$g^{ak}$ +2275,"$[\alpha_\epsilon,1]$" +2276,$t\ge 0$ +2277,$\mathsf{E}[g]\ge 1$ +2278,$p=1-g^{-1}(1-p)$ +2279,"$[t-dt, t]$" +2280,$s_l < s < s_u$ +2281,$x = F^{-1}(1-g^{-1}(1-\tilde p))$ +2282,$p-\nu-il$ +2283,$50 of the amount allowed on each claim in the classes under subsections 2 to 6 shall be deducted from the claim and included in the class under subsection 8. Claims shall not be cumulated by assignment to avoid application on the $ +2284,"$(s_{i}, g(s_{i}))$" +2285,$d\nu=d\mu/\alpha$ +2286,$\text{E}[X_i \mid X]$ +2287,$x\mapsto 1/x$ +2288,$\mathcal P$ +2289,"$[a,a+1)$" +2290,$\not =$ +2291,$a_{i-1} < a_i < a_{i+1}$ +2292,$\rho_g(X)=$ +2293,$V(X)>0$ +2294,$\rho(X)=\int_0^\infty g(S(x))dx$ +2295,${}^1S=S$ +2296,$^{2}$ +2297,$i=0.02$ +2298,$q(\epsilon)/(1+\epsilon)\approx (q+\epsilon\mathsf{E}_q(X_1) )(1-\epsilon)=q-\epsilon(q-\mathsf{E}_q(X_1))=q-\epsilon E_q(X_2)$ +2299,$Z_k \succeq_2 (Z_k\mid N)$ +2300,$\mathsf E[(a-X)^+]$ +2301,$= \rho(B(s_l)) (1 -g(s)) + \rho(B(s_u)) g(s)$ +2302,"ho=0.5, x=3, M=1.5,\sigma=0.85, K=8$" +2303,$X(t)=X(\mathbf{x})=(1-t)X_1 + tX_2$ +2304,$a=\infty$ +2305,$H(x)\not=H(y)$ +2306,"$\mathbf{x}=(x_1,x_2)$" +2307,$Q(x) = \nu(F(x))F(x)$ +2308,"$\mathcal{Z}=\{ Z\in L^\infty\mid \mathsf{E}[Z]=0, \mathsf{E}[Z^2]\le 1 \}$" +2309,$B=2.7\times 10^{-6}$ +2310,"$(lee.east |- lee.north)+(0.25,0.25)$" +2311,$Y=g(X)$ +2312,$p\delta(p)/p\nu(p)=\iota(p)$ +2313,$M(a)=\mathsf{E}(X\wedge a)+d_iN(a)+(v-\nu^*)\displaystyle\int_0^a \sqrt{F(x)S(x)}dx$ +2314,$\bar\delta(a)$ +2315,$\displaystyle\int_0^1 \mathsf{AVaR}(p)\mu(dp) = \displaystyle\int_0^1 \dfrac{1}{1-p}\displaystyle\int_{p}^1 q(s)ds \mu(dp) =\displaystyle\int_0^1\displaystyle\int_0^s \dfrac{\mu(dp)}{1-p}q(s)ds=\displaystyle\int_0^1\displaystyle\int_{1-s}^1 \dfrac{\mu(dp)}{p}q(s)ds=\displaystyle\int_0^1\phi(s)q(s)ds$ +2316,"$ϕ(s) = α^{-1}1_{[1-α, 1)}(s)$" +2317,$\rho_m(X)$ +2318,$X-b\le 0$ +2319,$\theta$ +2320,"$(\s,4.5-\s)$" +2321,$\rho(2X)= \rho(X+X)=\rho(X)+\rho(X)=2\rho(X)$ +2322,$g(S)\not=q\phi$ +2323,"$[0, t_1]$" +2324,$t\to 0$ +2325,$g'(t)dt < dt$ +2326,"$R,S$" +2327,$X\circ T$ +2328,$s = f/n$ +2329,$h_0$ +2330,$X=a$ +2331,$p=F(x)$ +2332,$r\times 1$ +2333,$D-N = \sum_{i\in I} (D_i-N_i) - N_a$ +2334,"$\bar\delta,\bar\nu$" +2335,$0.0625$ +2336,$\mathsf{TVaR}_p=\dfrac{1}{1-p}\displaystyle\int_p^1 F^{-1}(p)dp$ +2337,$\ll$ +2338,$s>0$ +2339,$E_i$ +2340,$O(\delta^2)$ +2341,"$(a,b)$" +2342,$n=\square^\square$ +2343,$m(x)=S(x)+\delta(p)F(x)=S(x)+dF(x)+(\delta^*-d)\sqrt{F(x)S(x)}$ +2344,$\zeta\in\mathscr{O}(\eta)$ +2345,"$f(x,y)=q_\alpha(x) - G(x,y)$" +2346,$f(X)$ +2347,$\rho(X)\le \liminf \rho(X_n)$ +2348,$\pi(X)=\int_a^{\alpha(X)} g(S(t))dt$ +2349,$X$ +2350,$\rho(X) = \mathsf{E}(X) + c\| X-\mathsf{E}(X) \|_p$ +2351,$2\square^2 + 2\square$ +2352,"$[0.2, 0.85]$" +2353,$v_i = a_i/a$ +2354,$a+da$ +2355,$Q=(P+P')/2$ +2356,$μ = w_1 δ_{α_1} + w_2 δ_{α_2}$ +2357,$G_0$ +2358,$(\bar P_{x+b} - \bar P_x)\bar a_{x+b}=\bar A_{x+b}-\bar P_x \bar a_{x+b}=: {}_b\bar V$ +2359,$L(a)=\mathsf{E}(X\wedge a)$ +2360,$Y_i=\partial Y/\partial x_i$ +2361,$\alpha=1$ +2362,"$B(b)\approx -b\mu_xv^b \approx {-}_bq_xv^b = -A^{\, 1}_{x:\lcroof{b}}$" +2363,$X\circ\tau$ +2364,"$(X^∗_1, \dots, X^∗_n)$" +2365,$\phi_{\bar x}$ +2366,$dF(X)$ +2367,$-1_{B_r}$ +2368,$p(1-\nu(p))=p\delta(p)$ +2369,$g(s)=a^\alpha$ +2370,"$u=(u_1, u_2)$" +2371,$\lambda^Q_t = \lambda^Q\mu_t$ +2372,$\rho(1)=1$ +2373,$u''<0$ +2374,$X(\mathbf{x})$ +2375,"$\langle X_i, \zeta \rangle$" +2376,$\mathsf{E}(X \mid X\le a)$ +2377,$D_\lambda$ +2378,$g(0)=r_0$ +2379,$p(1-p)/(\nu-l)^2=0.5(1-0.5)=0.25$ +2380,"$i=1,\dots,n$" +2381,$\displaystyle\int_0^\infty xf(x)dx = \displaystyle\int_0^\infty S(x)dx$ +2382,$\epsilon\to 0$ +2383,$\bar p$ +2384,$A^k=(g^a)^k$ +2385,"$g:[0,1]\to [0,1]$" +2386,$16\times 4=64$ +2387,$g(s)=d+vs$ +2388,$\omega\mapsto q(\omega)=F^{-1}(\omega)$ +2389,$\mathbf{x}=\mathbf{1}$ +2390,$\nu^*$ +2391,$q(p)+y$ +2392,$\mathsf{E}(X)=\int_0^\infty xf(x)dx = \int_0^\infty S(x)dx$ +2393,$c_k-G=\gamma_k$ +2394,$Z_1=Z\circ T$ +2395,$p\not=0.5$ +2396,${}_tq_x=1-\tpx$ +2397,$L_2(\Omega)$ +2398,$n:=\nabla_yG/\|\nabla_y G\|$ +2399,$\{X = a\}$ +2400,$\phi(s)\ge 0$ +2401,$g(s) = \Phi(\Phi^{-1}(s)+\lambda)$ +2402,$\mathbf{T}$ +2403,$\partial\bar P/ \partial a$ +2404,$X\not\equiv 0$ +2405,$\mathsf{E}_\mathsf{Q}(X_i \mid X=x)=\mathsf{E}(X_i \mid X=x)$ +2406,$k\ge 0$ +2407,$a(\mathbf{x}) =\mathsf{VaR}_p(X(\mathbf{x}))= q_p(\mathbf{x})$ +2408,$u''(z+t)$ +2409,$\rho(X) = \mathsf{E}(X) + V(X)$ +2410,$F^{-1}(1-s)$ +2411,$\rho_i(X_i) - \rho_i(F_i)$ +2412,$\mathsf{E}_Q(Y)=\tilde \rho(X)$ +2413,$R_i t^*$ +2421,$X_i=x_i$ +2422,"$(fun5.north west)+(-\smlspc,\smlspc)$" +2423,$\tpx \mu_{x+t}$ +2424,"$(s_{i+1}, g(s_{i+1}))$" +2425,$r_P < r$ +2426,$T_B$ +2427,"$X\in \mathcal A_{t,t+1} + \mathcal A_{t+1}\iff -\rho_{t+1}(X)\in\mathcal A_{t+1}$" +2428,$(1+\theta)\rho$ +2429,$\rho(A_k) \le \rho(A_0) + k \rho(N) \le \hat\rho(A_0) + k\rho(N)=\hat\rho(A_k)$ +2430,$EL=\mathsf E[X\wedge a]$ +2431,$dt=g'(1-s)ds=\phi(s)ds$ +2432,$A^∗_i$ +2433,"$I=[0,1]$" +2434,$\bar R'(x)=R(x)$ +2435,$X=X(\mathbf{x})$ +2436,$Y_n=-X_n$ +2437,$2^{256}\approx 10^{77}$ +2438,$P$ +2439,$\mathsf{Pr}(X\le a)=F(a)$ +2440,$g(t)=1$ +2441,"$Y=\max(X_1, \dots, X_N)$" +2442,$α$ +2443,$p=0$ +2444,$0\le\lambda \le 1$ +2445,$\phi(x)/x$ +2446,$=P + r(P+S)$ +2447,$\nabla g'$ +2448,$(f)$ +2449,$\iota^*=0.125$ +2450,$R_2 > C_2$ +2451,$\delta\bar a_{x:\lcroof{n}}+\bar A_{x:\lcroof{n}}=1$ +2452,"$(g^k, Km)$" +2453,$p=100043$ +2454,$6 \times 10^7$ +2455,"$[0,1]\to \mathbb{R}$" +2456,$b$ +2457,$2\square^2 + \square + 5$ +2458,$99a}$ +2504,$\hat p=1-g^{-1}(1-p)$ +2505,$\iota^*=0$ +2506,$\mathsf{E}\zeta=1$ +2507,$\epsilon^2$ +2508,"$(fun1.north west)+(-\smlspc,\smlspc)$" +2509,$\tau$ +2510,$\partial a / \partial x_i$ +2511,$\sum u_iX_i$ +2512,$S(X)$ +2513,$\log$ +2514,$\rho(X)=\mathsf{E}_\mathbb{Q}(X)$ +2515,$A_k = A_0 + kN$ +2516,$\sum_i X_i$ +2517,$L$ +2518,$a_i$ +2519,$X(x)$ +2520,"$x_c, x_n$" +2521,$\rho(X)=\int_0^1 q(s)\phi(s)ds=\int_0^1 q(s)g'(1-s)ds$ +2522,"$\langle \mu, X+a \rangle = \langle \mu, X \rangle + a$" +2523,$\mathsf{E}_\mathsf{Q}$ +2524,$kX$ +2525,"$\forall X,Y,t\ge 0$" +2526,$f_X(a)$ +2527,$\bar F(a) = a-\bar S(a)$ +2528,$1-p \le st$ +2529,$g(0) = 0$ +2530,$\phi_i$ +2531,$1$ +2532,$\mathsf{E}(N)=\lambda$ +2533,"$(B.south east) + (-0.07mm,0)$" +2534,$0\le a\le 2^{256}$ +2535,$\bar \nu$ +2536,$X:\Omega\to\mathbb R$ +2537,$1=\delta(p) + \nu(p)$ +2538,$s=S(x)=1-p$ +2539,"$\rho(X,a)=\rho(X\wedge a)$" +2540,$l_p<0$ +2541,$S(x)\approx 1$ +2542,$\mathsf{Var}(X+a)=\mathsf{Var}(X)$ +2543,$\mathsf{E}_Q(X_i)$ +2544,$\rho=\mathsf{AVaR}$ +2545,$d-d^2=v-v^2=dv$ +2546,$\mathsf{E}(X_i ; X \le a)$ +2547,$dt=g'(1-s)ds$ +2548,$1 \times 10^{17}$ +2549,${}_tp_x\mu_{x+t}$ +2550,$x_i=q(u_i)$ +2551,$P(x)=S(x)$ +2552,$P_i \le \rho_i(X_i)$ +2553,"$\bar x\mapsto G\circ F(\bar x, \omega)$" +2554,$r_i = (P_i - \mathsf{P}[X_i]) / P_Q$ +2555,$g(t)$ +2556,$m_i$ +2557,$5 \times 10^5$ +2558,$\tilde p=p_a$ +2559,$Z\circ\tau$ +2560,$s_i=1-p_i$ +2561,$\iota^*=0.15$ +2562,$\rho(X+\epsilon Y)=\mathsf{E}[h_\epsilon (X+\epsilon Y)]$ +2563,"$\langle \mu,X \rangle$" +2564,$p=2$ +2565,$\iota_{1/2}$ +2566,$\mu_{t}$ +2567,"$[1-\alpha, 1]$" +2568,"$(\s,4-\s)$" +2569,"$\bar P(\mathbf{x}, a)$" +2570,$(1-r_0)δ_1$ +2571,$10^{-3}$ +2572,$m'$ +2573,$n\ge 2$ +2574,$\rho_m(X) a]S(a)$ +2590,$R_2(t_2-\epsilon)s$ +2595,$\mathbb{P}(B)=0$ +2596,$v=1-d$ +2597,$g^{ak}=(g^k)^a=K^a$ +2598,$\hat X_i$ +2599,$s=k^{-1}(m + ra)$ +2600,$\rho_{(g)}(X)=$ +2601,$\mathsf E[u(X-\pi +R)]$ +2602,$M(s)=\mathsf{E}[X^s]=\mathsf{E}[e^{s\log(X)}]$ +2603,$\sigma=3$ +2604,$F^{-1}(s)$ +2605,"$C_2(t) < \bar P^a(0, 1)$" +2606,$T=T_B\circ T_A$ +2607,$1 = p(a) + (1-p(a))$ +2608,$})=1-\mathsf{Pr}(\text{No events $ +2609,$Q\ll P$ +2610,$B(p)=0$ +2611,"$(A, a)$" +2612,$m + ra$ +2613,$X_{2}$ +2614,$A_k=A_0 + kN \ge A_0 + k'N = A_{k'}$ +2615,$l(p)$ +2616,$V=(a - X)^+$ +2617,"$(\x*1.2, 2)$" +2618,"$(Alice)+(0,-1)$" +2619,$L_p/L_q$ +2620,$\nu(S(x) + \iota)$ +2621,$\mathsf{Q}_2$ +2622,$AR(2)$ +2623,$10^{1+6+12}=10^{19}$ +2624,$\log_g(n)=a$ +2625,$0\le p\le 1$ +2626,$I(p)$ +2627,$M_t$ +2628,$\rho(X) = \sup \{ \rho_\phi(X) \mid \phi\in A \}$ +2629,"$\rho(X)=\int g(S_X(t))\,dt$" +2630,$1-g^{-1}(U)$ +2631,$0\leq f \leq 1$ +2632,$\mathsf{Pr}(X>q(p))=1-p$ +2633,$x\mapsto xX$ +2634,"$(valu1.south east)+(\boundpad,-\boundpad)$" +2635,$\rho(X)=\displaystyle\int_0^1 q(p) \phi(p) dp$ +2636,$\mathsf{E}[X]=28$ +2637,$X\ge a$ +2638,$x=y$ +2639,$g(t)=t^α$ +2640,$x_i / \sum_i x_i$ +2641,$\rho(X)=\int_0^\infty x g'(S(x))f(x)dx$ +2642,$2^1+1\rightarrow 3^1+1-1=3^1 \rightarrow 4^1-1=3 \rightarrow 2 \rightarrow 1 \rightarrow 0$ +2643,$(\delta^*-d)\sqrt{S(x)F(x)}$ +2644,$\rho_t(-\rho_{t+1}(X))\le \rho_t(\rho_{t+1}(Y))$ +2645,$r=d/(1-d)$ +2646,$n = 2$ +2647,$p(1-p)/(v-l)^2$ +2648,$p$ +2649,$\rho(X)\ge X$ +2650,$\nu=\mu_X-\mu_Y$ +2651,$\lambda_t$ +2652,"$(p,q(\hat p))$" +2653,$F^{(2)}(\mu_X)$ +2654,$\mathsf{Q}$ +2655,"$X,\, X_i\in L^\infty$" +2656,"$[\alpha_0,1]$" +2657,$f_Y$ +2658,$Y_n\uparrow 0$ +2659,$1 \times 10^{14}$ +2660,$\tilde p=\tilde F(F^{-1}(p))$ +2661,$\mathsf{E}(X_i)$ +2662,"$k,a$" +2663,$\mathcal A$ +2664,$H(X)\le H(Y)$ +2665,$400 to over $ +2666,$F^{-1}(p)$ +2667,$\nu(p)=(1+\iota(p))^{-1}$ +2668,$\Phi_i(a)/a$ +2669,$\beta=\delta^*-d$ +2670,$(p-\nu)/(\nu-l)$ +2671,$\mathsf{E}[X^k-Y^k]=\int x^k\mu_X(dx)-\int y^k\mu_Y(dy)=\int x^k\nu(x)$ +2672,$\mathsf{E}[Y] = 50.4$ +2673,"$g(s) = \min(1, s/(1-\alpha))$" +2674,$B_r$ +2675,$\int_0^1 \mathsf{TVaR}_p(X)m(dp)$ +2676,$0\mapsto 0$ +2677,"$(Alice)+(0,-3.25)$" +2678,$p\nu(p)=p((\nu(p)-l(p))+l(p)) = \nu^*\sqrt{pq} + v(p-\sqrt{pq})$ +2679,$B=\Omega\setminus C$ +2680,$c:\mathbb{R}^n\to\mathbb{R}$ +2681,$g(u) = m'u / (r(u) - m'u)$ +2682,$\rho(X) = a = \mathsf{E}[X | A] = ES$ +2683,$(\delta^*-d)\sqrt{F(x)S(x)}$ +2684,$\sigma=1.667$ +2685,$\subset$ +2686,$\alpha=$ +2687,$\Rightarrow$ +2688,$\mathsf{E}(\cdot)$ +2689,$A=X_1+\cdots +X_N$ +2690,$p(v_p-l_p)$ +2691,$x_i$ +2692,$X\le b$ +2693,$\mathsf{E}(X_i \mid X=\hat x)=\mathsf{E}(X_i \mid X=F^{-1}(\tilde p))$ +2694,$5 trillion business. Property casualty insurers write $ +2695,$X=\sum_i X_i$ +2696,$\tilde Z\in\mathscr{P}$ +2697,$\rho_{(g)}(X)$ +2698,$\tau^{-1}$ +2699,"$, @Pichler2015a, 6.1. @Dentcheva2010 (DPR) goes to great lengths to prove represented by transforms (AVaR to spectral transform) with $" +2700,"$x=1, M=1.5,\sigma=0.75, K=6$" +2701,$\pi(X)=\mathsf{E}_g(X\wedge \alpha(X))=\int_0^{\alpha(X)} g(S(t))dt$ +2702,$X>Y$ +2703,$F_u^{-1}$ +2704,"$d(x,\omega)$" +2705,$\mathsf{E}(Y\sigma(U))$ +2706,$k \ge k'$ +2707,$X_n=0$ +2708,$<$ +2709,$\rho(X \circ T)=\rho(X)$ +2710,$\rho^*= (1-\alpha-\beta)^{-1}-1$ +2711,$\|Y\|_{\sigma} = \rho_\sigma(Y)$ +2712,$1/x^2$ +2713,$172.4\times \exp(2.7^2/2) = 6600$ +2714,"$\eta,\zeta$" +2715,$N_i$ +2716,$g(s) = d + (1-d)h(s)$ +2717,$\mathsf{E}[g]=1$ +2718,$\tilde S$ +2719,$\rho(X) + c = \rho(X+c)\ge \rho(X) + \mathsf{E}[cg]$ +2720,$M_i(t)\not=C_i(t)$ +2721,"$(fun5a.south west)+(-0.5*\wspcer,-0.5*\medspc)$" +2722,$c \le 0$ +2723,$(1+\epsilon)\mathbf{X}$ +2724,$\tilde\rho(X)=\mathsf{E}_Q(Y)$ +2725,$\le 1/N$ +2726,"$h_{i,\epsilon}-h_0\to 0$" +2727,$\text{AVaR}_\alpha(X)$ +2728,$N \mid G$ +2729,$q+\epsilon\mathsf{E}_q(X_1)$ +2730,$X_2=t$ +2731,$g(st)\le g(s)g(t)$ +2732,"$\langle \zeta, Z-\mathsf{E} Z\rangle$" +2733,$\mu(dp)=f(p)dp$ +2734,$\rho_i(F_i)$ +2735,$q(p)\phi(p)$ +2736,$\rho(X^{\oplus N})$ +2737,$\mathcal{X}$ +2738,$F:\mathbb{R}^n\to\mathcal{Z}$ +2739,"$(Bob) + (0,-3.5)$" +2740,$p=0.001$ +2741,$X_i/X$ +2742,$\bar P_2$ +2743,$k0$ +2802,$\rho_\gamma(X) = \gamma\rho(X/\gamma)$ +2803,$S_u(t)=\text{Pr}(X_u>t)$ +2804,$\rho(X)\le\rho(Y)$ +2805,${}_{dt}q_{x+t}\approx dt\mu_{x+t}$ +2806,"$(-\x*.8, 2*2)$" +2807,$\mathsf{E}(X_iX_i \mid X)\not=\mathsf{E}(X_i \mathsf{E}(X_i\mid X)\mid X)=\mathsf{E}(X_i\mid X)^2$ +2808,$i=2$ +2809,$P(a)=g(S(a))\ge S(a)$ +2810,$p=29$ +2811,$c_x$ +2812,$a=150$ +2813,$L=$ +2814,$H(n + \text{prev hash} + \text{value})g(x)$ +2841,$a=M(a)+Q(a)$ +2842,$\mathsf{VaR}_p$ +2843,$1 \times 10^{18}$ +2844,$1=F(x)+S(x)=\delta+\nu$ +2845,$I$ +2846,"$\mathsf{E}(C_1(a,c))$" +2847,"$(\x*0.65, 3.75*2)$" +2848,$10^{18}$ +2849,$p\approx 1$ +2850,$\delta(p)=\iota(p)/(1+\iota(p))=1-\nu(p)$ +2851,$01$ +2858,$\bar P(a) = \bar S(a) +\bar\delta(a) \bar F(a)$ +2859,$Y=W+Q$ +2860,"$X_i=F(0,\dots, x_i,\dots, 0)$" +2861,$\mathsf{E}[XZ]=\mathsf{E}[\mathsf{E}[XZ\mid X]]=\mathsf{E}[X\mathsf{E}[Z\mid X]]=\mathsf{E}[X\tilde Z]$ +2862,$c\ge 0$ +2863,"$(fun4a.south east)+(0.5*\wspcer,-\medspc)$" +2864,"$\mathcal{M}=\mathcal{M}[0,1]$" +2865,$4\times 10^{19}$ +2866,"$(P, R)$" +2867,"$\bar A^{1}_{x:\lcroof{n}}, \bar a_{x:\lcroof{n}}$" +2868,$X=q$ +2869,"$A = \{\zeta' \in L_q \mid \zeta'=1+\zeta-\mathsf{E}\zeta, \|\zeta\|_q\le c \}$" +2870,$\nu(F(x))F(x)dx$ +2871,$sqrt{st}$ +2872,$R^S=g^mA^R$ +2873,$X\not= Y$ +2874,$\rho_{(g)}(X)=\displaystyle\int_0^\infty g(S(x))dx$ +2875,$\Delta \tilde p$ +2876,$L_\sigma^*:=\{ Z\in L_1\mid \| Z\|_\sigma^*< \infty \}$ +2877,${}_tE_x=e^{-\delta t}{}_tp_x$ +2878,$(\lambda)$ +2879,$\mathsf{TVaR}_\alpha(X)=\dfrac{1}{1-\alpha}\displaystyle\int_{\alpha}^1 q(p)dp$ +2880,$\int_0^\infty g(S(x))dx = \int_0^1 q(t)\phi(t)dt$ +2881,"$(lee.west |- lee.north)+(0,-2.5)$" +2882,$\rho(X)=\mathsf{E}(X\theta)$ +2883,$\iota_p$ +2884,$dQ/dP=g'(S(X))$ +2885,"$\mathcal{M}_{X,r}$" +2886,$\rho(X)=50$ +2887,$\mu_x$ +2888,$\rho_0$ +2889,$\leftrightarrow\mathcal P\rightarrow \rho_t(X)=\max_{P\in \mathcal P}\mathsf E_P[X\mid mathcal F_t]$ +2890,$\mathcal{G}\subset\mathcal{F}$ +2891,$-\sqrt{x}$ +2892,$a=a(t)$ +2893,$u$ +2894,$\rho(X)=\displaystyle\int_0^\infty x g'(S(x))f(x)dx$ +2895,$\tilde p$ +2896,"$(de.east |- lee.north)+(0.375,0.25)$" +2897,$Z_0$ +2898,$(X)$ +2899,"$(p, x)$" +2900,$\mu_t = \begin{cases} 0 & t<1 \\ n & 1\le t\le 1+1/n\end{cases}$ +2901,$A_\cdot$ +2902,$\sigma=0.175$ +2903,$_p$ +2904,$\mathsf{TVaR}_{p_1}(X) = r$ +2905,$1-\delta=\nu$ +2906,"$\mathsf E[(X-K)^+] \le \mathsf E[(Y-K)^+],\ \forall K\in\mathbb R$" +2907,$\text{E}_{\Bbb{Q}}[Y\mid \mathcal{G}] \text{E}[Z \mid\mathcal{G}] = \text{E}[YZ\mid \mathcal{G}]$ +2908,$\zeta_{\bar x}$ +2909,$\rho^{ho}_c$ +2910,$p(1-p)/(p\nu_p)^2$ +2911,"$g(s) = \min(1,\exp(a+b\log(s)))$" +2912,"$\mathcal A_t\subseteq \mathcal A_{t,t+1} + \mathcal A_{t+1}\iff \rho_t(-\rho_{t+1}) \le \rho_t$" +2913,$q_1(t)=t$ +2914,$\inf_t t+\| (X-t)_+\|_p$ +2915,$\rho(X)\le 0$ +2916,"$(brR15 |- lee.south)+(-0.125,-0.25)$" +2917,$g'<1$ +2918,$\hat\rho(X)$ +2919,$\mathsf{FAT}$ +2920,$g(s)=s^{1/3}$ +2921,$R_2(t)>R_2(0)$ +2922,$t>0.5$ +2923,$p(1-p)/\nu^2$ +2924,$\mathsf{E}(B(p))=p$ +2925,$X_n=-e^{-nx}$ +2926,$1\mapsto 1$ +2927,$F^{(-2)}=[F^{(2)}]^*$ +2928,$\mathsf{Pr}(Agg > x) \approx \text{frequency}\times \mathsf{Pr}(Occ > x)$ +2929,$. Definition of normal cone to $ +2930,$\rho_\alpha(X)=\mathsf{E}(X\mid X\ge q_\alpha(X))$ +2931,$R_1=C_1$ +2932,$c=(1-\alpha)^{-1}$ +2933,$=64 \times 4 = 256$ +2934,$=g(s)-s$ +2935,$X\le \rho(X)$ +2936,$P_i = L_i + \iota K_i$ +2937,$\rho(X)\le b$ +2938,$\nu=1/(1+\iota)$ +2939,$\mathsf{E}_\mathsf{Q}(X_i \mid X)=\mathsf{E}(X_i \mid X)$ +2940,$\mathsf Q$ +2941,$\mathrm{COC}$ +2942,$g_i$ +2943,$X_k$ +2944,$Z\ge \tau$ +2945,$t = 2$ +2946,$c=\text{Var}(G)=\nu^2$ +2947,$ϕ_s(X)$ +2948,$X^{\oplus n-1}$ +2949,$\bar S(a):=\mathsf{E}(X\wedge a)$ +2950,$C_i = m_i - X_i$ +2951,$\bar a_{75}=9.81$ +2952,$\rho(A_0) \le \rho(A_0) + \mathsf E[A] \le \rho(A)$ +2953,$X> 0$ +2954,$g'(s)$ +2955,$a-\bar S(a)=\bar R(a)+\bar Q(a)$ +2956,"$\langle X(\epsilon), \zeta_{x+\epsilon} \rangle$" +2957,$0< m\le 1$ +2958,$\tilde Z$ +2959,$\partial\rho(X)=\{Q_0\}$ +2960,$q=S$ +2961,$r_i$ +2962,$\phi_i = 1/n$ +2963,$K-1$ +2964,$W=0$ +2965,$(\rho_t)_t$ +2966,$X:\mathbb{R}\to\mathbb{R}$ +2967,$c_h(1-\alpha)$ +2968,$p\gg n$ +2969,$N=\sum_i N_i$ +2970,$\hat Z\tilde Z_{xn}$ +2971,"$k=1,2,\dots,m$" +2972,$\mathsf{P}[\cdot]$ +2973,$X\ge Y$ +2974,$m\in\mathbb R$ +2975,$M_G(\zeta):=\text{E}(e^{\zeta G})$ +2976,$P_i$ +2977,"$(\x*.75, -2)$" +2978,$(p-\nu-il(p))/(\nu-l(p))$ +2979,$\mathcal A=\mathcal A_\rho$ +2980,$T$ +2981,$\rho(X)=\rho_\phi(X):=\displaystyle\int_0^1 q(p)\phi(p)dp$ +2982,$\rho=0.9$ +2983,$p<0.5$ +2984,$\log(x)\le x-1$ +2985,"$\mathcal{A} = \{ X\mid \exists \alpha\ge 0, \exists Y : \rho(Y)=0, X=Y+\alpha \}$" +2986,$p(\nu(p)-l(p))$ +2987,"$\mathbf{r}=(1,r_1,\dots,r_k)$" +2988,$\ln(10)=2.302585$ +2989,"$(fun5a.south east)+(\medspc,-0.5*\medspc)$" +2990,$f_{\mathbf{x}}$ +2991,$g(0.25) < 1$ +2992,$L_t$ +2993,$k=st$ +2994,$(1-t)\mathsf{E}[X_1]$ +2995,"$(fun6.north west)+(-\smlspc,\smlspc)$" +2996,$\tpx$ +2997,$K$ +2998,$(1+r) = (1+rP)(1+m)$ +2999,$\Phi_i(a) = \int_0^a \phi_i(t) dt$ +3000,$eL + \rho S$ +3001,$q(p)\phi(p)dp$ +3002,$K=\mathsf{xTVaR}_p(X) = \mathsf{TVaR}_p(X) - \mathsf{E}(X)$ +3003,"$(\sqrt k, \sqrt k)$" +3004,$lsc(\rho)$ +3005,$0\le t\le 1$ +3006,$g^{m+ra} = g^m (g^a)^r = g^m A^r$ +3007,$\int g(S)$ +3008,$f=0$ +3009,$N:\mathbb{R}^n\to\mathcal{X}$ +3010,$0 \ge \rho(X_n) \ge -\rho(-X_n) \uparrow 0$ +3011,$skew(G)=skew(G')$ +3012,$\rho(X)=\sup\{\mathsf{E}[hX] \mid h \in \mathscr P \}$ +3013,$S_Y$ +3014,"$\sum_i h_{i, \epsilon}=h_0$" +3015,$L_\sigma$ +3016,"$(\mathsf{E}(X_i)-\mathsf{E}(X_{i,2}(a))/\mathsf{E}(X_i)$" +3017,$tC_i$ +3055,$X^n\to X$ +3056,$\mathscr{S}(X)$ +3057,"$(Bob) + (0,-3)$" +3058,$p=0.50$ +3059,$D_n$ +3060,$g(S(x))>S(x)$ +3061,$G=const_j$ +3062,$g(0+) \gt 0$ +3063,$a=\alpha(X)$ +3064,$f(0.x_1x_2x_3...) = 0.x_1x_3\dots$ +3065,$\mathsf{TVaR}_{p_1}(X)\le r$ +3066,$R_1(t)\approx R_1(0)$ +3067,$R_2(t) \ge \mathsf{E}[X_2]$ +3068,$a/X$ +3069,$t=0.4$ +3070,$O(n\log(n))$ +3071,"$[0, 0.25]$" +3072,$\rho(X)$ +3073,$P=Pg^{ak}/g^{ak}$ +3074,"$\langle \nabla X,\zeta \rangle$" +3075,$\partial Y/\partial X_i$ +3076,$m(a)=S(a) + \delta F(a)$ +3077,$g(p)=\displaystyle\int_0^p\phi(1-t)dt=\displaystyle\int_{1-p}^1 \phi(t)dt$ +3078,$g(p)=p$ +3079,$d=r/(1+r)$ +3080,"$q_X(U), q_Y(U))$" +3081,$0 1/n$ +3119,$0\le f(x)-f(y)\le x-y\ \forall 0\le y < x$ +3120,$(1-p)/(p\nu(p)^2)$ +3121,$\bar M_i(a)>0$ +3122,"$104 million pretax writeoff, resulting in a $" +3123,"$\min(\delta, \max(X-x))$" +3124,$q(p)=-\log(1-p)/\mu$ +3125,$O(mn^2)$ +3126,$r(u) - m'u$ +3127,$\pi(p)$ +3128,$V=(a-X)^+$ +3129,$\pi_X(t_{2j})\le \pi_Y(t_{2j})$ +3130,$1/x^3$ +3131,"$L_a=[a, a+da]$" +3132,$\\{N=n\\}$ +3133,${}_0V = 1$ +3134,$g(t) = r_0 + (1-r_0)t$ +3135,$\rho(X)=E_Q(X)$ +3136,$5 \times 10^{19}$ +3137,$-(\nu-l)-l=-\nu$ +3138,$\rho=0.4$ +3139,$1-\tilde p=\tilde p(1)-\tilde p(p)=\int_p^1 (d\tilde p/dp)(s)ds = \int_p^1 g'(1-s)ds = \int_0^{1-p} g'(s)ds = g(1-p)-g(0)=g(1-p)$ +3140,$E(X\wedge a)=\int_0^a tf(t)dt + aS(a)$ +3141,$E2=0$ +3142,"$\partial\rho(X+\epsilon X_i)=\{Q_{i, \epsilon} \}$" +3143,"$\mathcal A_t = \mathcal A_{t,t+1} + \mathcal A_{t+1}$" +3144,$u'''\ge 0$ +3145,$g^{-1}(p)=p^2$ +3146,$\$ +3147,$g'(1-s)ds$ +3148,$S(x)=1-F(x)=\mathsf{Pr}(X>x)$ +3149,$\rho(X\mid \mathcal F_1)$ +3150,$\mathsf{E}[Z^*\mid X] = n^{-1}\sum_{T\in\mathscr{S}(X)} Z^*\circ T = n^{-1}\sum_i \alpha_i \sum_T Z\circ T_i\circ T = n^{-1}\sum_i \alpha_i \sum_T Z\circ T=\sum_i \alpha_i\tilde Z =\tilde Z$ +3151,"$(x_1, \dots, x_n)$" +3152,$ipl(p)$ +3153,$g_2$ +3154,$k_i=a_i/x_i$ +3155,$x+b$ +3156,$p\nu(p)$ +3157,"$\max(x,0)$" +3158,$\bar F(a)=\int_0^a F(x)dx = a-\bar S(a)$ +3159,$=\dfrac{s}{g(s)}$ +3160,$q=q_j$ +3161,"$\rho(X) = \mathsf{E}(\zeta X) = \langle \zeta, X \rangle$" +3162,$\psi_i(a)=\mathsf{E}(X_i/Y \mid Y>a)$ +3163,$L_\sigma=L_1$ +3164,$W=99$ +3165,$a=100$ +3166,$f^*=(L^t)^+m$ +3167,"$\Delta_{i,\epsilon}$" +3168,$1-p < s$ +3169,$\{N=n\}$ +3170,"$5,000) to as much as \$" +3171,$(X(\omega_1)-Y(\omega_1))(X(\omega_2)-Y(\omega_2))\ge 0$ +3172,$X(\omega)$ +3173,$x^+$ +3174,$0\le x\le 1$ +3175,$\sum Y_i=S$ +3176,$k= \mathsf{E}(X) + (\rho_m(X) - \mathsf{E}(X)) + (k-\rho_m(X))$ +3177,$f_u$ +3178,$k\ge k_0$ +3179,"$(C.north east)+(1.5, 0)$" +3180,$E_Q(Y)$ +3181,$\mathbf{u}=0$ +3182,$\mathsf{E}(X_i \mid X=a)$ +3183,$=E(X_i \mid X\le a)$ +3184,$C(u)$ +3185,$t\not=0.5$ +3186,$\log(\sqrt{2\pi})=0.399090$ +3187,$\rho_{m'}(Y) > \rho_m(Y)$ +3188,$\log(\phi(x)) = -\log(\sqrt{2\pi}) - \frac{x^2}{2\ln(10)}$ +3189,$X\wedge 1$ +3190,$1-F(x)=1-p$ +3191,$F^{(-2)}\int_0^p F^{-1}$ +3192,$P=\rho_{g}(X)$ +3193,"$\rho(X)=\langle \zeta, X \rangle$" +3194,$H_k((X)\le H_k(Y)$ +3195,$\rho(Y)$ +3196,$G:\mathbb{R}^n\to\mathcal{X}$ +3197,$\phi(p')\ge\phi(p)$ +3198,"$\rho^*(\mu)\ge \sup_{a\in\mathbb{R}} \{ \langle \mu,X+a \rangle - \rho(X+a) \} = \sup_{a\in\mathbb{R}} \{ a\mu(\Omega) -a+ \langle \mu,X \rangle - \rho(X) \}$" +3199,$\mathscr{P}=\{ \mathsf{Q} \mid \mathsf{Q} \ll \mathsf{P} \}$ +3200,$\exists$ +3201,"$1,2$" +3202,$q_Z(U)\in\mathscr{P}$ +3203,$\hat\rho(X_1)\le\hat\rho(X_2)$ +3204,$A_0$ +3205,$T_i\in\mathscr{S}(X)$ +3206,$L^1$ +3207,"$X\wedge a=\min(X,a)$" +3208,$x+\tau$ +3209,$GF(\bar x)$ +3210,$\mathsf{E}(X-k)_+$ +3211,$1-\tilde p=g(1-p)$ +3212,$S(x)$ +3213,$\mathsf{E}(W|X\ge a)$ +3214,"$50 of the amount allowed on each claim in the classes under subs. (3) to (6), except for claims of the federal government under subs. (3) and (3c), shall be deducted from the claim and included in the class under sub. (8). Claims may not be cumulated by assignment to avoid application of the $" +3215,$P_i/x_i$ +3216,$0.085$ +3217,$\mathsf{E}(Z\mid X)=Z$ +3218,"$\{(1-\alpha)^{-1}1_A\mid \mathsf{Pr}(A)=1-\alpha, X(\omega)\ge a,\ \forall \omega\in A \}$" +3219,$\epsilon(t-\mathsf{E}_q(X_2))$ +3220,"$X, Y$" +3221,$B\subset E$ +3222,$p'>p$ +3223,$\pi(X)$ +3224,$\sigma=0.3$ +3225,$n^2$ +3226,$c+\mathsf{E}(X-c)_+ = E(X-c)_- + \mathsf{E}(X)$ +3227,$\iota a + \mathsf{E}_Q(X-a)^+$ +3228,$c_x/c_{\text{Nov 1}}-1$ +3229,$g(s)\ge s$ +3230,$g'(1)=0$ +3231,$f(\lambda) = \mathbf{Tm}(\lambda)-\mathbf{r}$ +3232,${}^nS^{-1}(t) = \displaystyle\int_0^t {}^{n-1}S^{-1}(p)dp$ +3233,$10 million I **must care at least as much** about a loss of $ +3234,$Z_k$ +3235,$\mathsf{VaR}_p>2000$ +3236,$Q=100$ +3237,$0.25/0.75$ +3238,$\delta_{p}$ +3239,$\mathsf{E}(X_i g'S)$ +3240,$1+c\zeta-c\mathsf{E}\zeta$ +3241,$a\mapsto \sum_i a_iX_i$ +3242,$1 \times 10^9$ +3243,$X\circ T(\omega)=X(T(\omega))$ +3244,$B_\cdot$ +3245,$\lambda\in\mathbb R^+$ +3246,$\tpx^{(\tau)}$ +3247,$\bar A_{x+b} - \bar P_{x+b}\bar a_{x+b}=0$ +3248,$t_2<0.5$ +3249,$g(S)$ +3250,$\mathsf{E}_g(X\wedge a)$ +3251,$\sum \rho_k$ +3252,$0 \le f(X) \le X$ +3253,$\mathbf{X}$ +3254,$\rho_\phi(X)=\displaystyle\int_0^1 q(p)\phi(p)dp$ +3255,$\rho(X)=\log\mathsf{E}(\exp(\alpha X))/\alpha$ +3256,$s = s_l (1 - s) + s_u s$ +3257,$=\mathsf{E}(X-c)+=\int_c^\infty S(x)dx=\int_0^\infty (x-c)f(x)dx$ +3258,$1-p=S(x)$ +3259,$\rho(X) = \mathsf{E}(X)$ +3260,$\rho(X)\le\liminf\rho(X_n)$ +3261,$N=2^{256}=10^{77}$ +3262,"$p\in [1,\infty)$" +3263,$\sigma\in L_\infty$ +3264,$a_i=a_{i+1}=\dots=a_{i+l}$ +3265,"$G=P,Q,R,S$" +3266,"$=\mathsf{E}(X\wedge c)=\mathsf{E}(\min(X,c))=$" +3267,"$\rho(X) = \max(\mathsf{E}_{\mathsf{Q}_1}(X), \mathsf{E}_{\mathsf{Q}_2}(X))$" +3268,$1 = m(a)+\nu F(a) = (S(a) + \delta F(a)) + \nu F(a)$ +3269,$\rho(X) = \rho(\mathsf{E}[X | A]1_A + E[X | A^c] 1_{A^c})$ +3270,$\lambda=0.421$ +3271,$dp=dF(x)$ +3272,$\zeta:=1/(1-\alpha)1_A$ +3273,$\mathsf{E}(W\mid X\ge 100) = 99/4=19.8$ +3274,$\int_\Omega \zeta=1$ +3275,"$\mathsf{cov}(X_1, g'(S(X(t)))$" +3276,$\mathsf{Pr}(X(\mathbf{x})>a) = S(\mathbf{x}; a)=S(a)$ +3277,$F^{(2)}(x)$ +3278,$\mathscr{O}(\zeta)\subset\mathcal{A}$ +3279,$1/(1-p)$ +3280,$S(x)=e^{-\mu x}$ +3281,$E(G)= f + E(G') = 1$ +3282,"$(\nodespc/2, -\nodespc/2%)$" +3283,$\Delta\tilde p$ +3284,$\log(ROL) = a + b\cdot ln(EL)$ +3285,$α_1 < α_2$ +3286,"$(fun3a.south -| fun4a.south east)+(\smlspc,-\smlspc)$" +3287,$\rho_t(X) \le \mathsf E[\rho_{t+1}(X)\mid \mathcal F_t]$ +3288,$g(0)=0$ +3289,$\int xg'(S(x))f(x)dx$ +3290,$=1-g(s)$ +3291,$E(X_i \mid X=x)f_X(x)$ +3292,$\mathsf{E}_\mathsf{Q}(X_i)$ +3293,$K = (B)^{a} = g^{ba}$ +3294,$\phi(p)=g'(1-p)$ +3295,$v^{(\mathrm{time\ to\ payout})}\rho(\mathrm{risk now})$ +3296,$/$ +3297,$E_Q$ +3298,$x_0$ +3299,$\mathsf{E}[X|A]= n^{-1}\sum_T X\circ T$ +3300,$\bar B\setminus B$ +3301,$X_i=X_i\sum_i \partial C/\partial x_i + \partial N/\partial x_i$ +3302,"$\mathsf{TVaR}_p(Y)\in R_{Y:X,r_X}$" +3303,$\rho(X)=\mathsf{E}(XZ)$ +3304,"$\mathsf{E}(\min(X,a))=\mathsf{E}(X\wedge a)$" +3305,$R_i$ +3306,$\delta(p)=1-\nu(p)$ +3307,$\bar F'(x) = F(x)$ +3308,$d=iv=1-v$ +3309,$X=h(Z)$ +3310,$F^{(-2)}(p)$ +3311,"$cos(0)*sin(90)*(1,1)$" +3312,$f_i$ +3313,$\rho(X)\ge \rho(Y)$ +3314,$1/4$ +3315,$\rho>1$ +3316,$a = \sum_i a_i$ +3317,$\nabla \zeta$ +3318,"$(Bob)+(0,-3.25)$" +3319,$\displaystyle\int_0^a xf(x)dx \not= \displaystyle\int_0^a S(x)dx$ +3320,$\mathcal X$ +3321,$t_i$ +3322,$\nu_p=(1+\rho_p)^{-1}$ +3323,$P(x)=g(S(x))$ +3324,"$\beta(X) = \int\check g(S_X(x))\,dx$" +3325,$\rho(Y)\le\rho(0)=0$ +3326,$g(S(x))=1-\tilde p$ +3327,"$\min(X, a)=X\wedge a$" +3328,"$\alpha,\beta$" +3329,$\rho(-X) \ge -\rho(X)$ +3330,$a/x$ +3331,"$a \in_{R} \{2,\dots,p-2\}$" +3332,$\sum f_i$ +3333,$C_1(t) > C_1(t)$ +3334,$s =$ +3335,$\mathsf{E}_Q(X_iY_i)=\mathsf{E}_Q(X_i\partial Y/\partial X_i)$ +3336,$\rho_m(X) = \mathsf{E}(X) + (\rho_m(X) - \mathsf{E}(X))$ +3337,$\text{Var}(G)=c$ +3338,$t<0.25$ +3339,$\mathscr{P}$ +3340,$p-1$ +3341,$h_\epsilon\to h$ +3342,$p=F$ +3343,$G = C + \sum N_i$ +3344,"$\mathcal{M}_{X,c}=\mathcal{M}$" +3345,$\delta F(a)$ +3346,$p-p\nu_p = p\delta_p$ +3347,$c=\lambda$ +3348,$X\in L_p$ +3349,$PQ = P/Q$ +3350,"$f: [0,1]\to [0,1]$" +3351,$X_i(a)$ +3352,$23.81 / 34.05 = 70$ +3353,$\displaystyle\int_0^1 q(p)dp$ +3354,$E_g[Y] = \int g(S_Y(t))dt$ +3355,$1-s$ +3356,$\mathsf{E}[g(-Y)]\ge 0$ +3357,$\beta_i(t)$ +3358,$st$ +3359,$f_i(X)$ +3360,$Z_1=q_Z(F_X(X))$ +3361,$a(\mathbf{x}) =\mathsf{TVaR}_p(X(\mathbf{x}))$ +3362,$A = fX + Y$ +3363,$\rho(L) = F^{-1}(1-g{-1}(1-p)) dp > \mathsf{E}(L)$ +3364,$\delta^2 p + \nu^2 q-(p-\nu)^2=p(1-p)$ +3365,"$(x,y)\mapsto (x,y)$" +3366,$9 = 2^3 + 1 = 2^{(2^1 + 1)} + 1$ +3367,$\rho(X) = \mathsf{E}(X) + c\mathsf{E}(X-\mathsf{E}(X))_+$ +3368,$n = 1$ +3369,$=\mathsf{E}(X_i \mid X=q(\alpha))$ +3370,$\lfloor pN\rfloor$ +3371,$\iota(a)$ +3372,$\rho(Y) = \rho(Y-X + X) \le \rho(Y-X) + \rho(X)$ +3373,$A=g^a$ +3374,$<\alpha$ +3375,$\lambda(p=1)=0$ +3376,$\tilde \rho(X)=\inf\{ \alpha \mid X+\alpha \in\mathcal{A} \}$ +3377,$\mathcal{G}$ +3378,$X_t=X-t\bar X \le X$ +3379,$X_k=X_0+k$ +3380,$Y=Y(\mathbf{X})$ +3381,$X^{\oplus n} -\mathsf E[X] \succeq_2 X^{\oplus n-1}$ +3382,$\mathbf{Tm} = \mathbf{r}$ +3383,$t q(p)] \ge q(p)$ +3392,$Z=AX + (1-A)Y$ +3393,$c=q(\alpha)=VaR_\alpha(X)$ +3394,$L_i$ +3395,"$u_1,u_2$" +3396,$F^{-1}$ +3397,"$X\wedge \alpha(X):=\text{min}(X, \alpha(X))$" +3398,$\mathcal F_1=\sigma(X)$ +3399,$X_i = F(e_i)$ +3400,$t_1 > t_2$ +3401,"$\mathsf{cov}(N, Z_0)<0$" +3402,$N=20$ +3403,$X\le Y$ +3404,$^{***}$ +3405,$0.5 < t_1 < t_2$ +3406,$-1_{B_l}$ +3407,$\nu(p)-l(p)= \nu^*\sqrt{(1-p)/p}$ +3408,$g(S(X))$ +3409,$u^{(4)}<0$ +3410,$\preceq_k$ +3411,$E(X^k)\le E(Y^k)$ +3412,$g(0.x_1x_2x_3...) = 0.x_2x_4\dots$ +3413,$\lim_{\gamma\to\infty} \rho_\gamma$ +3414,$a=(1-f)^2/\nu^2=(1-f)^2/c$ +3415,$g(s) = d + sv$ +3416,$x_1=q(p)$ +3417,$\mathsf{E}(\Pi)$ +3418,$\rho(X)=E_\mathsf{Q}(X)$ +3419,$\dfrac{q(\epsilon)}{1+\epsilon}$ +3420,$\| Y \|_{\sigma_2} \le c \| Y \|_{\sigma_1}$ +3421,$g(s)=s^\alpha$ +3422,$dx\to 0$ +3423,$\int^x H(s)ds \ge 0$ +3424,$m\in \mathbb R$ +3425,$u^{iv}<0$ +3426,$G'$ +3427,$S(\mathbf{x}; a)$ +3428,$LR = L/P$ +3429,$\rho F$ +3430,$a=q_X(0.99)$ +3431,${}^nS(t) = \displaystyle\int_t^\infty {}^{n-1}S(u)du$ +3432,$\rho(X) = \rho(Y)$ +3433,$p\approx 0.01$ +3434,$\square \phi_i$ +3435,$u=a$ +3436,$t=1$ +3437,$\mu_t$ +3438,$h$ +3439,$2^1\rightarrow 3^1-1=2 \rightarrow 1 \rightarrow 0$ +3440,$\lambda_t=\lambda$ +3441,$N=2$ +3442,"$D_i(X_1,\dots,X_n; a)$" +3443,$\hat\rho(A_k) = \hat\rho(A_0) + k \rho(N)$ +3444,$\|Y\|_{\sigma}=\int_0^\infty \tau_\sigma(F_{|Y|}(y))dy$ +3445,$t-dt$ +3446,$0\le X_n\le 1$ +3447,$g^{ak} = (g^a)^k$ +3448,$X\le Y\implies f_t(X)\le f_t(Y)$ +3449,$(1-t)X_1 + tX_2$ +3450,$-1_{A^c}$ +3451,$M_G(\zeta) = (1-\theta\zeta)^{-a}$ +3452,$l(p) = v(1-\sqrt{(1-p)/p})$ +3453,$dN(a)=d(a-\mathsf{E}(X\wedge a)$ +3454,$A(c)=c$ +3455,$\mathsf{E}(X_i \mid X\le a)$ +3456,$f(x)dx$ +3457,$\tilde \rho_t = \rho_t(-\tilde\rho_{t+1})$ +3458,$\partial f(x_0)$ +3459,"$t\in[0.12, 0.25]$" +3460,$q_{\mathbf{x}}=F_{\mathbf{x}}^{-1}$ +3461,$C_2(0)\approx \mathsf{E}[X_2]$ +3462,$A - \mathsf E[A] = A_0 + (\mathsf E[X]N - \mathsf E[A])$ +3463,$\lambda_i$ +3464,$\rho(X) \ge \rho(Y)$ +3465,$i= \alpha/(1-\alpha)$ +3466,$dF=-dS$ +3467,$\iota(p)=\delta(p)/\nu(p)$ +3468,$\rho(X+Y)\ge$ +3469,$s=1-p$ +3470,$\delta(p)=\iota(p)/(1+\iota(p))=1-\iota(p)$ +3471,$t>0$ +3472,$q(p)=F^{-1}(p)$ +3473,$F(x)=1-e^{-\mu x}$ +3474,$> r$ +3475,$B(b)<0$ +3476,$t_1 p$ +3481,$B(p)=1$ +3482,$X(t)=X$ +3483,$\mathsf{E}_Q(Y)$ +3484,$s_f$ +3485,$>2$ +3486,$A_k$ +3487,$g'(S(x))f(x)dx$ +3488,$q_\alpha$ +3489,$2^{10}$ +3490,$1/\mu$ +3491,$a=\sum_i a_i$ +3492,$(\mathsf{E}(X_i)-\mathsf{E}(X_i(a))/\mathsf{E}(X_i)$ +3493,$\rho(X+Y)\ge\rho(X)+l(Y)$ +3494,$V(a) = 1_{X > a}$ +3495,$f_t(X+m)=f_t(X)+m$ +3496,$\mathbb{P}$ +3497,$x^\\alpha$ +3498,$\rho(\tilde X)=34/9$ +3499,$g(s)-s$ +3500,$\mathscr{P}=\{\mathsf{Q} \mid d\mathsf{Q}/d\mathsf{P} \le 1/(1-p) \}$ +3501,$0 p$ +3523,$\text{E}(G)=1$ +3524,$R_1(0)=\bar P^a_1(0)$ +3525,$()_+$ +3526,"$\langle \nabla\zeta, N \rangle + \langle \zeta, \nabla N \rangle$" +3527,$L_a$ +3528,"$\mathcal{A}=\{\mu\in \mathscr{P} \mid \langle \mu,X \rangle \le \rho(X) \ \forall X\in\mathcal{X}\}$" +3529,$\nu < p$ +3530,$x:3x:9x$ +3531,$x=\lambda y + (1-\lambda)z$ +3532,$p(1-\nu_p)$ +3533,$x_i\mapsto x_i f_i(x_i)$ +3534,${}^nS_X(t)\le {}^nS_Y(t)$ +3535,$\bar\iota>0$ +3536,$\tilde X=X\wedge a$ +3537,$id\times\pi$ +3538,$Y_i$ +3539,$\mathsf{E}(X)=\int_0^\infty xf(x)dx$ +3540,$E_2 = 0$ +3541,$Y=-X$ +3542,$\rho(X)=\mathsf{E}(Xg'(S(X)))=\mathsf{E}_Q(X)$ +3543,$a\le \rho(X)\le b$ +3544,$L_\sigma=F_L^{-1}(\tau_\sigma^{-1}(U))$ +3545,$\delta^*$ +3546,$(1-s) - (1-g(s)) = g(s)-s$ +3547,$=q-\epsilon\mathsf{E}_q(X_2)$ +3548,"$(I_1,\dots,I_n)$" +3549,$0.5 < t < 1$ +3550,$P(a)$ +3551,$(-2N\log(1-p))^{1/2}=22.49$ +3552,$X^n$ +3553,$\mathbf{n}$ +3554,$\phi'(s)=f(s)/(1-s)\ge 0$ +3555,$q(1)=\infty$ +3556,"$y,z\in X$" +3557,$(1-\nu_p-il_p)/(\nu_p-l_p)=\rho_{1/2}$ +3558,$p_i$ +3559,$g'(t)=\phi(1-t)\ge 0$ +3560,$a-EL$ +3561,$\ge$ +3562,$k_i$ +3563,"$(1-t,t)$" +3564,$P(x) = g(S(x))$ +3565,$t=-\log(s)$ +3566,$q_\zeta$ +3567,$\mathsf{MON}'$ +3568,$S(x)+R(x)$ +3569,$S_Z$ +3570,$M_i(t)=C_i(t)$ +3571,$T_s$ +3572,"$M(X_1, a_1)+M(X_2, a_2)=M(X_1+X_2, a_1+a_2)$" +3573,$\mathsf{PH}$ +3574,$B(p)$ +3575,$\sigma=1.333$ +3576,$. Therefore $ +3577,$\mathsf{E}(X-x)_+$ +3578,"$c\in[0,1/2]$" +3579,$\dfrac{d}{da}$ +3580,$q(0)$ +3581,$g\in\nabla\rho(X)$ +3582,$(\delta_p - il_p)/(\nu_p-l_p)$ +3583,$=\mathsf{E}(X_i(a))$ +3584,$\mathsf{E}(X ; B)$ +3585,$\delta = g(s)g(t)-g(st)$ +3586,$\rho(\lambda X)=\lambda \rho(X)$ +3587,$\zeta=1+c(1-\mathsf{Pr}(Z>\mathsf{E} Z)$ +3588,"$(x, S(x))$" +3589,$k+1$ +3590,$E(G-E(G))^3 = 2a\theta^3$ diff --git a/greater_tables/data/tex_list.csv b/greater_tables/data/tex_list.csv index 436e08c..e3768bc 100644 --- a/greater_tables/data/tex_list.csv +++ b/greater_tables/data/tex_list.csv @@ -1,5849 +1,16720 @@ ,expr -0,$\mathbf {s_3}$ -1,$\bar M$ -2,$(1+r)\lambda \mathsf{E}[X]$ -3,$m(1)=m_3=0$ -4,$X_2=2$ -5,$a=1$ -6,$e^{-kX}/\mathsf{E}[e^{-kX}]$ -7,$U < s$ -8,$n \le pN < (n+1)$ -9,"$\mathsf{TI,\ MON}$" -10,$\log(g')$ -11,$\mathsf{E}_{\mathsf Q}[X\mid \mathcal F]=\mathsf{E}[XZ\mid \mathcal F]/\mathsf{E}[Z\mid \mathcal F]$ -12,$(.*?)\$ -13,$\rho(X)=\infty$ -14,$F(x-) = \lim_{t\uparrow x} F(t)$ -15,"$\mathsf{MON,\ TI,\ PH}$" -16,$Y\succeq Z$ -17,$|S|$ -18,$\mathsf{CONVEX}$ -19,$s^{1/2}$ -20,$1000e^{\mu}$ -21,$p^* =0.7501$ -22,$X=\sum_j X_j$ -23,$\beta_{2}$ -24,$\sigma=0.50$ -25,$Z(s)=\Phi^{-1}(s)$ -26,$\hat p=1-g^{-1}(1-p)$ -27,$\kappa_i(X)=\mathsf{E}[X_i\mid X]$ -28,$\mathsf{E}[X_i\mid X](\omega)$ -29,$\sigma^2 t$ -30,$\uparrow\uparrow$ -31,$F(x)=1-e^{-x/\mu}$ -32,$g(S(X))$ -33,$0<\rho\le 1$ -34,$P = \mathsf{E}[X] + \pi\mathsf{E}[X]$ -35,$\bar Q_{0}=a_{0}-\bar P_{0}$ -36,$s\downarrow 0$ -37,$X=\frac{1}{n}\sum_i X_i$ -38,$>(s_0/2^{n+1})2^n\bar q(s_0)=s_0\bar q(s_0)/2$ -39,$\rho(X)>\max(X) g(0+)=\infty$ -40,$\mathsf{E}[Z]\le 1$ -41,$\lambda\to\infty$ -42,$\mathsf{j}(a)=6$ -43,"$g(s)=w+(1-w)s, s>0$" -44,$\mathsf{TVaR}_{0.65}$ -45,$c(S)=g(\mathsf{Pr}(S))$ -46,$c(S\cup\{i\})=c(S)+c(i)$ -47,$\mu(\{p_j\})$ -48,$\mathsf{Pr}(E')+\mathsf{Pr}(E)=\mathsf{Pr}(\Omega)=1$ -49,$q(Y)$ -50,"$(\Omega, \mathcal F, \mathsf{Pr})$" -51,$Z_A$ -52,$\mathcal D(X)\ge 0$ -53,$p=\text{Pr}[L^* > A]$ -54,"$\beta_H:=\mathsf{cov}(r_H, r_M)/\mathsf{var}(r_M)$" -55,$X_{t+dt}=X_t + \mu dt + \sigma dW_{dt}$ -56,$\rho(X)\ge \mathsf{E}[X]$ -57,$u(x)=-v(-x)$ -58,$g(x)=1$ -59,$F_{\mathbf{v}}(x)=s$ -60,${n}-X_2$ -61,$U_X > p$ -62,$b_i$ -63,$\rho(\nu Z) \le \nu\rho(Z)$ -64,$\Phi(x):=\int_{-\infty}^x \phi(t)dt$ -65,$\mathsf{E}[X]=27.5$ -66,$U = A$ -67,$X\le l$ -68,$U_X < p$ -69,$g'(1-p) \frac{q\wedge \alpha}{q}$ -70,$rpq$ -71,$c>0$ -72,$Y=0$ -73,$\mathbf \Omega$ -74,$\rho(X)=\max_k \mathsf{E}_{\mathsf Q_k}[X]$ -75,$1-p_0$ -76,$L(X)=k(X-\mathsf{E} X)$ -77,$P = \mathsf{E}[Xe^{\pi X}]/\mathsf{E}[e^{\pi X}]$ -78,"$(p, 1-g^{-1}(1-p))=(p,\hat p)$" -79,$\mathit{MV}(a)$ -80,$Z_4$ -81,"$\kappa_i(\mathbf{v}, x)$" -82,"$x=A,L,S$" -83,$c(S)=\rho(\sum_{i\in S} X_i)$ -84,$S_X(a)$ -85,"$a,b=\pm 1/n$" -86,$\mathbf {X_{2}(a)}$ -87,"$x_{1,i}, x_{2,i}$" -88,$1_{X>a}$ -89,"$\int_0^\infty -z(x)\,dF(x)=-1$" -90,$k\mapsto k\rho(X)$ -91,$\rho_g(X)=\mu+\lambda\sigma$ -92,$\hat q$ -93,$F_X^{-1}(V)=q_X(V)$ -94,$0\le\beta<1$ -95,$p>S(x^*)$ -96,$a\le X\le b$ -97,$P(x)=A(1_{X>x})=g(S(x))$ -98,$g(S)\Delta X'$ -99,$1<\lambda=k+f$ -100,$\rho(X)=\mathsf{E}[X] + c\mathsf{Var}(X)$ -101,$1./16=0.0625$ -102,"$\alpha>1,0\le\beta\le 1$" -103,$\mathsf{Pr}(A)=1-p$ -104,$g''(s)\le 0$ -105,$S(x_{max})=0$ -106,$\{X=x\}$ -107,$\rho_g(X\wedge a)$ -108,$Z=(1-p)^{-1}1_{\tilde X>q_{\tilde X}(p)}$ -109,$Z_1$ -110,"$X_{t-1,1}$" -111,$X_2(10)$ -112,"$X_{t,3}$" -113,$X\le x$ -114,$r = (g(s)-s)/(1-g(s))$ -115,$1_A/\mathsf{Pr}(A)$ -116,$\mathsf{TVaR}_1(X)$ -117,$\rho(Y)=\rho(X)g(p)=g(q)g(p).$ -118,$M(x)=g(S(x))-S(x)$ -119,$Y_{1}$ -120,$\mathsf{Pr}(X0$ -124,$X_n(\omega)\to X(\omega)$ -125,$^{***}$ -126,$\bar S(a)$ -127,$\mathsf{E}[X_i g'(S(X))]$ -128,$\sum (X\wedge a)p$ -129,"$\{1,2,\dots, N\}$" -130,$D\rho_{X_g}(X_c)$ -131,$(g(s)-s)/(1-g(s))=\iota$ -132,"$P_X(a,b] = F(b)-F(a)$" -133,$k > 0$ -134,$X_n\downarrow X$ -135,$x\to \infty$ -136,$\Phi(Z(s))=s$ -137,$q^-(p) = \inf\ \{ x\mid F(x) \ge p\}$ -138,$Y(\omega_1)\le Y(\omega_2)$ -139,$v(A)\le v(B)$ -140,$\mathbf {1_{X>x}}$ -141,$\alpha_i(a) S(a)$ -142,$\mathsf{E}[X]=\mathsf{TVaR}_0(X)$ -143,$\mathbf {Z_2}$ -144,$\hat{\tilde p}=1-g^{-1}(1-[1-g(1-p)])=p$ -145,$\pi(X)=\log(m_X(\alpha)) / \alpha$ -146,$\log(\mathsf{E}[e^{\pi X}])/\pi$ -147,$E[s|W=t]$ -148,$S(x)\gg 0$ -149,$1-\beta_i(x)g(S(x))$ -150,$S_X(x)=\Phi(-(x-\mu)/\sigma)$ -151,$\pi(X) = \rho(X\wedge \alpha(X))$ -152,$a(\mathbf{v}) =\mathsf{VaR}_p(X(\mathbf{v}))= q_{\mathbf{v}}(p)$ -153,$\mathsf Q \in \mathcal Q$ -154,$a=D+S$ -155,"$\bar P_{t,0}$" -156,"$0, 8, 10$" -157,$Q(x)/(1-S(x))$ -158,$p=1/6$ -159,"$\rho_2(X)=\mathsf{E}[X] + \mathsf{cov}(X,Z)$" -160,$\mathbf {g(S)}$ -161,$\rho=\mathsf{TVaR}_{0.95}$ -162,$f(S_t)=\log(S_t)$ -163,$\int_0^\infty xdF(x) =\int_0^\infty xf(x)dx$ -164,$u_j(x)$ -165,$f_{xx}=-1/S_t^2$ -166,$\mathbf {M_{2}\Delta X}$ -167,$\mathsf{E}[X\mid \mathcal F_t]$ -168,$X$ -169,$t+2$ -170,$n\ge m$ -171,$\mathbf {Z_4}$ -172,$|f|$ -173,$b$ -174,$g'(S(x))$ -175,$\mathsf{var}(Y_{d})=\sum_{s>d} \sigma_s^2$ -176,$r_l$ -177,$\mathbf {Z_8}$ -178,"$\rho(Y_{2,0})$" -179,$1+\iota^*=(1+\iota)(1+\tau)$ -180,$r_f/(1+r_f)$ -181,$L^r$ -182,"$\mathsf{E}[(X_i-\mathsf{E} X_i)(X-\mathsf{E} X)]/\mathsf{SD}(X)=\mathsf{cov}(X_i,X)/\mathsf{SD}(X)$" -183,$u(0)=0$ -184,$(ng)$ -185,$\tilde Z = \mathsf{E}[Z\mid X]$ -186,$E[X|X>qp]$ -187,$\rho(X) + c = \rho(X+c)\ge \rho(X) + \mathsf{E}[cZ]$ -188,$1-g(S)$ -189,$a_{0}$ -190,$\bar M_t = \bar P_t - \mathsf{E}[Y_{t}]$ -191,$\rho_g(X \wedge a)$ -192,$\rho(0)=\rho(0 \times X)=0\times \rho(X)=0$ -193,$\rho_g(X)$ -194,$\mathbf {\mu}$ -195,$\displaystyle\int_\Omega X(\omega)p(\omega)\mathsf{Pr}(d\omega)$ -196,"$n={{n}}, p=1/{{p}}={{pf}}$" -197,$\Delta Q_{gc}(a) = a_{gc}-P(X_{0}(a_{gc}))-a$ -198,"$\bar S_i = \sum_{j} X_{i,j}p_j$" -199,$\mathcal G\subset\mathcal F$ -200,$\tilde X_2 = X_2 - \mathsf{E}[X_2]$ -201,$10^{-12}$ -202,$\rho(X)=\mathsf{E}[XZ]$ -203,"$x\in[0,\infty)$" -204,$\mathsf{Pr}(S_t > a)=\mathsf{Pr}(X_t > a/S_0)=1-\Phi\left([\log(a/S_0)-(r-\sigma^2/2)t]/\sigma\sqrt{t} \right)=\Phi(d^*-\sigma\sqrt{t})$ -205,$F_0 = \bar P_{act}-\bar P = R-\bar M$ -206,$\mathsf{E}_\mathsf{Q}[X+c]=\mathsf{E}_\mathsf{Q}[X]+c$ -207,$X_{-3}$ -208,$\bar\delta$ -209,$t>0$ -210,"$(\Omega, \mathcal F, \mathsf{P})$" -211,$\mathit{LGD}$ -212,$\mathsf{E}[L\wedge A]$ -213,$\mu_c$ -214,$p<0.5$ -215,$a_h=2-a_l<2-b_l=b_h$ -216,"$F(p)=\mu([0,p])$" -217,$\mathsf{E}_\mathsf{Q}$ -218,$\lambda dt\to 0$ -219,$0 < p_0 < p_1 < 1$ -220,$\mathsf{E}[X] + d(\max(X)-\mathsf{E}[X])$ -221,$p\mapsto g'(1-p)$ -222,$\omega=0.\omega_1\omega_2\dots$ -223,"$\mathbf {X\,\Delta S}$" -224,$BCD$ -225,$\beta_i(x)<\alpha_i(x)$ -226,$\nu=\nu(p)$ -227,$a_1 = a(Y_{1})$ -228,$\mathit{NPV}_{\infty}=2\times 2.5=5$ -229,$dG/dF$ -230,$\mathbf {X(a)}$ -231,$M = P - \mu_U= 0.505$ -232,$H_k(X)=H_k(Y)$ -233,$l(p)$ -234,$\bar Q$ -235,$\mathsf{E}[N]=2.0$ -236,$L_0^{l_1} + L_{l_1}^{l_1+l_2} = L_0^{l_1+l_2}$ -237,$\mathsf{E}[X_d]$ -238,$X''$ -239,$\mathsf{VaR}_{0.7}(X)=2.439 > 2 \times 1.204=2.408$ -240,$\mathsf{CTE}^+$ -241,$0 < p < 1$ -242,$\displaystyle\int_0^\infty xg'(S_X(x))dF_X(x)$ -243,$\pi=0$ -244,$h(p)=1-g(1-p)=1-(1-p)^{1/3}$ -245,$\alpha(\mathsf Q)=\infty$ -246,$\gamma$ -247,$x\in A$ -248,$p_j=\mathsf{P}(X=x_j)$ -249,"$F_n,F$" -250,$\mathsf{Pr}(\cup_i E_i)=\sum_i \mathsf{Pr}(E_i)$ -251,$\rho(\lambda X)=\lambda\rho(X)$ -252,$\nu^{-1}\mathsf{E}[\nu(X)]$ -253,$A(1_{X>x})$ -254,$g(s)=(\iota+s)/(\iota+1)$ -255,"$\max(x, 0)$" -256,$x\mapsto x^{n}$ -257,$\mathsf{E}_{\mathsf Q}[X_i\mid X\le a](1-g(S(a))) + a\mathsf{E}_{\mathsf Q}[X_i/X\mid X >a]g(S(a))$ -258,$E[G]=1$ -259,$\Lambda = \dfrac{E( r_{U} ) - r_{f}}{\sigma_{r_{U}}}$ -260,"$\{90,\dots,99\}$" -261,$P = 3.103$ -262,$g(s) \ge s$ -263,$\mathsf{MONETARY}$ -264,$\mathsf{TVaR}_{0.95}(X)=\mathsf{E}[XZ]$ -265,$p(\omega)=0$ -266,$a(X_i;X) = \lim_{t\to 0} (\rho(X+tX_i)-\rho(X))/t$ -267,$\sigma_{U} = \sqrt{1 - 2p - p^{2}} = 0.973$ -268,$\sigma_A$ -269,$\mathsf{E}[X_1Z]$ -270,$\beta$ -271,$\mathbf {x}$ -272,$\mathit{NPV}_1 = \bar Q - \bar Q = 0$ -273,"$X_4, X_5$" -274,"$g:[0,1]\to[0,1]$" -275,$X+Y$ -276,$\sup_\mathsf{Q} \mathsf{E}_\mathsf{Q}[X]$ -277,$Y=1-X$ -278,$A\subset\Omega$ -279,$g'(s)\ge 1$ -280,$K_h(t):=k(h+t)-k(t)$ -281,$\mathscr{E}_i$ -282,$\rho_2$ -283,$y_c$ -284,$\mathsf{E}[X\mid t]$ -285,$1-F(q(p));\alpha)$ -286,$w(X)=1_{X>X_p}$ -287,$\delta=0$ -288,$q(0)$ -289,$|x|$ -290,$Y_n$ -291,$X_1+({n}-X_2)$ -292,$w=0.06405$ -293,$\sum_j Y_j = 0$ -294,"$P_X(a,b]=\mathsf P(X\in (a,b])=F(b)-F(a)$" -295,$e^{kx}S(x)\to\infty$ -296,"$f(\cdot, \omega)$" -297,$N_i$ -298,$\lambda S(x)$ -299,$\mathbf {M=g(S)-S}$ -300,$t=2$ -301,$\mathsf{E}[X_2(a)\mid X_1(a)=x] \le a-x$ -302,$0\le s\le 1$ -303,$\rho(X) \le 0$ -304,$x_{i-1}$ -305,$Y_{0}$ -306,$\infty-\infty$ -307,$\mathsf{j}(a) = \max\{j:X_j < a \}$ -308,$s \ne s^\ast$ -309,$\mathbf {d=1}$ -310,$\sigma_d^2$ -311,$P=L + \iota Q = \nu L + \delta a=L(1+\rho)$ -312,$\rho(X)=x_p$ -313,"$\mu=7.4, \sigma=1.9$" -314,$\mathsf{E}[X]+kR(X)$ -315,$\bar q(s/2)\le 2\bar q(s)$ -316,$Q_1=0.125$ -317,$\mathsf{E}[Z_j\mid X]$ -318,"$D_n, D_n^*$" -319,$\rho(X)=\mathsf{E}_{\mathsf{Q}}[X]=\mathsf{E}_{\mathsf{Q}}[\sum_i X_i]=\sum_i \mathsf{E}_{\mathsf{Q}}[X_i]$ -320,$a>b_h$ -321,$\sum_t Q_t$ -322,$0\le \lambda < 1$ -323,$\mathbf {t+2}$ -324,$-u''(w)/u'(w)$ -325,$q(p)=-\log(1-p)\mu$ -326,$\mathsf{E}_Q[X_i\mid X]=\mathsf{E}[X_i\mid X]$ -327,$1=v+d$ -328,$n=2$ -329,$\mathsf{P}(1_{U < s}=1)=\mathsf{P}(U < s)=s$ -330,$X=U$ -331,$X(\omega') = \sum_\omega X(\omega)1_\omega(\omega')$ -332,$a'$ -333,$U_i$ -334,"$\bar P_{0,1}$" -335,$g_i=u_i^{1/b} < u_i$ -336,$\rho(X\wedge a)=\bar P(a)$ -337,$E(X\wedge a)=\bar S(a)$ -338,$1-g(0^+)$ -339,$\alpha\not\equiv 0$ -340,"$[0,1]\times [0,1]$" -341,"$X_{i,j}\Delta g(S_j)$" -342,$c_i=\displaystyle\sum_{i\not\in S\subset\Omega}\dfrac{|S|!(N-|S|-1)!}{N!}\times$ -343,$\mathbf {\sigma}$ -344,"$\mathit{MV}(X, a) = a - \rho(X\wedge a)$" -345,$u'(0)=1$ -346,$S(x)=0.1$ -347,$s=0.01$ -348,$\int_a^{a+y} g(S(x))dx$ -349,$\sum X_i(a)p$ -350,$\beta(x)\le \alpha(x)$ -351,$X_1=18$ -352,$g(s)$ -353,$Z'(s)=1/(\Phi'(Z(s)))=\sqrt{2\pi}\exp(Z(s)^2/2)$ -354,$D/L$ -355,"$S\,\Delta X$" -356,$a=11$ -357,$\rho(X+tY)\ge \mathsf{E}_{\mathsf Q_X}[X+tY]$ -358,$\mathsf{E}[X_1]=4.75$ -359,$\log(1-1/n)<-1/n$ -360,"$, which he describes as the standard way to obtain the $" -361,$\phi(p) = g'(1-p)$ -362,$\mathsf{VaR}_p(X_1+X_2)\le \mathsf{VaR}_p(X_1)+\mathsf{VaR}_p(X_2)$ -363,$P(X_i(a_{gc}))$ -364,$n$ -365,$t > 1/3$ -366,$\mathsf{E}[u(P-X)]=0$ -367,$\mathsf{Var}(\pi)$ -368,$g'(S(x))f(x)$ -369,"$(lee.west |- lee.north)+(0,-2.5)$" -370,"$D^n\rho_X(X_{i,\cdot})$" -371,$-x^2$ -372,$X_n\to X$ -373,$r_f/(1+ r_f) = 0.0196$ -374,$\mathbf {g_4(s)=s^{0.9}}$ -375,$D\rho_{X_n}(X_c)$ -376,$f_{opt} =(pb - q)/b$ -377,$\{n\mid X(n)\not =0\}$ -378,$\mathsf{TVaR}_0(\cdot)=\mathsf{E}[\cdot]$ -379,$\mathbf {\iota}$ -380,$\rho(X_0+Y) \ge \rho(X_0) + \mathsf{E}[YZ]$ -381,$\ge 1$ -382,$n-3$ -383,$Q = C + lg$ -384,"$(1-p, 1]$" -385,$\tilde X-X$ -386,$\Delta Q_{ro}(a)$ -387,$\mathsf{E}[Z_1]=\mathsf{E}[Y]$ -388,$\lim_{x\to\infty}F(x)=1$ -389,$\mathsf{E}[X_i]=14$ -390,$g^{-1}$ -391,$p=0.9973$ -392,$M=P-s$ -393,$f(x_i)$ -394,$\mathcal F'_0\subset\mathcal F_0$ -395,$M/EL$ -396,$\mathit{EER}$ -397,$a(c_1;X) = c_1$ -398,"$\delta = 34/39, \nu=5/39$" -399,$\mathsf{P}(\{\omega\})$ -400,$A(X)-B(X)$ -401,$\rho(X\wedge a) = \sum\rho(X_i(a))$ -402,$q(0)=0$ -403,$k=c/(e^c-1)$ -404,$\Lambda = \dfrac{M - K r_f}{\sigma_U}$ -405,$\nu < 1$ -406,$\rho_g(X) = \infty$ -407,$U''(x)<0$ -408,$M = P \mu_U = 0.3$ -409,$\bar S_i(a)$ -410,$y=$ -411,$g'(S(x))=v$ -412,$\mathsf{Pr}(\{\omega_1\})=1/3$ -413,$\bar Q(a)$ -414,$\mathsf{j}(a)=4$ -415,$\mathsf{TVaR}_{0.8}(X)$ -416,$L/P$ -417,$\bar P(a+da)-\bar P(a)$ -418,$t+d$ -419,$g(0+)M$ -420,$Z(\omega)\mathsf{P}(\omega)$ -421,$\mathsf{E}[X_0]=80$ -422,$\mathbf {X_{1}(a)}$ -423,$t > 0$ -424,$g'(S(x))f(x)dx$ -425,"$k\mathsf{E}[(X_i-\mathsf{E} X_i)(X-\mathsf{E} X)]=k\mathsf{cov}(X_i,X)$" -426,$v_f(\mathsf{E}_\mathsf{Q}[X_i] - \dfrac{\mathsf{E}_\mathsf{Q}[X_i]}{\mathsf{E}_\mathsf{Q}[X]}\mathsf{E}_\mathsf{Q}[(X-a)^+])$ -427,"$(\x*0.65, 3.75*2)$" -428,$\rho$ -429,$\mathsf{E}_\mathsf{Q}[X_i]$ -430,$\hat p = F(x) = 1-g^{-1}(1-p)$ -431,"$\min(x_1,x_2)$" -432,${\mathsf{Q}}$ -433,$0=\rho(0)=\rho(X-X)\le \rho(X) + \rho(-X)$ -434,$c(\mathsf{var}nothing)=0$ -435,$f'_-(x)\le f'_-(y)\le f'_+(y)$ -436,$v_f\mathsf{E}_Q[X_i]$ -437,"$(x_{1,1}, x_{1,2})$" -438,$\sum_n 1/n$ -439,"$\displaystyle\int_0^a \alpha_i(x)S(x)\,dx$" -440,"$\beta(X,M)=\mathsf{cov}(X,M)\sigma_M^2$" -441,$X_{-1}$ -442,$\mathcal Q=\{\mathsf Q\mid \alpha(\mathsf Q)=0 \}$ -443,$A_i$ -444,"$a(X,p)$" -445,$r\lambda\mathsf{E}[X]$ -446,"$(s,\iota)$" -447,$a-L_0^a(X)$ -448,$\tilde Z=\mathsf{E}[Z\mid X]$ -449,$S(a+x)=d/dx(\mathsf{E}[X \wedge (a+x)-X \wedge a)$ -450,"$[p_{-},p_{+}]$" -451,$y=x$ -452,$\inf_x \{ x + \alpha\mathsf{E}[(X-x)^+] + \beta\mathsf{E}[(X-x)^-] \}$ -453,$af$ -454,$M$ -455,$\mathsf{Pr}(\mathsf{var}nothing) =0$ -456,$\mathsf{TVaR}_{p^\ast}$ -457,$\mu=0.107$ -458,$E(X_{-1}(a))$ -459,$g'(S_X)$ -460,$j > 0$ -461,$a=\sum_i a\alpha_i(a) = \sum_i\kappa_i(a)$ -462,$\mu=0$ -463,$x>1$ -464,$F(p)=p$ -465,$X_i$ -466,$q_{\tilde X}$ -467,$a\le \dfrac{P-S}{\iota} + P\approx \dfrac{P-\mathsf{E}[X]}{\iota} + P$ -468,$\omega\in \Omega$ -469,$Y_c=(Y\mid Y > y_c)$ -470,$(m_1-m_0)/s_1$ -471,$q_B(p)=\sup B$ -472,$\mathsf{E}[X]+k\mathsf{var}(X)$ -473,$M_1\Delta X$ -474,"$(a,b]$" -475,$\rho(m)=\rho(0)-m$ -476,$\mathbf v$ -477,"$\omega=(1,0,0,1,0,0,\dots)$" -478,$g(S(x))=1$ -479,$0 < s < 1/4$ -480,$r_h$ -481,$X\ge a$ -482,$Q$ -483,$p\delta_p$ -484,$y^{\ast}$ -485,$\nu=1/(1+\iota)$ -486,$\mu=0.1$ -487,$s_1=0$ -488,$p=0.4$ -489,$g(S_{X}(x))$ -490,$\mathsf{Q}(B_k)=\mathsf{P}(B_k)/\mathsf{P}(B_k)=1$ -491,$m(t^\star)=3m/4$ -492,$n_s(1-g(s))$ -493,"$g,h:[0,1]\to [0,1]$" -494,$x_{(j)}-x_{(j-1)}$ -495,$\mathsf{SRM}$ -496,$v\in V_X$ -497,$a(X_i)$ -498,"$\mathsf{var}(W)=\sum_{d\ge 0} \mathsf{var}(Y_{-d,d})$" -499,$\mathsf{E}_{\mathsf{Q}}[X]$ -500,$A/L$ -501,$a_{2}$ -502,$\rho_g(X)=\bar P$ -503,$\arg \min_{q \in \mathbb{Q}} E_q[U(a)]$ -504,$X=X_1+X_2$ -505,"$n=(0.702, 1.163)$" -506,$\sum_i$ -507,$\phi'(p)$ -508,"$(X_{1,j},\dots,X_{m,j})$" -509,$E(X\wedge a)$ -510,$1/6$ -511,$\mathsf{Pr}(\{\omega_2\})=2/3$ -512,"$\Omega=\{\omega_1,\omega_2,\omega_3,\omega_4\}$" -513,$a(X_i;X)\ge \mathsf{E}[X_i]$ -514,$\nu = 1/\lambda$ -515,$\alpha \le 1$ -516,$n\times m$ -517,$\mathsf{Q}$ -518,${6 \choose 2}=15$ -519,$\mathsf{E}[X \mid U]$ -520,$\sup(\lambda X)=\lambda \sup(X)$ -521,$P+Q=a$ -522,$k=2$ -523,$f(x) \to 0$ -524,$X=1$ -525,$v_1X_1(1)$ -526,$\pi=\Pi/p\nu(p)$ -527,$\mathcal{N}_X(X_i(a))$ -528,$\mathcal B_p$ -529,$S(x)\le s^*$ -530,$q_A \le q_B$ -531,"$A_2=[\epsilon, \epsilon]$" -532,$X=\sum_i X_i$ -533,$K = A - P$ -534,"$(1-g(s), 1-s)$" -535,"$r=1,2,3,4$" -536,$0=x_0 1-p) \}$ -549,$X=g(Z)$ -550,$P = \mathsf{E}[X] + \pi\mathsf{Var}^+(X)$ -551,$Y\mid Y > y_c$ -552,$a_1' = a_0-X_1$ -553,"$X_{t-1,3}$" -554,$\mathbf{B}(t)$ -555,$\mathsf Q\in\mathcal Q(X)$ -556,$g''<0$ -557,$g(w s_1 + (1-w)s_2) \le w g(s_1) + (1-w) g(s_2)$ -558,"$k=1,\dots,m$" -559,$S_t=S_0 X_t$ -560,"$G=\mathrm{cl}\{\, (\mathsf{E}_\mathsf{Q}[X_i], \mathsf{E}_\mathsf{Q}[X]) \mid \mathsf Q\in\mathcal Q \, \}$" -561,$\rho(-X)$ -562,$\mathsf{E}[X]\le \mathsf{E}[Y]$ -563,"$[s_1,1]$" -564,"$[0, 1-p]$" -565,$X(\omega)=1-\omega$ -566,$1-g(S(x))$ -567,$T = \min\{ t:U(t)\le 0 \}$ -568,$x_0=q^-(p_0)$ -569,"$\beta_i(t\mathbf{v}, x)$" -570,$\lambda=g(\lambda_{obj})$ -571,"$[-2\pi, 2\pi]$" -572,"$\mathsf{E}[X_i\,\mathsf{E}[Z\mid X]]$" -573,$X(\lambda\mathbf{v})$ -574,"$\bar P_{t,0} = D\rho_{W_t}(Y_{t,0})$" -575,$a>1$ -576,$a=R+Q$ -577,$k-L_0^k$ -578,$p\ge 0$ -579,$\mathsf{E}[\iota Q] = \mathsf{E}[\iota]\mathsf{E}[Q]$ -580,$\int g(S)$ -581,$\mathcal E(X)=\mathsf{E}[(p X^+ + (1-p)X^-)/(1-p)]$ -582,$0\le f<1$ -583,"$I(q,p)=0$" -584,$1_{X < q(1-s)}$ -585,$g - s$ -586,$x_i=1$ -587,$x\ge q(1-s^*)=:x^*$ -588,$\mathsf{TVaR}_0(X)=\mathsf{E}[X]$ -589,$X\succeq Z$ -590,$0\le w\le 1$ -591,$\mathsf{CTE}$ -592,$\iota = \dfrac{\delta}{1-\delta}$ -593,$X=x$ -594,$g^{-1}(s)$ -595,$U(0)=2$ -596,$\alpha = 0.642.$ -597,$s>1-p$ -598,$M_i := \beta_ig-\alpha_iS$ -599,${}^2$ -600,$C_c$ -601,$ROL = a + b\ \mathit{EL} + c \ C(t)$ -602,$X_2=0$ -603,$M=\delta a'$ -604,$\alpha(x) S(x)>\beta(x) g(S(x))$ -605,$P(X_{-1}(a_{gc}))$ -606,$L = \text{E}[L^*\wedge A]$ -607,$c(S)$ -608,$A\cap B\subset B$ -609,$g(s) = 1 - (1 - s)/(1 + r_f + Ck(s))$ -610,$X-b\le 0$ -611,$a=\mathsf{E}_\mathsf{Q}[X]$ -612,$f(x)=(\sqrt{2\pi}x)^{-1}\exp(-(\log(x)-\mu)^2/2\sigma^2)$ -613,$r_f=0$ -614,$\mathsf{VaR}_p(X)-f(\mathsf{VaR}_p(X))$ -615,$MX$ -616,$\mathsf{E}_\mathsf{Q}[\lambda X] = \lambda \mathsf{E}_\mathsf{Q}[X]$ -617,"$\displaystyle\int_0^{1-g(S(a))} \kappa_i(q(1-g^{-1}(1-p)))\,dp + a\beta_i(a)g(S(a))$" -618,$X(\omega)=\exp(10 + 2\Phi^{-1}(\omega))$ -619,$g(s)=\nu s + \delta$ -620,$\mathsf{E}[W\tilde X] \le \rho(\tilde X)$ -621,$W$ -622,$\mathsf{var}nothing$ -623,$f=f_x=f_{xx}$ -624,$1_A$ -625,$\wedge$ -626,$g'(s)$ -627,$a$ -628,$\mathsf{E}[Y]$ -629,$\rho(X)=\rho(\mathsf{E}[X]+X-\mathsf{E}[X])=\mathsf{E}[X] + \rho(X-\mathsf{E}[X])$ -630,$\mathsf{E}_\mathsf{Q}[X]$ -631,$X\wedge l$ -632,"$X_{t-d,d}$" -633,$\alpha(\mathsf Q)=0$ -634,$\bar q_{X_1+X_2}(s) \approx \bar q(s/2)$ -635,$X_2$ -636,"$(s,g(s))=(0.2,0.36)$" -637,$\mathsf{E}[kX]=k\mathsf{E}[X]$ -638,$ \& $ -639,$\inf_x\{ x + c{(X-x)^+} \}$ -640,$P(X\wedge a)$ -641,$x_2(S(x_1)-S(x_2))=x_2\mathsf{P}(X=x_2)$ -642,$1-g(S(a))$ -643,$\mathsf{E}_\mathsf{Q}[X_i \mid X]=\mathsf{E}[X_i \mid X]$ -644,$\| Z \|^*= \sup\ \{ \mathsf{E}[YZ] \mid \| Y \| \le 1 \}$ -645,"$Y_{1,0}$" -646,$\nu^{\ast}$ -647,$A(\lambda X)=A(\lambda X)$ -648,$dF$ -649,$\downarrow\downarrow$ -650,$\rho_2(X_1)=1$ -651,$-X$ -652,"$[x_1, x_2]$" -653,$v_f(\mathsf{E}_\mathsf{Q}[X_i] - \mathsf{E}_\mathsf{Q}[X_i/X(X-a)^+])$ -654,$\kappa_i(x)$ -655,$\mathbf {g_2(s)=s^{0.5}}$ -656,$r-r_L$ -657,$\mathbf {S\Delta X}$ -658,$\alpha_i(x) S(x)$ -659,$(g(s_0)-g_0)/s_0 = g'(s_0)$ -660,"$\mathbb{Q} = \left \{ q:I(q,p) \le I^* \right \}$" -661,$\rho=0$ -662,$\mathsf{E}_{\mathsf Q}[\cdot]$ -663,$\mathbf {Q}$ -664,$s=f'(x_0)$ -665,$\rho(X)=\sup(X)$ -666,$g(0+)>0$ -667,$S(x)=e^{-\beta x}$ -668,"$s_g, s_b$" -669,$1000$ -670,$da>0$ -671,$\mathbf {\beta_{2}g(S)\Delta X}$ -672,$\mathsf{P}(X=0)=0.4$ -673,$u'''\ge 0$ -674,$0\le \lambda_1 \le 1$ -675,$\rho(X+tY)\ge \mathsf{E}_{\mathsf Q_X}[X+tY]=\mathsf{E}_{\mathsf Q_X}[X]+\mathsf{E}_{\mathsf Q_X}[tY]=\rho(X)+t\mathsf{E}_{\mathsf Q_X}[Y]$ -676,$P_X$ -677,$x_1+x_2=x$ -678,$=\mathrm{MV}(X\wedge a)$ -679,$M_i(x)+Q_i(x)=\alpha_i(x)F(x)$ -680,$\delta = \iota/(1+\iota)$ -681,$a_1'=a_0-X_1$ -682,$X=\sum X_i$ -683,$\mathbf {S\Delta X'}$ -684,$X\le b$ -685,$\delta=\iota/(1+\iota)$ -686,$(\delta_p - il_p)/(\nu_p-l_p)$ -687,$x=\mathsf{VaR}_p(X)$ -688,$\mathbf {\alpha_2}$ -689,$1200/1800=0.667$ -690,$\sigma_0=\sigma_1$ -691,$a(f + (1-f)/q) -1$ -692,$g \cdot dX$ -693,$\beta_i(a)/\alpha_i(a) < 1$ -694,$\xtext$ -695,$Q_{1}\Delta X$ -696,$X_g$ -697,"$X=X(x_1,\dots,x_n)=x_1X_1 + \cdots + x_nX_n$" -698,$s\leftrightarrow 1-s$ -699,$\mathcal Q_i(X)$ -700,$\mathsf{E}[X] +\lambda\mathsf{E}[(X-\mathsf{E} X)^+]$ -701,$V_j$ -702,$X'=X\wedge a$ -703,$20+8t$ -704,$\mathsf{Pr}(X < x)\le \mathsf{Pr}(X\le x)$ -705,$\Delta_{2}$ -706,$\alpha_{2}$ -707,"$(1,1)$" -708,$4$ -709,"$Q_{i,j} = M_{i,j}/\iota_j$" -710,$L^\infty$ -711,$f(1)=1$ -712,"$0,10,40$" -713,$\rho(X+c)=\rho(X)+c$ -714,$H[Y_j]$ -715,$Z=(1-p)^{-1}1_A$ -716,$\beta_i(x)g(S(x))$ -717,"$A_3=[0, \epsilon-k]$" -718,$\mathsf{TVaR}_{0.95}$ -719,"$dx,dt,ds$" -720,$f(\omega)\ge 0$ -721,$\beta=0.57$ -722,$(X\wedge a)$ -723,$X < a$ -724,$\lambda<1$ -725,"$X_{0,1}$" -726,$\omega'\not=\omega$ -727,$X_0< X_1 < \dots < X_m$ -728,$\tilde X_1 + \tilde X_2 = X_1 + X_2$ -729,$\mathbf {X_1(a)}$ -730,$\mathsf{VaR}\_p(X\_0)$ -731,$-(1-s)g''(1-s) + g(0+)\delta_1 + \sum_s s(g'(s-)-g'(s+))\delta_{1-s} + g'(1)\delta_0$ -732,$a>a_{ro}$ -733,$g'(0)=\infty$ -734,$(X\wedge a)/X$ -735,$P_g\ll P_X$ -736,$Z\le (1-p)^{-1}$ -737,$F_g$ -738,$\bar P(x)$ -739,$d^*=(\log(A/L) + (r_h-\mu_L + \sigma^2/2))/\sigma\sqrt{t}$ -740,"$g(s)= \displaystyle\int_0^s g'(t)\,dt = (s/(1-p)) \wedge 1$" -741,"$(s_j=0,g_j>0)$" -742,$P'<\rho(W_1\wedge a_1)$ -743,$\mathsf{COHERENT}$ -744,$\Delta g(S_0)=1-g(S_0)$ -745,$\rho_g(V)$ -746,$X_t$ -747,$X_1+X_2=X=x$ -748,$m=1$ -749,$X_n\uparrow X$ -750,$v_1$ -751,$a\ge 10$ -752,$\mathbf {X_{1}}$ -753,$\gamma=0.633$ -754,$r=0.038$ -755,$1000(1+t)$ -756,$\mathbf {x_2}$ -757,$f(0)=0$ -758,$\mathcal M(\mathsf{P})$ -759,$p(\nu(p)-l(p))$ -760,$B(X)$ -761,$h(0.9)/0.9 = 0.76$ -762,"$\int_{[0,p]} \dfrac{\mu(dt)}{1-t}$" -763,$\mathsf{TVaR}_{0.5}(X_1)=9$ -764,${}^nS(t)$ -765,$Q(a)=\nu F(a)$ -766,$\rho(X_i)$ -767,$S(x_5)$ -768,$h_x$ -769,$\mathbf {Z_1}$ -770,$Y\le 0$ -771,$\mathsf{E}[X] + \pi \mathsf{E}[(X-\mathsf{E}[X])^+]$ -772,$(I/a + U/R)$ -773,$v=1/1.1<1$ -774,$0 < r \le 1$ -775,$\{ p \mid q^-(p) \le x \}=\{ p \mid p \le F(x) \}$ -776,"$(s,g(s))$" -777,$R_f=0$ -778,$\alpha_i'(x)>0$ -779,$\lim_{s\downarrow 0} g_\tau(s) = \tau / (1+\tau)$ -780,$\mathit{NPV}_1=0$ -781,$X\wedge a\Delta S$ -782,$\mathsf{TVaR}_{0.75}(X_2)=90$ -783,$K = A-P$ -784,$A\in\mathcal F'$ -785,$\le 0$ -786,$Z'(g(s))g'(s)=Z'(s)$ -787,"$\sum_i a(X_i, p^*)=a(X)$" -788,$a_{gc}:=\mathit{VaR}_{p}(X)=18000.0$ -789,$v=1/(1+i)$ -790,"$\alpha, \beta, \kappa$" -791,$S_{X\wedge a}(x) = S_X(x)$ -792,$W_0=Y_{0} + W_1$ -793,"$s_0, s_1, s_2$" -794,$AR$ -795,$S_j:=S(X_j)$ -796,$f'_-$ -797,"$ is average invested assets, equal to $" -798,$\mathsf{VaR}_{0.99}(X_2)=100$ -799,$X_t:=\mathsf{E}[X\mid \mathcal F_t]$ -800,$q(F(x))$ -801,$a_i$ -802,$X_1=t$ -803,$q=ps_g$ -804,$X>Y$ -805,$M=g(S)-S$ -806,$X=1800$ -807,$g_2(s)=s^{0.5}$ -808,$xS(x)|_0^\infty$ -809,$x_h(1-p)$ -810,$\nu+\delta=1$ -811,$\rho_i$ -812,$\mathbf {Q_{2}\Delta X}$ -813,$\mathsf{SSD}$ -814,$X_i\dfrac{X\wedge a}{X}$ -815,$r(X)=g'(S(X))$ -816,$X\wedge d$ -817,$1_{X>x_1}$ -818,"$\int g(S(x))\,dx$" -819,"$c(1,3)-c(3)$" -820,$0.5$ -821,$A(\lambda X)=\lambda A(X)$ -822,$\mathsf{Pr}(X=y_j)$ -823,$\mathsf{E}[u(R - X)]=0$ -824,$\rho(X\wedge a)=\mathsf{E}_\mathsf{Q}[X\wedge a]$ -825,$\mathbf {Z_5}$ -826,$c=(1-\alpha)^{-1}$ -827,$\mathsf{TVaR}_p(X)=1=\mathsf{E}_\mathsf Q[X]$ -828,$M_2dX$ -829,$\mathit{EGL}_{ro}(a)=P(X_{-1}\wedge a) - P(X_{-1}\wedge a_{ro}) \ge 0$ -830,$2\le x\le 8$ -831,$\mathsf{CTE}_p$ -832,$f(\mathsf{VaR}_p(X))$ -833,$\mathsf{E}_{\mathsf Q_k}[X'']=\mathsf{E}_\mathsf{P}[X'']$ -834,$X_n=X$ -835,"$Y_{t',d}$" -836,$\mathsf{E}[F_2]=\mathsf{E}[F_0]$ -837,$\mathsf{E}[e^{hX}] = \exp(h\mu+\sigma^2h^2/2)$ -838,$D\rho_X(X_i)=D\rho_i = x_i\dfrac{\partial\rho}{\partial x_i}$ -839,$a(X)\le a(Y)$ -840,$g'(s)<1$ -841,$\beta > \alpha$ -842,$\bar\iota=\iota$ -843,$\int_a^{a+y} S(x)dx$ -844,$0.125 \cdot 8 = 1$ -845,$h\left(\displaystyle\int_\Omega g(X(\omega))\mathsf{Pr}(d\omega)\right)$ -846,$\bar\delta(x)$ -847,$P_{act}-P$ -848,"$\rho(X, p^\star)=a(X)$" -849,$q(0.75)$ -850,$s=S_X(y)$ -851,$\rho l = \iota C$ -852,$\alpha(1-\alpha)(1-s)^{\alpha-1} + \alpha\delta_0$ -853,$Y_s$ -854,$\eta\nu$ -855,$(g_j-s_j)/(1-g_j)$ -856,$Z=g'(S_X(x))$ -857,$\mathsf{E}_{\mathsf Q}[Y]=\mathsf{E}[Yg'(S(X))]$ -858,$\Delta S_5$ -859,$F(x)$ -860,$D=(X-a)^+$ -861,$\sigma^2/2$ -862,$i=1$ -863,$h(p)\le p$ -864,$b = g/(1-g)$ -865,"$d=d(X_1,\dots,X_n)$" -866,$X=\max(X)$ -867,$v$ -868,$F(q(p))=p$ -869,$\mathsf{E}[Z]=1$ -870,$g(0+)=\mu(\{1\})$ -871,$\mathsf{E}[X\mid \mathcal F_{\tau}]$ -872,$X_i(a)$ -873,$p=0.999$ -874,$m\ge 1$ -875,$X_1(a)$ -876,$\Delta_s=g'(s-)-g'(s+)$ -877,$\mathsf Q \ll \mathsf P$ -878,$k/n$ -879,$L(X)=w(X)/\mathsf{E}[w(X)]$ -880,"$X_{t-1,2}$" -881,$\mathsf{Pr}(X\ge x)\ge 1-p\ge \mathsf{Pr}(X> x)$ -882,$d=1-v$ -883,"$f(t)=a(tx_1,\dots, tx_n)=ta(x_1,\dots, x_n)$" -884,$\partial a/ \partial v_i$ -885,$-g''$ -886,$g'(1)=0$ -887,$\mathsf{E}[X_ih(X)]=\mathsf{E}[\kappa_i(X)h(X)]$ -888,$\mathsf{E}[XZ(X)]$ -889,$P(a)=g(S(a))\ge S(a)$ -890,$x\mapsto x$ -891,$x^{\ast}=\mathsf{VaR}_p(X)$ -892,$\mathsf{E}[X] \le \bar P \le \sup X$ -893,"$(1,\dots,1)$" -894,$\mathsf{Pr}(X=x_i)=\lambda_i/\lambda$ -895,$Y=-X$ -896,$\lim_{y\downarrow x} f(y)$ -897,$\iota=0.1$ -898,$A_Y = 2.155$ -899,$g(S)=1$ -900,$X:=Y$ -901,$0.05$ -902,"$\mathbf {j, p, S, \kappa_1, \Delta X, \Delta(X\wedge a)}$" -903,$\mathsf{Pr}(M=m)=\frac{r}{1+r}\frac{1}{(1+r)^m}$ -904,$xS(x)\vert_0^\infty =\lim_{x\to\infty} xS(x)=0$ -905,$k!$ -906,$\kappa_i(x)=\mathsf{E}[X_i\mid X=x]$ -907,$602.6 billion and converted to net premium based on $ -908,$q(p)\phi(p)\times dp$ -909,$B_t$ -910,$ABC$ -911,$\lim_{x\to-\infty}F(x)=0$ -912,$\mathsf{E}[X_2\mid X=20]=6$ -913,$\mathbf {M_2\Delta X}$ -914,$a = 0.6565$ -915,$\mu(ds)$ -916,$p<\infty$ -917,$X_n(2/3)$ -918,$X_s$ -919,$x=q(p)$ -920,$q_X(p)=\mu+\sigma z_p$ -921,"$Y_{0,t}:=\sum_{d>t} X_{0,d}$" -922,$\mathsf{E}[X1_A] / \mathsf{E}[1_A]$ -923,$Z_{a}(a)$ -924,$\le p$ -925,$dx$ -926,$A = 8.14864$ -927,$L(X)=1_{X=x_p}(X)/f(x_p)$ -928,"$\{0, 8, 10\}$" -929,$\mathcal D(X)=c\mathsf{TVaR}_p(X-\mathsf{E}[X])$ -930,$P = \mathsf{TVaR}_\pi(X)$ -931,$w=w f(1)=w f(1)+(1-w)f(0) \le f(w 1 + (1-w)0)= f(w)$ -932,$Z_\mathit{lin}$ -933,$X_t=\mu t + \sigma W_t$ -934,$\alpha S$ -935,$f(x)=\sin(x)$ -936,$\mathbf {X_{2c}}$ -937,"$\Omega=\{\omega_1,\dots,\omega_n\}=\{\text{Ada}, \text{Bernhard}, \dots, \text{Zeno} \}$" -938,$\alpha(1+fg/(1-g))$ -939,$s > s_1$ -940,$t=2/3$ -941,$\int_0^s \phi(1-t)dt$ -942,$\rho(U)=\mathsf{E}_\mathsf Q[U]$ -943,$H_k(X) \le H_k(Y)$ -944,$X\preceq Y$ -945,$1-1/c$ -946,$0 < s < 1$ -947,$-\rho(-X)\le \mathsf{E}[X]$ -948,$\infty$ -949,$q(\hat p)$ -950,$Z=g'(S(X))$ -951,$n+1=N$ -952,$P=L/(1+R_L)$ -953,$\rho(X_n)\not\to \rho(X)$ -954,$X'\Delta g(S)$ -955,$\mathbf {x_1}$ -956,$\beta_i(X_4)$ -957,$s>0.2$ -958,$q_{X+c}(p)=c+q_X(p)$ -959,$X=q(F(X))$ -960,$0.2 < s < 1$ -961,$\mathsf{E}[X\mid \mathcal F'](\omega)$ -962,$t>0.5$ -963,$0 \le t \le 1$ -964,"$\mathsf{TVaR}_p(X(x_1,x_2))=(x_1 + x_2)\mathsf{TVaR}_p(Y)$" -965,$X_1\le X_2\implies a(X_1;X)\le a(X_2;X)$ -966,$\rho(X_j)=\max_k \mathsf{E}_\mathsf{Q_k}[X_j]$ -967,$\rho_c(X)=\mathsf{TVaR}_{0.8}(X)=8.5$ -968,$\mathbf {\Delta S}$ -969,$V_X$ -970,$\mathsf{E}[g'(S(X))]=\int_0^\infty g'(S(x))f(x)dx=\int_0^\infty -\frac{d}{dx}g(S(x))dx=g(S(0))-g(S(\infty))=g(1)-g(0)=1$ -971,$\rho(1)=1$ -972,"$(3,2)$" -973,$a_2'$ -974,$x_{i-1}\le x'_i\le x_i$ -975,$\mathsf{E}[ X_i \mid X(x) = q_{x}(p)]$ -976,$\mathsf{TVaR}_p(X)=(12(0.9-p) + 2.5)/(1-p)$ -977,$V$ -978,"$D^f\rho_{W_t\wedge a, W_t}(Y_{0})$" -979,$\mu$ -980,$\beta_i(x) =\mathsf{E}_{\mathsf Q}[X_i/X\mid X>x]$ -981,$y=(\log(x)-\mu)/\sigma$ -982,$\sup(X)<\infty$ -983,$+\infty$ -984,$F^{-1}(p)=q(p)$ -985,$Z(y_j)$ -986,$\bar Q_{d}=a_{d}-\bar P_{d}$ -987,$\rho(X_n) \uparrow \rho(X)$ -988,$\bar P_0>\mathsf{E}[Y_{0}]$ -989,$S(a)$ -990,$(1-g(s))(1-q)$ -991,$\Delta \mathit{MV}_{gc}(a)$ -992,"$X_1,\dots,X_m$" -993,$da1_{X>x}$ -994,$g_1F$ -995,"$\mathsf{E}[X_i(a)\,g'(S_{X\wedge a}(X\wedge a))]$" -996,"$\bar P_{0,t}:=\rho(Y_{0,t})$" -997,$x_0+x_1+x_2$ -998,$\bar S(a)=\displaystyle\int_0^a S(x)dx$ -999,$S(X_j)>0$ -1000,$f(s)=\alpha(1-\alpha)(1-s)^{\alpha-1}$ -1001,"$1_A:\Omega\to \{0,1\}$" -1002,$g(S(\infty))=0$ -1003,"$\alpha_i(a) = \dfrac{\sum_{j:X_j>a} (X_{i,j}/X_j)p_j}{\sum_{j:X_j>a} p_j}$" -1004,"$P_i,M_i, Q_i$" -1005,$C'_i$ -1006,$l_i$ -1007,$A(c)=c$ -1008,$I$ -1009,$X\preceq_m Y$ -1010,"$(-\x, 2)$" -1011,"$\rho(X),\rho(Y)\le 0$" -1012,$a_{d} = \mathsf{E}[Y_{d}]+4\sigma(Y_{d})$ -1013,"$X_{0,t}$" -1014,$a-X\le 0$ -1015,$m_3=0$ -1016,$\mathsf{E}[Z]\ge 1$ -1017,$\mathsf{E}[X_iZ_j]$ -1018,$\rho(W_1\wedge a_1 \wedge a_1')$ -1019,$\mathsf{E}[XZ]$ -1020,"$\mathsf{CONVEX,LI}$" -1021,$1_{X>x}$ -1022,$\tau a$ -1023,$E\in\mathcal F$ -1024,$a/Q = 1 + R/Q$ -1025,$F_Y$ -1026,$\mathbf {\Delta g(S)}$ -1027,$X(T(U))$ -1028,$\esssup(X)=\sup\{x\mid \mathsf{Pr}(X>x)>0 \}$ -1029,$\le 1/(1-p)$ -1030,$0\le \lambda\le 1$ -1031,$r\times 1$ -1032,"$(0,1,2,3,4,5,6,7,8,9)$" -1033,"$(3,1)$" -1034,$M=\mathsf{var}nothing$ -1035,$\mathcal F_0\subset\mathcal F_1\subset \cdots\subset \mathcal F_N$ -1036,$v_f\mathsf{E}_\mathsf{Q}[X_i]$ -1037,$\mathsf{Pr}(X=2)=0.5$ -1038,$\dots$ -1039,$R_C$ -1040,$k = 3.3 s^{0.82}$ -1041,"$X_n=1_{\{0,1,\dots,n-1\}}$" -1042,$X(\omega)=x$ -1043,$R_L$ -1044,$X=10$ -1045,$Q_i$ -1046,$P(a)$ -1047,$\mathsf{E} X + c{(X-\mathsf{E} X)^+}_p$ -1048,$\rho(X)\ $ -1049,$U(1)=1$ -1050,$g(S_{X\wedge a'}(x))$ -1051,"$ occurs, i.e., those with the value 1 in the $" -1052,$\Delta X_m$ -1053,"$(0,0,0,0,0,0,0,5,0,5)$" -1054,$D=1$ -1055,$\rho(X)=\max_i \rho_i(X)$ -1056,$a_h=2-a_l$ -1057,$0 < \alpha \le 1$ -1058,"$i=1,\dots,N$" -1059,$-norm equal to 1. (Note that $ -1060,$g(0.1)=\sqrt{0.1}=0.316$ -1061,$\rho_g(X)=\mu+\lambda$ -1062,$0.5 + U/2$ -1063,$-g'(S(x))f(x)$ -1064,$\mathsf{E}[Y \mid U]$ -1065,$1-(p_R+p_Y)$ -1066,$(1+\epsilon)v_1$ -1067,$\Vert X-Y\Vert := \sup_{\omega\in\Omega} |X(\omega) - Y(\omega)|$ -1068,"$(\partial a/\partial x_1)(tx_1,tx_2)= 3tx_1 /a(tx_1, tx_2) = 3x_1 /a(x_1, x_2)=\partial a/\partial x_1$" -1069,$\mathsf{TVaR}_p$ -1070,$U\le u$ -1071,$-dS=f(x)dx$ -1072,$\mathsf{E}_\mathsf{Q}\left[\dfrac{X_i}{X}(X\wedge a)\right] + \tau a \mathsf{E}_\mathsf{Q}[X_i/X\mid X > a]$ -1073,$\mathsf{COM}$ -1074,$1_\omega$ -1075,$\alpha=0.5$ -1076,"$\mathsf{biTVaR}_{p_0,p_1}^w(X)=\mathsf{TVaR}_{p^\ast}(X)$" -1077,$\mathbf{x}=\mathbf{1}$ -1078,$\beta_i(x)/\alpha_i(x)$ -1079,$d^*$ -1080,"$\mathbf {\omega_1},\dots,\mathbf {\omega_n}$" -1081,$X_2-X_1$ -1082,$q_{X_i}(p)=\Phi^{-1}(p)$ -1083,$\mathsf{Q}\in\mathscr{P}$ -1084,$Z_a$ -1085,$\mu(\{p_0\}) = 1-w$ -1086,$Z(\omega)> 0$ -1087,$r=0.045$ -1088,$h(s)=s^m$ -1089,$X\_{1}$ -1090,$cv=0.557$ -1091,$du = -g'(S(x))dF(x)$ -1092,$g(0)=r_0$ -1093,$\sup_\mathsf{Q} (\mathsf{E}_\mathsf{Q}[X] - l(Q))$ -1094,$M_i=\beta_ig(S)-\alpha_iS$ -1095,$j$ -1096,$g-s$ -1097,$\mathsf{E}[X_i \mid X=q(1-g^{-1}(1-p))]$ -1098,$w_u=1+c(1-\gamma)$ -1099,$a:=\rho(X)$ -1100,$g\Delta X \wedge a$ -1101,$\beta_i(a)g(S(a))=\mathsf{E}_{\mathsf{Q}}[(X_i/X) 1_{X>a}]$ -1102,$M=rQ$ -1103,"$X,X_i$" -1104,$Y_c$ -1105,$($ -1106,$S_{X\wedge a}$ -1107,$\rho(1_A)$ -1108,$g_4(s)=s^{0.9}$ -1109,"$(4,1)$" -1110,$f(L)=0$ -1111,$\mathsf{Q}'(\Omega_a) =\mathsf{Q}(\Omega_a)$ -1112,$E[(X-qp)^+]$ -1113,$I/a + U/R > 0$ -1114,$g'(S(x))=(1-p)^{-1}$ -1115,$a\le 1$ -1116,$a-b_h<0$ -1117,$\mathcal V(X)=\frac{1}{1-p}\mathsf{E}[X^+]$ -1118,$\mathsf{TVaR}_p(X) := (1-p)^{-1}(T_1+T_2)/N$ -1119,$0.417 < p < 0.791$ -1120,$1-\nu p$ -1121,$\sqrt{0.9}=0.95$ -1122,"$c(1,2)-c(2)$" -1123,$\lambda X$ -1124,$r_A$ -1125,$\dfrac{\iota}{1+\iota} p$ -1126,$a = a(W) = \mathsf{E}[W]+4\sigma(W)$ -1127,$\rho_g(X)=452.98$ -1128,$a < \infty$ -1129,$\alpha(\mathsf Q) = 0$ -1130,$\mathsf{VaR}_p$ -1131,$X_1=X_2=Y$ -1132,$P = 1.5$ -1133,$\mathsf{E}_\mathsf{P}[X']$ -1134,$S(x)dx$ -1135,$L_a^{a+y}$ -1136,"$\mathsf P,\mathsf Q_2,\dots,\mathsf Q_r$" -1137,$F(t)$ -1138,"$P((1+\epsilon)v_1, v_2, a+da)=P^a((1+\epsilon)v_1, v_2)$" -1139,$\mathsf{E}[X] + \pi\mathsf{var}(X)$ -1140,$\tau=0+d$ -1141,$Y=f(X)$ -1142,$a_1 = 5.991$ -1143,$\mathbf {\iota=M/Q}$ -1144,$X=X\wedge a + (X-a)^+$ -1145,"$s\wedge p=\min(s,p)$" -1146,$a=30$ -1147,$1_{U_X\ge p}$ -1148,$g(s)\ge s$ -1149,$\mathsf Q(A)>0$ -1150,$\mathsf{COH}$ -1151,$D f(x_0)$ -1152,$r_H$ -1153,$d=iv$ -1154,$U>p$ -1155,$p<0.1$ -1156,"$\mathsf{biTVaR}_{0,0.9}^{0.3138}$" -1157,$(g(s)-s)/(1-s)$ -1158,$P/L$ -1159,$\mathsf{E}_Q[X]$ -1160,$j=7$ -1161,$\mathbf{v}'$ -1162,$0< p <1$ -1163,$\psi(u)=\mathsf{Pr}(Y > u)$ -1164,$\mathsf P(A)=0$ -1165,$X_{-1}=x$ -1166,$x=q^-(p)$ -1167,$(\lambda S(x))$ -1168,$Q=1-g(S)$ -1169,$\mathsf{E}[X_i/X|X>a]$ -1170,$1^+$ -1171,$X \wedge a$ -1172,$\mathsf{E}[Y_i\mid X_n]$ -1173,$\delta(s)$ -1174,"$[x, y]$" -1175,$\omega>0$ -1176,"$t \in (0,1)$" -1177,$1=1_{X\le a}+1_{X>a}$ -1178,$\rho(X_n)$ -1179,$Y\equiv 1$ -1180,$(dt)^{3/2}$ -1181,$m_0=0$ -1182,$\iota=\dfrac{M}{Q}$ -1183,$X\circ f$ -1184,$\rho_c(X)=\mathsf{E}[X]+c\sigma(X)$ -1185,$g(s)=s^\lambda$ -1186,"$\mathsf{MON,\ NORM}$" -1187,$\sum_i \kappa_i'(x)=1$ -1188,$ax$ -1190,$p'\ge p$ -1191,"$H_k(X):=\mathsf{E}[\max(X_1\dots, X_k)]$" -1192,$\bar P_i(a)$ -1193,$\sum_\omega \mathsf Q(\omega) =\mathsf{E}[Z] / \mathsf{E}[Z]=1$ -1194,$\mathsf{P}(X=0)$ -1195,$B$ -1196,$Np=67.45$ -1197,"$X_n,X$" -1198,$(1-p)\gamma(dp)$ -1199,$X'=X$ -1200,$(1-p)/(p(\nu_p-l_p)^2)$ -1201,$0.33$ -1202,$\mathsf{E}[X] = \mathsf{E}[\mathsf{E}[X\mid Y]]$ -1203,$\mu_U = 1-p = 0.995$ -1204,$j+1$ -1205,$q_{X+Y}=q_X+q_Y$ -1206,$\mathsf Q_{X}$ -1207,"$u_{X,r}(p)=\psi_{X,r}^{-1}(p)$" -1208,$L_a^{a+da}=L_0^{a+da}-L_0^a$ -1209,$c(X(\mathbf{v}))=c(\mathbf{v})$ -1210,$\mathsf{MRM}$ -1211,$^{*}$ -1212,"$s=0,1$" -1213,"$X(x,-x)\equiv 0$" -1214,$F(x):=\mathsf{P}(X\le x)$ -1215,$\mathbf {X_{g}}$ -1216,$\max X$ -1217,"$\{\mathsf{E}[X_i\,Z] \mid \rho(X)=\mathsf{E}[XZ] \}$" -1218,$\rho(X)=\mathsf{E}_{\mathsf Q_X}[X]$ -1219,$q=q(p)$ -1220,$\rho(\mathsf{E}[X_2\mid X_1])\le \rho(X_2)$ -1221,$1/m>0$ -1222,"$B\subset [0,1]$" -1223,$g(S(x))=1-p$ -1224,"$f:(0,1)\to (0,1)$" -1225,$\mathbf {S}$ -1226,"$p_0,\dots, p_{n'}$" -1227,$X_1-X_0$ -1228,$\bar P = \bar S + \bar M$ -1229,$\mathbf {X_1}$ -1230,$\rho(\tilde X)=\rho(X) + \rho(\tilde X-X)$ -1231,$\mathsf{E}[(X-a)^+]$ -1232,"$u\in D_n=\{ u \mid u^{(k)} \ge 0, k=1,\dots,n-1, u^{(n-1)}\text{ nondecreasing} \}$" -1233,$l(\mathbf X)=(\sum_i X_i^2)^{0.5}$ -1234,$\mathsf{E}_{\mathsf{Q}}[\tilde X-X] \le \rho(\tilde X-X)$ -1235,$s=S(x)$ -1236,$\mathsf{E}_{\mathsf Q}[Y]=\mathsf{E}[YZ]$ -1237,$s_j < 1$ -1238,$\bar S(a+da)-\bar S(a)\approx \bar S'(a)da = S(a)da$ -1239,$t-1$ -1240,$\mathcal D(X+c)=\mathcal D(X)$ -1241,"$s\in[0,1]$" -1242,$\mathsf{E}[Yg'(S(X))]$ -1243,$p=1-1/n$ -1244,$X(\omega)=X_1(\omega)+X_2(\omega)$ -1245,"$S(x) + d\,F(x) + (\delta^{\star}-d)\sqrt{S(x)F(x)}>1$" -1246,$\bar S_i(a) := \mathsf{E}[X_i(a)]$ -1247,$S(x_#4)$ -1248,$1-e^{-\lambda S(x)}$ -1249,$\mathcal V$ -1250,$\beta>1$ -1251,$X_n=n1_A$ -1252,$d-1$ -1253,$g(S(x))\approx S(x)\approx 1$ -1254,$t_0$ -1255,$D_1$ -1256,$\mathcal E$ -1257,$s\uparrow 1$ -1258,$Mg(0+)$ -1259,$S/L\ge A/L-1$ -1260,$\succeq$ -1261,$2\mathsf{VaR}_p(X_1) - \mathsf{VaR}_p(X)$ -1262,$Y = X + Z$ -1263,$)$ -1264,$\rho(X)=\mathsf{VaR}_{0.995}(X)-\mathsf{E}[X]$ -1265,$\tilde X_2 = X_2 -\mathsf{E}[X_2\mid X_1]$ -1266,$p\to 1$ -1267,$1-(1-s)^m$ -1268,$\mathsf P(T^{-1}(A))=\mathsf P(A)$ -1269,$-zf(x)=(d/dx)g(S(x))$ -1270,$\rho_X(X_i)$ -1271,$P=\rho(X \wedge a)$ -1272,$s=0.02$ -1273,$F(q^-(p_0))=p_+>p_0$ -1274,$\Delta g(S)$ -1275,$\Delta$ -1276,"$\mu=10, \sigma=2$" -1277,$t=3$ -1278,$0\le q\le 1$ -1279,$L_a^y$ -1280,$l=\sum_i l_i$ -1281,$X=30$ -1282,$f:I\to\Omega$ -1283,$\mathsf{E}[X_2\mid X=x]$ -1284,"$f(x,y)=x^3/(x^2+y^2)$" -1285,$g(0+)=\delta$ -1286,$S_i(x)$ -1287,$h=2$ -1288,$g'_\tau(s) = g'(s)/(1+\tau)\ge 0$ -1289,$1-\mathsf{P}(X=0)$ -1290,$t \ne 0$ -1291,"$\mathbf {D^f\rho_{X\wedge 30,X}(X_1)}$" -1292,$\rho=\mathsf{TVaR}_p$ -1293,$\kappa_j(x)\approx \mathsf{E}[X_j]$ -1294,$\tilde M_i(a) = \bar M_i(a)-\tau_i a_i$ -1295,$a>10$ -1296,$x^+$ -1297,$\pi^{-1}\log\mathsf{E}[e^{\pi x}]$ -1298,$A(-X)=-A(X)$ -1299,$g(s)=s^{1/3}$ -1300,$\{X = x\}$ -1301,"$p_1,p_1$" -1302,$0\le x \le 1000$ -1303,$U_s$ -1304,$\mathsf{Pr}(X< x)\le 0.75 \le \mathsf{Pr}(X\le x)$ -1305,"$\{1,2,3\}$" -1306,"$i=0,1$" -1307,$\mathsf{Var}(\Pi)$ -1308,$\mathsf{TVaR}_{0.75}(X_1)=10$ -1309,$g_k(s)=1-(1-s)^k$ -1310,$g'(S_{X}(X))$ -1311,$(8t+10t)/2$ -1312,$g(S(x_i-))=g(S(x_{i}))$ -1313,$\nu + \delta = 1$ -1314,$1-1/n$ -1315,$\Omega_1$ -1316,$\mathsf{Pr}(A\cup B)=\mathsf{Pr}(A)+\mathsf{Pr}(B)$ -1317,$\Delta g(S_j)$ -1318,$x\leftrightarrow u(x)$ -1319,$\eta=0.49$ -1320,$X=q(p)$ -1321,$\log(\mathit{EER}) = \gamma + \eta \log(\mathit{PFL}) + \beta \log(\mathit{LGD})$ -1322,$Y=-X_0$ -1323,$g'\circ S_{X\wedge a}$ -1324,$s_2 - s_1$ -1325,$y < q_A(p)$ -1326,$\Delta\mathit{MV}$ -1327,$g'(s+)$ -1328,$\mathsf{Q}(A)=\mathsf{E}[1_AZ]=0$ -1329,$w=E[w|s=0.1]=0.06405$ -1330,$f'_+$ -1331,$f_x=1/S_t$ -1332,$S(X(\omega))$ -1333,$\rho_2(X)$ -1334,$\mathsf{E}[X\mid \mathcal F_t](\omega)=\sum_{i \le t} \omega_i/2^i+2^{-(t+1)}$ -1335,$L$ -1336,$\partial a/\partial x_1=3x_1/a$ -1337,$g(s)\ge 0g(0) + sg(1)=s$ -1338,$T:\Omega\to\Omega$ -1339,$t>x$ -1340,$L^1$ -1341,$(a-X_{\mathsf{j}(a)})$ -1342,$\alpha=d_i$ -1343,"$A=\mathbb Q\cap [0,1]$" -1344,$\mathsf{E}[F_1] > \mathsf{E}[F_0]$ -1345,$Q_1\Delta X$ -1346,$f(L) \ge 0$ -1347,$\rho(X_1)=\rho(X_2)$ -1348,$\rho(\tilde X)$ -1349,$\mathsf{E}[X] + \pi \mathsf{Var}(X)$ -1350,$F_3$ -1351,$\mathsf{CTE}_p(X)$ -1352,$1_{U < s}$ -1353,$Q_2dX$ -1354,$p\to S\to gS \to \Delta gS$ -1355,$P\ge (\mathsf{E}[X] + \iota a)/(1 + \iota)$ -1356,$\Delta Q_{gc}(a)$ -1357,$g(s) = s^a$ -1358,$\mathsf{P}(X=1)=0.6$ -1359,$d^\ast = 1-(1-g^\ast)/(1-s^\ast)$ -1360,$g(s)=g(1-p)$ -1361,$\alpha_{Cat}$ -1362,"$D^f\rho_{X\wedge a,X}(X_i(a))$" -1363,$\mathsf{E}[e^{kX}]$ -1364,$\tilde{\mathsf{Q}}$ -1365,$r_f$ -1366,$X = \sum_i X_i$ -1367,$x_3(S(x_2)-S(x_3))=x_3f(x_3)$ -1368,$\preceq_2$ -1369,$\Delta \bar Q$ -1370,$m_0$ -1371,$(\alpha_i S)'(x)=-\mathsf{E}[X_i\mid X=x]f(x)/x=-\kappa_i(x)f(x) / x$ -1372,$Q(a)=1-g(S(a))$ -1373,$\bar P_i(x)$ -1374,$S\subset T$ -1375,$f(L)$ -1376,$D_n$ -1377,"$\{1+\lambda(f-\mathsf{E} f) \mid f\ge 0, \|f\|_q\le 1 \}$" -1378,$R_M$ -1379,$Z_5$ -1380,$\mathbf {s_1}$ -1381,$q^-=q^+$ -1382,$\mathsf{E}[X_i \mid X = x]$ -1383,$-\int xd(g\circ S)=\int g(S(x))dx$ -1384,$y\not=z$ -1385,$1-g_\tau(s)$ -1386,$a=\mathsf{E}[X \mid X > q(p)]$ -1387,$\rho(aX+bY) = a\rho(X) + b\rho(Y)$ -1388,$\rho L = \iota Q$ -1389,$W \equiv T_{(1)}=min_k{T_k}$ -1390,$\lambda \rho(X)$ -1391,$Y=h(Z)$ -1392,$y^{\ast}-x^{\ast} < \epsilon$ -1393,$\mathsf{E}[X] + \pi \mathsf{E}[((X-\mathsf{E}[X])^+)^p]^{1/p}$ -1394,$U/4$ -1395,$D\rho(X_0)=\{Z \}$ -1396,$X > A$ -1397,$\pi-\lambda\mathsf{E}[X]$ -1398,"$\mathsf{Pr}(A)\in [0,1]$" -1399,$1=\mathsf Q(\Omega)\not=\sum_n \mathsf Q(\{n\})=0$ -1400,$\sigma=0.25$ -1401,$\Delta \mathit{MV}_{gc}(a)$ -1402,$G(x)=\mathsf{Q}(\{X\le x\}) = 1-g(1-F(x))$ -1403,$\Phi'(Z(s))Z'(s)=1$ -1404,$\bar q_{X_1+X_2}(s) \ge \bar q(s/2)$ -1405,$K = 5.029$ -1406,$1_{X>x_2}$ -1407,$S\Delta X$ -1408,$\mathsf{Pr}(X > x)$ -1409,"$G(X_1,\dots, X_n)'=(Y_1,\dots, Y_r)'$" -1410,$\mu_L=r_L +\pi$ -1411,$X=20$ -1412,$\mathsf P(X=\max(X))=0$ -1413,$r_a+r_l$ -1414,$S_1$ -1415,$\mathbf X / l(\mathbf X)$ -1416,"$w, 1-w$" -1417,$\mathcal D$ -1418,"$ (range.south)+(0, -1) $" -1419,$\mathsf{P}$ -1420,$X=\sum_{i=1}^n X_i$ -1421,$X_j=x$ -1422,$\Omega_a$ -1423,$S_j$ -1424,$\beta>\alpha$ -1425,"$f(W_t,t)$" -1426,$Z=d\mathsf{Q}/d\mathsf{P}$ -1427,$\mathbf {Q_{1}\Delta X}$ -1428,$p\le S(x^*)$ -1429,$\phi(t)$ -1430,$S(x)=p$ -1431,$U/2$ -1432,$\int Zd\mathsf P=1$ -1433,$1+t$ -1434,$a_{1}'$ -1435,$r_h=-0.025$ -1436,$\mathsf{E}_{\mathsf{Q}}[\cdot]$ -1437,"$(x_A,g(S(x_A)))$" -1438,$p(1-\nu(p))=p\delta(p)$ -1439,$\beta_i$ -1440,$1-S$ -1441,$p_{\mathit{pr}}$ -1442,$g(0+)=\lim_{t\downarrow 0} g(t)\ge 0$ -1443,$0\le \pi\le 1$ -1444,$\mathsf{E}[cZ]=c\mathsf{E}[Z]=c$ -1445,$Z=Z(X)$ -1446,$r_a$ -1447,"$\int_a^\infty g(S(x))\,dx$" -1448,$\prec X$ -1449,"$\{2, 3\}$" -1450,"$(0,1,2,3,4,8,8,8,8,9)$" -1451,$n\ge 3$ -1452,$=\mathrm{MV}(a-X)^+$ -1453,$g(s)/(1-g(s))$ -1454,$\bar P_i(a)=\mathsf{E}_{\mathsf{Q}}[X_i(a)]=\mathsf{E}[X_i(a)g'(S(X))]$ -1455,"$E[Y\,dG/dF]$" -1456,$g(S_X(x))=1$ -1457,$q(p)=\inf\{x \mid F_X(x)\ge p \}$ -1458,$\mathit{NPV}_{\infty}$ -1459,$E[X_1 | X]$ -1460,$\beta_D$ -1461,$\mathbf {X_{2}}$ -1462,$\alpha_i(x)S(x)=\mathsf{E}[(X_i/X)1_{X>t}]$ -1463,$\sigma=0.1246$ -1464,$F(x;\alpha)$ -1465,$D_\infty$ -1466,"$(1,3)$" -1467,"$X, Y$" -1468,$\mathsf{E}_{\mathsf{Q}}$ -1469,$q^-(p)=\mathsf{VaR}_p(X)$ -1470,"$i=1,\ldots,n$" -1471,$P/l-1 =\rho= \iota Q / l = \iota(C/l + g)$ -1472,$c(x)=\rho(\sum_i x_iX_i)$ -1473,$\omega_1=0$ -1474,$E_{\mathsf{Q_X}}$ -1475,$M_{2}\Delta X$ -1476,$S(x_#5)$ -1477,"$(\nu,\nu,\dots,\nu,\nu+10\delta)$" -1478,$\mathsf{E}[X\wedge a(X)]$ -1479,$\mathcal F'\subset \mathcal F$ -1480,$\Delta S_0$ -1481,$a_{d}$ -1482,$\tilde X(x) = x$ -1483,$A/L<1$ -1484,$X_n(\omega)$ -1485,$\mathsf{E}[X_{d}]$ -1486,$\bar P^a(\mathbf{v})$ -1487,$\int_0^1 f(s)ds = 1 - \alpha < 1$ -1488,$\mathcal{N}_{X}(X_i(a))$ -1489,$a-P$ -1490,$\rho(X)=\sup_{\mathsf Q\in\mathcal Q} \mathsf{E}_\mathsf{Q}[X]$ -1491,$\mathsf{Q}(A)\le g(\mathsf{P})(A))$ -1492,$d=0$ -1493,$x\mapsto g(s)+g'(s)(x-s)$ -1494,$\mathsf{VaR}_{1-s}$ -1495,$\rho_g(X\wedge a)=(\bar L + ra)/(1+r)$ -1496,$(a-X)$ -1497,$\omega'=1$ -1498,$1/6 + 2 /6 + 4/2 + 9/6$ -1499,$\rho_a(kX) = \rho(kX \wedge a(kX)) = \rho(kX \wedge ka(X)) = \rho(k(X\wedge a(X))) = k\rho(X\wedge a(X)) = k\rho_a(X)$ -1500,"$500mm, enough to materially impair their franchise, is judged to be 0.4%. This has a corresponding risk-neutral value of 2.5%. However, they believe that a loss over $" -1501,$(a_1'-a_1)^+$ -1502,$X\wedge a=\sum_i X_i(a)$ -1503,$\mathbf {a}$ -1504,$\int_0^a g(S(x))dx$ -1505,"$Q,\iota,M$" -1506,$\mathsf{E}[p]=1$ -1507,$p>p^*$ -1508,$\{X\ge q(p)\}=\{X \ge 12\}$ -1509,$g(1)-g(0)=1$ -1510,$g(s)(1-q)$ -1511,$(g(S(x^-)-g(S(x)))/(S(x^-)-S(x))$ -1512,"$\sum_j X_{i,j}(a)\Delta g(S_j)$" -1513,"$\mathsf{P}(a,b]=b-a$" -1514,"$j=1,\dots,d$" -1515,$Z(\omega)=0$ -1516,$l(p)= \nu(p)-\sqrt{(1-p)/p}$ -1517,$\int_0^1 g(s)ds - 0.5$ -1518,$\rho_{g}$ -1519,$\prec_1$ -1520,$S\ge (1-\epsilon)\mathsf{E}[X]$ -1521,$\alpha(\mathsf{Q})$ -1522,$\mathsf{E}[\mathsf{E}[Z\mid X]]=\mathsf{E}[Z]$ -1523,$\epsilon v_1$ -1524,"$\phi(p) = (1-\alpha)^{-1}1_{[1-\alpha, 1)}(p)$" -1525,$S(M)=0$ -1526,$c\ge 0$ -1527,$p_1=1$ -1528,"$x_{1,i}+x_{2,k(i)}$" -1529,"$(x_1, x_2)$" -1530,$\alpha_i'(x) \to 0$ -1531,"$\displaystyle\int_0^{F(a)} \kappa_i(q(p))\,dp + a\alpha_i(a)S(a)$" -1532,$\bar P(a)$ -1533,$q(U)$ -1534,$\iff\rho$ -1535,$F_g(x)$ -1536,$Q(a) = 1-P(a)= \nu F(a)$ -1537,$\mathsf P(\{x\})=0$ -1538,$\mathsf{E}[X_2]=22.75$ -1539,$ = \mathsf{E}_{\mathsf{Q}}[X_i\mid X= x]$ -1540,$1_V$ -1541,$R_Q$ -1542,$\mathcal D:=\{X\mid X\preceq_2 Y \}$ -1543,"$X_{j,i}$" -1544,$g(1-F(x))=1-\tilde p$ -1545,$p'$ -1546,$\beta_i(a)g(S(a))$ -1547,"$A\subset[0,\infty)$" -1548,$X_1/X$ -1549,$x$ -1550,$q_{\mathbf{v}}(p)$ -1551,$\rho(X) = \rho(X\wedge a) + \rho((X-a)^+)$ -1552,$q^-(p)=\sup\ \{ x\mid \mathsf{Pr}(X < x) < p \}$ -1553,$1\not\in S$ -1554,$\mathsf{VaR}_{0.99}(X)=1100$ -1555,$X_n=1/n$ -1556,$\rho_g(X)=\mu/b>\mu$ -1557,$<1$ -1558,$S(X)$ -1559,$a=kP+Q$ -1560,$X\wedge a = \sum X_i(a)$ -1561,$\mathsf{TVaR}_{p_0}(X)=\mathsf{E}[X \mid A]$ -1562,$A\subset \{ Z=0 \}$ -1563,$Z\circ T_i$ -1564,$a(X_i; X)\le \sup(X_i)$ -1565,"$Y_{1,2}$" -1566,$M_{2}$ -1567,$x \le 300$ -1568,$\implies c_i\ge 0$ -1569,$F(x)=1-s$ -1570,$h(0.9) = 1-\sqrt{0.1} = 0.684$ -1571,"$\alpha = 1, \kappa = 0.2$" -1572,$(8)(0.25)+(10)(0.25)=4.5$ -1573,$W_0=0$ -1574,$Q=S$ -1575,$X^{(d)}_i(a):=(X_i-d)^+$ -1576,${\mathcal{M}}$ -1577,$X = X_1 + X_2$ -1578,$V_t$ -1579,"$\mathsf P(\{ \omega\mid X(\omega)=X(\omega_0), \omega \le \omega_0 \})$" -1580,$m_3 := m_2$ -1581,$g(s)=(s+\iota)/(1+\iota)$ -1582,$\iota = \delta/\nu$ -1583,$r_X= r_f + \beta_X(r_m-r_f)$ -1584,$Z\circ T\in \mathcal Q$ -1585,$\mathbf {s}$ -1586,$Z\succeq_2 \mathsf{E}[Z\mid X]$ -1587,$\rho(X_1) \ge P_1$ -1588,$a-X$ -1589,$P(A)=1-p$ -1590,$10+0$ -1591,$\phi'(p)=-g''(1-p)>0$ -1592,"$\mathsf{TI,\ MON,\ SA,\ PH}$" -1593,$\Delta_1=a_1'-a_1$ -1594,$\mathit{RDS}_k$ -1595,$t=-ln(1-p)$ -1596,$C_i=c_i$ -1597,$\lim_{s\to 1} (g(s)-s)/(1-s) = \lim_{s\to 1} 1-g'(s)$ -1598,$\rho_i(X)$ -1599,$v(A\cap B) + v(A\cup B)\le v(A)+v(B)$ -1600,$\mathsf{TVaR}_{0.5}$ -1601,"$X_1, X_2$" -1602,$\rho=\sup$ -1603,$\mathsf{E} X + c\mathsf{E}[((X-\tau)^+)^p]^{1/p}$ -1604,$m_i$ -1605,$\mathsf{E}[g(X_n)]\to \mathsf{E}[g(x)]$ -1606,$k\in\mathbb{R}$ -1607,$g'(s) = as^{a-1}$ -1608,$q(p)=F^{-1}(p)$ -1609,$E_4$ -1610,"$\psi_{X, m}(u)$" -1611,$f=(1-p)^{-1}1_A$ -1612,$<0$ -1613,$X=X_1 + X_2$ -1614,$G=g$ -1615,$-q_{-Y}^-(1-p)$ -1616,"$\rho(\lambda P,\lambda R,\lambda a)=\lambda\rho(P,R,a)$" -1617,$1+bf$ -1618,$Y_j$ -1619,$dP_g/dP_X$ -1620,$\mathsf{E}[X|X>x]=x+\mathsf{E}[X]$ -1621,$M=g-S$ -1622,$FL$ -1623,$\int gS(x)dx=\int xg'(S(x))P_X(dx)$ -1624,$\mathit{MV}_{ro}(a) = a-\rho(X_{-1}\wedge a)$ -1625,$\mathcal V(X)=\mathsf{E}[X]+c\mathsf{E}[X^2]$ -1626,$n+1$ -1627,$g'(s)=\phi(1-s)$ -1628,$X_i(a)\not= X_i\wedge a_i$ -1629,$\lim_{x\downarrow x_0} F(x)=F(x_0)$ -1630,$F(w) = 1-\exp(-w)$ -1631,$\mathsf{E}[X(1_{U_X\ge p}-B)]=\mathsf{E}[(X-m)(1_{U_X\ge p}-B)]$ -1632,$B_i^c$ -1633,$\Omega_a := \{\omega\in \Omega \mid (X\wedge a)=a \}$ -1634,$1/10$ -1635,$\mathsf{Q}_k$ -1636,$Q_i(a)$ -1637,$Q>0$ -1638,$r_h-\mu_L$ -1639,$s_j$ -1640,$\beta g(S)$ -1641,$\rho(W)=\mathsf{E}[W]+\lambda\sigma(W)$ -1642,$\ge 0$ -1643,$E[u_j(W_j - X_j)]$ -1644,$\phi((x-\mu)/\sigma)/\sigma$ -1645,$X_{2}$ -1646,$E[X \wedge x+a]-E[X \wedge a]$ -1647,$\mathsf{TVaR}_p(X)=25$ -1648,$X-(1+r)T$ -1649,"$\int_0^1 a'(tx)\,dt=\int_0^1 a(1)\,dt = a(1)=a'(x)$" -1650,$ (#1)+(#3) $ -1651,$g=F_G^{-1}(p_{\mathit{pr}})-1$ -1652,$X_{2}(a)$ -1653,$g(s)=s(1-s)$ -1654,$\mathsf{VaR}_{0.995}(U)-0.5=0.495$ -1655,$\kappa_2(10)$ -1656,$\lambda < 0$ -1657,$\mathit{ROE}(s) = fs/(1-f-s)$ -1658,$p_i$ -1659,$X_m$ -1660,$g(t) = r_0 + (1-r_0)t$ -1661,"$Y_{1,1}$" -1662,$s > s^*$ -1663,$\theta$ -1664,$g(s)=s^{1/2}$ -1665,$X\wedge a=a$ -1666,$\mathsf{Pr}(X < x)=1/6=\mathsf{Pr}(X\le x)$ -1667,$P=l + \iota Q$ -1668,$X-Y$ -1669,$\log(\mathit{ROL}) = a + b \log(\mathit{EL}) + b X$ -1670,$q_{X_1+X_2}(p) \le q_{X_1}(p) + q_{X_2}(p)$ -1671,$k\ge 0$ -1672,$\Phi'(z)=\phi(z)$ -1673,$q^-(p)=\inf \{ x \mid F(x) \ge p \}$ -1674,$\rho_X(X_i) \ge \mathsf{E}[X_i]$ -1675,"$g'(s)=(1-p)^{-1}1_{[0,1-p]}$" -1676,$X(\mathbf{v})=\sum_i v_iX_i$ -1677,$s_0$ -1678,"$t=0,1$" -1679,$d^\ast = 2g^\ast-1$ -1680,"$(s_1,g(s_1))$" -1681,$g(s)=s$ -1682,$0\times\infty=0$ -1683,"$\bar Q_{0,t}:=a_{0,t}-\bar P_{0,t}$" -1684,$q_X(p)$ -1685,$\rho_c$ -1686,$\mathbf {X\wedge a}$ -1687,$M(a)=g(S(a))-S(a)$ -1688,$\rho(X_n)=\rho(0)=0$ -1689,$\mathbf {X}$ -1690,"$\displaystyle\int_0^a \kappa_i(x) f(x)\,dx + a\alpha_i(a)S(a)$" -1691,$\bar\iota = 0.12$ -1692,$\mathsf P(X=\sup(X))=0$ -1693,$\mathsf{E}[Y\mid\mathcal F']=\mathsf{E}[Y]$ -1694,$\alpha_2(98)=0.9$ -1695,$p\delta(p)/p\nu(p)=\iota(p)$ -1696,$g_\tau(1)=1$ -1697,"$H(A, L, t)=LH(A/L, 1, t)$" -1698,$g_2F$ -1699,$X=X_0+X_1$ -1700,"$697.6 billion in 2016, $" -1701,$\bar Q=53.031$ -1702,$\mathsf{P}(\{n\})>0$ -1703,$c(S\cup\{i\})=c(S\cup\{j\})$ -1704,$\mu_L=0.03$ -1705,$Q_0=\rho(V_0)=\rho(X_1)$ -1706,$g'(s-)=g'(s+)$ -1707,$U = X + Y$ -1708,$B=B(p)$ -1709,$9+1=10+0$ -1710,$n=67$ -1711,$a(X(\mathbf{v}))$ -1712,$v(\Omega)=1$ -1713,$p_Y=1-p_R$ -1714,"$p\,da$" -1715,$t\mapsto \rho(X+tY)$ -1716,$Y^S$ -1717,$g'(S(x)) = (1-p)^{-1}1_{x >\mathsf{VaR}_p(X)}$ -1718,$E_{\mathsf{Q_X}}[X_i(a)]$ -1719,$\rho(X)\le \rho(Y)$ -1720,$1-\tilde p=g(1-p)$ -1721,$R_f-R_L>0$ -1722,$P = \log(\mathsf{E}[e^{\pi X}])/\pi$ -1723,$\rho_c(X)$ -1724,$X^\star$ -1725,$X\wedge a'$ -1726,$\mathsf{E}[\Pi]$ -1727,$0.675=(6.258/7.613)^2$ -1728,$q<1$ -1729,$\alpha_1(90) = (0.0909 \times 0.0625 + 0.1 \times 0.0625)/(0.0625+0.0625)=0.0955$ -1730,$g(Q)$ -1731,"$X_2=0,0,0,0,1,1,1,4,24, 500$" -1732,$\bar P_i$ -1733,$Z=\mathsf{E} Z$ -1734,$a(X)=3.769$ -1735,"$\rho(P,R,a)=\sqrt{(0.4P)^2+(0.25R)^2+(0.1a)^2}$" -1736,$\exp(x)$ -1737,$X_j$ -1738,"$(anch.west |- lee.north)+(-0.125,0.25)$" -1739,$\ge\mathsf{E}[X_i]$ -1740,$g(s)=20s\wedge 1$ -1741,$f(x_p)$ -1742,$\{X=q_X(p) \}$ -1743,$\mathsf{E}[X_i\mid X=x]$ -1744,$EL(a)$ -1745,$30-11=19$ -1746,$x\in\mathbb{R}$ -1747,$p_R<0.5$ -1748,$\beta_{1}$ -1749,$g(S(a))\ge S(a)$ -1750,$r=16$ -1751,$\beta_i(a)$ -1752,$N=71$ -1753,$\rho(X_1+X_2)\le \rho(X_1)+\rho(X_2)\le 0$ -1754,$a_{gc}$ -1755,"$1 between any of the layers, then $" -1756,$\mathcal{M}$ -1757,"$\sum_i \rho(X_i, p^*)=a$" -1758,$\int_0^\infty g(S(x))dx$ -1759,$t=1-p$ -1760,$\rho'(x)=U'(-x)$ -1761,$D\rho_X(X_i) \ge \mathsf{E}[X_i]$ -1762,$\mathsf{Pr}(B)=\mathsf{Pr}(A)$ -1763,$x=\mathsf{VaR}_{0.99}(X)$ -1764,$\alpha_i(x)-\kappa_i(x)/x=0$ -1765,$x\mapsto |x|$ -1766,$\mathsf{Pr}(X_{-1}\sigma(Y)=0$ -1771,$D\rho_X(X_2)$ -1772,$\beta_i(a)g(S(a))=\mathsf{E}_{\mathsf{Q}}[(X_i/X) \mid X>a]g(S(a))=\mathsf{E}_{\mathsf{Q}}[(X_i/X) 1_{X>a}]$ -1773,$\rho_g(X)=\mathsf{E}[X]$ -1774,$L_d^l(x)$ -1775,$\beta_1g(S)dX$ -1776,$p_j=\Delta S_j$ -1777,$x1$ -1779,$E[s|t]$ -1780,$\mathsf{Q}(A)=\mathsf{E}_\mathsf{Q}[1_A]$ -1781,"$C(a)=\int_a^\infty S(x)\,dx + \tau a$" -1782,$\beta=d^\ast-d$ -1783,$-0.00002$ -1784,$y=0$ -1785,$L_X$ -1786,$\lambda=0.5$ -1787,$g(s)=(1-p)^{-1}s\wedge 1$ -1788,$\sum M_i\Delta X$ -1789,$1\le x \le 2$ -1790,$f(x) \ge f(x_0) + f'(x_0)(x-x_0)$ -1791,"$1,\dots,m$" -1792,$X\in L_p$ -1793,$n\mathsf{Pr}(Y\le y_c)$ -1794,$x=1.5$ -1795,$u^{iv} \le 0$ -1796,$1_{X > x}$ -1797,$S_{X_i}$ -1798,$xS(x)\to 0$ -1799,$(a-X)^+=a-(X\wedge a)$ -1800,"$j=0,1,\dots, n'$" -1801,$\mathsf{P}(\omega)$ -1802,$\bar Q=a-\bar P$ -1803,"$\mathbf {X\,p}$" -1804,$SdX$ -1805,$\sqrt{p}$ -1806,$L^p$ -1807,$\mu<0$ -1808,$\mathsf{E}[Y_{d}]=\sum_{s>d} \mu_s$ -1809,"$X_{i,i}(a)=X_{i,j}\dfrac{X_j\wedge a}{X_j}$" -1810,$\mathscr{M}$ -1811,$ so $ -1812,$1/4$ -1813,$\mathsf{E} X+\lambda\sigma(X)$ -1814,$\lambda\ge 0$ -1815,$d\bar S(a)/da=S(a)$ -1816,$(\alpha S)'(x)=-\kappa_i(x)f(x)/x$ -1817,$\sup f=1$ -1818,"$X_{t-2,3}$" -1819,$\beta_i(x)/\alpha_i(x) 0$ -1821,$\bar\nu a$ -1822,$a(1-f)$ -1823,$X\succeq Y$ -1824,$p_R$ -1825,$\mathsf{E}[p]\not=1$ -1826,$s_1 < s_2$ -1827,$1$ -1828,$\mathbb{Q}$ -1829,$a_x=1/\lambda$ -1830,$f:\mathbb{R}\to\mathbb{R}$ -1831,$\mathsf{E}[1_{U < s}]=s$ -1832,"$I=[0,1]$" -1833,$\rho(X)\le 0$ -1834,$B(0.5)$ -1835,"$i=1,2,\dots$" -1836,$r_D=1-D/L$ -1837,"$\min(X,a)$" -1838,$\mathbf {t-1}$ -1839,$\Delta S$ -1840,$ is the total return on invested assets and $ -1841,$\mathsf{E}[(A-L)^+]/\mathsf{E}[L]$ -1842,$X(\psi)=X(\omega)$ -1843,$X_j\ge 0$ -1844,$\mathcal{S}$ -1845,"$i=1,\dots, n$" -1846,"$\rho_{a,\tau}(X)=v\rho(X\wedge a) + da$" -1847,"$(brR15 |- lee.south)+(-0.125,-0.25)$" -1848,$n\ge N$ -1849,$x_1 \wedge x_2$ -1850,$X_s = X_{s_1} + X_{s_2}$ -1851,$0$ -1909,$x_0 \in \{ x \mid F(x) \ge p \}$ -1910,"$\bar P(\mathbf{v}, a)$" -1911,$x\mathsf{E}[X_i/X\mid X>x]$ -1912,$x_2(S(x_1)-S(x_2))=x_2f(x_2)$ -1913,$r_h=0$ -1914,"$S=[0,2\pi]$" -1915,$gn$ -1916,$p=F(x)$ -1917,$1/g'(s)$ -1918,$z(x)$ -1919,$-\sigma^2u''(w)\approx -cu'(w)$ -1920,$r=0.1$ -1921,$\mathsf{CTE}_p(X) := \mathsf{E}[X \mid X \ge \mathsf{VaR}_p(X)]$ -1922,$\beta_1$ -1923,"$i=1,\dots, M$" -1924,$\mathsf{E}_\mathsf{P}[X]$ -1925,$S^{-1}(g_i)$ -1926,$\mathbf {\Delta X'}$ -1927,$d =\iota/(1+\iota)$ -1928,"$\mathsf{E}[X_{t,d}\mid \mathcal F_{\tau}]$" -1929,$Z=g'(S_X(X))$ -1930,$E_i\cap E_j = \mathsf{var}nothing$ -1931,$i\not\in S$ -1932,$s+\delta p$ -1933,"$X_1=1+cos(X_3), X_2=1-cos(X_3)$" -1934,"$\mathcal F'=\{\mathsf{var}nothing, \Omega \}$" -1935,$(1-p)^{-1}1_A$ -1936,$\rho=P/L-1=M/L$ -1937,$F(X)$ -1938,$\lambda=$ -1939,$\rho_g(X)=352$ -1940,$x=0.5$ -1941,$A = -\log(p) = 5.298$ -1942,$\rho(X_{-1}\wedge a)$ -1943,$g'(S)dF(x)$ -1944,$-norm by integrating against a function with $ -1945,$(X-d)^+$ -1946,"$x=1000,2000,\ldots$" -1947,$\mathsf{E} X +\lambda {(X-\mathsf{E} X)^+}_1$ -1948,$\int_0^\infty S(x)dx$ -1949,$a=100$ -1950,$+ \mathit{PV}_{r_f}(\text{Inv Inc tax})$ -1951,$S(x_1)(x_2-x_1)$ -1952,$\mathsf{E}[(X-m)(1_{U_X\ge p}-B)] = 0$ -1953,$m=q(p)$ -1954,$wx + (1-w)y\in C$ -1955,$m_X$ -1956,$A(\text{Bernoulli})$ -1957,"$X,Y$" -1958,$\tilde Q$ -1959,"$Y_{0,2}$" -1960,"$\mathbf {X\,\Delta g(S)}$" -1961,$E[T]=s$ -1962,$\max(X)<\infty$ -1963,$\rho(Z_2)$ -1964,$\alpha_2SdX$ -1965,$\mathbf {x_0}$ -1966,$c\ge 1/2$ -1967,$g(s)=\dfrac{s+\iota}{1+\iota}$ -1968,"$X_i(\mathbf{v}, a)$" -1969,$X \prec_n^* Y$ -1970,"$X\wedge a'=\min(X, a')$" -1971,$d=2$ -1972,$s^\alpha$ -1973,$X(x)=\sum_i x_iX_i$ -1974,$Z(\omega):=(d\mathsf{Q}/d\mathsf{P})(\omega)$ -1975,"$\{\, (\mathsf{E}_\mathsf{Q}[X_i], \mathsf{E}_\mathsf{Q}[X]) \mid \mathsf Q\in\mathcal Q \, \}$" -1976,$1/6\le x < 2/6$ -1977,$p\ge r\ge 1$ -1978,$\mathbf{B}(0)=\mathbf{P_0}$ -1979,$Q=(a-EL)/(1+\iota)$ -1980,"$\rho(P,R,a)$" -1981,$t\mapsto v^t$ -1982,$\{ X=x\}$ -1983,$\omega \in \Omega$ -1984,"$j, p, S, \kappa_1, \Delta X, \Delta(X\wedge a)$" -1985,$0.375/1.5 = 0.25$ -1986,"$a(v_1(1+\epsilon),v_2)=a(v_1,v_2)+da$" -1987,$M_i$ -1988,$\alpha_i$ -1989,$p=1-\exp(-t)$ -1990,$\rho(X - b)=\rho(X)-b\le 0$ -1991,"$\boldsymbol{j, p, S, \kappa_1, \Delta X, \Delta(X\wedge a)}$" -1992,$x\ge 0$ -1993,$\rho(\lambda X) \le\lambda\rho(X)$ -1994,"$(1,1,\dots,1,1)$" -1995,$\mathbf {\Delta X}$ -1996,"$1-p, p$" -1997,$\mathsf{Pr}(X_n\in A)\to\mathsf{Pr}(X\in A)$ -1998,$S(x)=(k/(k+x))^\beta$ -1999,$p = 0$ -2000,$x_1$ -2001,$x=X(1-g^{-1}(1-\tilde p))$ -2002,$s < 1$ -2003,$\cdot$ -2004,$a'=a(1+r)$ -2005,$\phi(\cdot)$ -2006,"$i \in \{1,\dots,4\}$" -2007,$\gamma=r_f$ -2008,$\Delta A$ -2009,$P(X_{-1}(a))$ -2010,$0\le\lambda\le 1$ -2011,$\max$ -2012,$\Omega_0$ -2013,$0\le v\le 1$ -2014,$Y(\omega)=1$ -2015,$Q=A-P$ -2016,$0.75$ -2017,$a+y$ -2018,$\mathsf{Pr}$ -2019,$0.25$ -2020,$s=\mathit{EL}$ -2021,"$(1-g(S(x)),x)$" -2022,$\nu+10\delta$ -2023,$1=ps_g + (1-p)s_b$ -2024,$U(1)=2$ -2025,$\Phi(-d^*)>0$ -2026,$P = \mathsf{E}[X] + \pi \mathsf{E}[|X-\mathsf{E}[X]|^p]^{1/p}$ -2027,$x\to\infty$ -2028,$g(pq)=g(p)g(q)$ -2029,$\frac{d}{dp}(1-p)^{-1}=(1-p)^{-2}=q^{-2}$ -2030,$\rho(X)<\infty$ -2031,$X_0=\mathsf{E}[X]$ -2032,$\mu_L=r_L + \pi$ -2033,$\mathsf{E}[X_i(v_i)]=v_i\mathsf{E}[X(1)]$ -2034,"$k=(0.04, 0.4)$" -2035,"$A,B$" -2036,$\Delta S=p$ -2037,$N(1-p)$ -2038,"$(\omega'=1, \omega'')\in B_k$" -2039,$\sum\mathsf{E}[C_i^2]=\sum m_i(1+v_i^2)$ -2040,"$p_0,\dots, p_m$" -2041,$\tilde Z$ -2042,$\tilde X+X$ -2043,$dF(x) = dp$ -2044,$x_0 < \mathsf{TVaR}_{p_0}$ -2045,$\lambda\sigma$ -2046,$Z_j$ -2047,$m'(1) \to -1$ -2048,$g(S_j)$ -2049,$g(s(t)) = m(t)+s(t)$ -2050,$A\subseteq \mathbb{R}^N$ -2051,$f(x)\ge f(x_0) + s(x-x_0)$ -2052,$p=0.9982$ -2053,$a=10$ -2054,$\mu + \lambda\sigma$ -2055,$\beta<\alpha$ -2056,$Z\ge 0$ -2057,$\mathsf{E}_{\mathsf Q}[X]=\mathsf{E}[XZ]$ -2058,$\bar\nu(x)$ -2059,$6.258$ -2060,$\rho(X)=-\rho(-X)$ -2061,$-\sigma^2/2$ -2062,$k>0$ -2063,$r = 0.12$ -2064,$\mathsf{E}[Z \mid X]\preceq_2 Z$ -2065,"$(3,4)$" -2066,$dG/dF=r(x)$ -2067,$F_0=2.5$ -2068,$F_g(b)-F_g(a)=g(S(a)) - g(S(b))$ -2069,$P_g$ -2070,$\bar S$ -2071,$p=F(a)=1-s$ -2072,$Z(\omega)<1$ -2073,$\alpha\equiv 0$ -2074,$Var(G)=c^2$ -2075,$a = a(X)$ -2076,"$x\in\Omega=[0,1]^N$" -2077,$1_{U_X\ge p}=1$ -2078,$r_h<0$ -2079,$g(S(x_i)-g(S(x_i-))$ -2080,$F(a)$ -2081,$\mathbf {q}$ -2082,$\mathbf {d}$ -2083,$L_d^{d+l}(x)=(x-d)^+ \wedge l$ -2084,$\psi(0)=1-\mathsf{Pr}(Y=0)=1-\mathsf{Pr}(M=0)=\frac{1}{1+r}$ -2085,$X_3$ -2086,$\mathsf{E}[XB]$ -2087,$\bar P(a+y) - \bar P(a)$ -2088,$\bar P$ -2089,$x_{i+1}$ -2090,$-X_2$ -2091,$M_2\Delta X$ -2092,$(1+r)\mu$ -2093,$\bar P^a$ -2094,$\ge p$ -2095,$\displaystyle\int_0^\infty u(x) g'(S_X(x)) dF_X(x)$ -2096,$\mathsf{E}_{\mathsf{Q}}[X_i \mid X]$ -2097,"$\omega\in [k2^{-m}, (k+1)2^{-m}]$" -2098,$p=2$ -2099,$X=98$ -2100,"$0\le U, V\le 1$" -2101,$Y'$ -2102,$\mathbf {\mathsf{P}(X_1)}$ -2103,$\displaystyle\int_0^\infty xf(x)dx$ -2104,$(1-g(s))/(1-s)$ -2105,$00$ -2113,$a/X$ -2114,$q_1(t)=t$ -2115,$\mathbf{B}'(1) = -3\mathbf{P_2}+3\mathbf{P_3}$ -2116,$k\ge 1$ -2117,$X_{1}(a)$ -2118,$\Delta(X\wedge a)$ -2119,$P = S + M$ -2120,$(0.304-0.2)/(1-0.304) = 15$ -2121,$\omega_2$ -2122,$\mathsf{E}[h(X_i)L(X)]$ -2123,$P/S-1$ -2124,$g(s)/s$ -2125,$C(t)$ -2126,"$h(x)=\sup_{s\in[0,1]} g(s)-sx$" -2127,$t=4$ -2128,$\rho(X)=\max_\mathsf{Q} \mathsf{E}_\mathsf{Q}[X]$ -2129,$i^*$ -2130,$g(1)=1$ -2131,$C'_1+\cdots + C'_n$ -2132,$s_1$ -2133,$BY \succ AR$ -2134,$0.8 \times 1.2 = 24/25$ -2135,$(g(s)-s)/(1-g(s))$ -2136,$a = 8.1484$ -2137,$Y\circ T_i$ -2138,$p=0.9999$ -2139,$Z_X$ -2140,"$X_{0,1},X_{0,2},\dots, X_{0,N}$" -2141,$Z=0$ -2142,$-k$ -2143,$v(A)=g(\mathsf{P}(A))$ -2144,"$\bar P_i(\mathbf{v}, a)$" -2145,$B_p$ -2146,$a_i=x_i(\partial a/\partial x_i)$ -2147,$N$ -2148,$\sup$ -2149,$q_X(p)\le q_Y(p)$ -2150,$S(x)=s$ -2151,$X\preceq_n Y$ -2152,"$y,z\in X$" -2153,$\Omega_0 \times \Omega_1$ -2154,$P = \mathsf{E}[X] + \pi \mathsf{E}[((X-\tau)^+)^p]^{1/p}$ -2155,$df/dx=f$ -2156,$\mathsf{TVaR}_p(X)$ -2157,$X=8$ -2158,$\mathbf {\mathsf{P}(X_2)}$ -2159,$Q\in\mathcal{Q}$ -2160,$0.125$ -2161,$s < p$ -2162,$P(X_{-1}\wedge a)$ -2163,"$n=1,2,\dots, m-1$" -2164,$S(x)\approx 1$ -2165,"$X_2=(0,1,2,3,4,8,6,4,0,9)$" -2166,$1.5$ -2167,$q_X(p) = X(T(p))$ -2168,$1-m\le 1$ -2169,$v_f(\mathsf{E}_Q[X_i] - \dfrac{\mathsf{E}_Q[X_i]}{\mathsf{E}_Q[X]}\mathsf{E}_Q[(X-A)^+])$ -2170,"$k=1,\dots, n-1$" -2171,$\rho_g(X)=\mathsf{E}_{\mathsf{Q}}[X]$ -2172,$X_{-1}+X_{0}$ -2173,$p<0.05$ -2174,$\delta$ -2175,"$\gamma([0,p])=C(p)$" -2176,$10$ -2177,$T(U)$ -2178,$\rho_a(X+c) = \rho((X+c)\wedge a(X+c)) = \rho((X+c)\wedge (a(X)+c)) = \rho((X\wedge a(X))+c) = \rho((X\wedge a(X))) + c=\rho_a(X)+c$ -2179,$\bar M_t$ -2180,"$x~\text{Unif}[0,1]$" -2181,$g'(S(X))$ -2182,$\tilde Z=\mathsf P(X=\sup(X))^{-1}1_{X=\sup(X)}$ -2183,$\bar P(a+da) -\bar P(a)$ -2184,$X(x)=1/x$ -2185,$x=\mathsf{VaR}$ -2186,$\beta_2g(S)dX$ -2187,$\sigma(X_d)$ -2188,$\mathsf Q(X>a)/\mathsf P(X>a)$ -2189,$\mu(dp)$ -2190,$c=(g-s)/(g(1-g))$ -2191,$X\wedge a=a=90$ -2192,$\sigma(W)$ -2193,$1\le p\le \infty$ -2194,$X=4$ -2195,"$\sigma(L^\infty, L^1)$" -2196,$\mathsf{E}[X^n]$ -2197,$p_0\not= p_1$ -2198,"$a_{0,0}'=a_{0,0}$" -2199,$\mathsf{E}_{\mathsf{Q}}[X_i \mid X=x] = \mathsf{E}[X_ig'(S_X(X)) \mid X=x]/\mathsf{E}[g'(S_X(X)) \mid X=x] = \mathsf{E}[X_i \mid X=x]$ -2200,$\mathbf {K}$ -2201,$\{\omega\mid X(\omega) > x\}$ -2202,$P_i$ -2203,$\lambda_2\not=1$ -2204,$p>0.9$ -2205,$E(X^k)=E(Y^k)$ -2206,$\mathsf{E}[X_i \mid X=q(p)]$ -2207,$\mathsf{E}[Z \tilde X]$ -2208,$\bar P_t$ -2209,"$\Omega=\{ 1,2,3,4,5,6 \}$" -2210,$p<0.7$ -2211,"$a=10,20,40,50,60$" -2212,$-\infty+\lambda=-\infty$ -2213,$\mathbf {g(s)}$ -2214,$x=y$ -2215,$d=0.1/1.1$ -2216,$\beta_2>\alpha_2$ -2217,$k(h):=\log\mathsf{E}[e^{hX}]$ -2218,$=\displaystyle\int_0^\infty x dF(x)$ -2219,$\mathcal Q=\{\mathsf Q_k\}$ -2220,$a(f + (1-f)/q)$ -2221,$\lfloor x \rfloor$ -2222,$A\in\mathcal F$ -2223,$v(A)=\lambda(\pi_1(A))$ -2224,$\mathsf{E}_{\mathsf{Q}}[(X - a)^+] = \rho((X - a)^+)$ -2225,$n\to\infty$ -2226,$\mathsf{EPD}$ -2227,$\Longleftarrow$ -2228,"$\eta_{p,\alpha}$" -2229,$\Omega$ -2230,$\mathsf{QCX}$ -2231,$\omega=\omega'$ -2232,$\mathbf {M_{1}\Delta X}$ -2233,$g(S_{\mathsf{j}(a)})(a-X_{\mathsf{j}(a)})=(0.5)(80-11)=34.5$ -2234,$z_p=\Phi^{-1}(p)$ -2235,$g_1(s)=s^{0.4}$ -2236,"$1-e^{-\lambda S(\mathsf{PML}_{n, \lambda})}=1/n$" -2237,$q^-(U)$ -2238,$s=\exp(-a/b)$ -2239,$F(x)\ge p\iff q^-(p)\le x$ -2240,$(P-L)/L=P/L-1$ -2241,"$[p,1]$" -2242,$F_2$ -2243,"$\{H,T\}$" -2244,$a(1-p) + \mu p - \sigma\phi(z_p)$ -2245,$\rho(b-X)=b+\rho(-X)$ -2246,$s<1$ -2247,$g''(s)=-s^{3/2}/4$ -2248,$D^n\rho_X(X_1)=6.2048$ -2249,$\Delta X\wedge a$ -2250,$v=1/(1+r)$ -2251,$(1-p)^{-1/2}/4$ -2252,$T(X):=y\wedge (X-r)^+$ -2253,$x=S^{-1}(g^{-1}(u))$ -2254,$\mathsf{E}_\mathsf{Q}[X\mid A]$ -2255,$\mathsf{Q}(\{\omega_i\})=0$ -2256,$A(X+c)=A(X)+c$ -2257,$P \le \dfrac{S}{\lambda} \approx \dfrac{\mathsf{E}[X]}{\lambda}$ -2258,$\mathit{EGL}_{gc}(a)$ -2259,"$c\in[0,1/2]$" -2260,$\mathbf {X_{1c}}$ -2261,$\sigma=2.58$ -2262,$dp=\exp(-t)dt$ -2263,$a_x=4$ -2264,"$\beta_i(a) = \dfrac{\sum_{j:X_j>a} (X_{i,j}/X_j) \Delta g(S_j)}{\sum_{j:X_j>a} \Delta g(S_j)}$" -2265,$X = X\wedge a + (X - a)^+$ -2266,$(1-p)/(p\nu_p^2)$ -2267,$u$ -2268,$\omega$ -2269,$\mathsf{TVaR}_{0.8}(X+tX_1)$ -2270,"$\rho_g(X)= \sum_j X_j\,\Delta g(S_j)$" -2271,"$X_1,\dots,X_n$" -2272,$D\rho_{X}(Y) \subset D\rho_{X\wedge a}(Y)$ -2273,$\lambda=\dfrac{1}{1+\rho}$ -2274,$q^-(s)=\mathsf{VaR}_s(X)$ -2275,$v_i$ -2276,$\mathsf{E}_{\mathsf{Q}}[X\wedge a] \le \rho(X\wedge a)$ -2277,"$p=0.01, 0.02, \dots, 0.99$" -2278,$\mathsf{VaR}\_p(X)$ -2279,$a_0$ -2280,$0\le b\le 1$ -2281,"$A=(a,b]$" -2282,$a(\mathbf{v}) =\mathsf{TVaR}_p(X(\mathbf{v}))= (1-p)^{-1}\int_p^1 q_{\mathbf{v}}(s)ds$ -2283,$-g$ -2284,$q^-(p) := \sup\ \{x \mid F(x) < p \} = \inf\ \{ x \mid F(x) \ge p \}$ -2285,$p(\omega)\ge 0$ -2286,$D/L>1$ -2287,$-m_2/(1-s_2)$ -2288,$g(1-F(x))=1-p$ -2289,$h(1_{X\le a})$ -2290,$E(\pi)$ -2291,$\mathsf{TVaR}_{0.95}(X)$ -2292,$b-X\ge 0$ -2293,$Z = \sum_j X_j$ -2294,$X+Z$ -2295,$\mathsf{E}_{\mathsf{Q}}[X\wedge a] = \rho(X\wedge a)$ -2296,$\mathsf{VaR}_{0.75}(X)=90$ -2297,$\mathsf{E}[X_0] + \mathsf{VaR}_p(X_1)$ -2298,$QR_Q = aR_A + PR_L$ -2299,$x=\lambda y + (1-\lambda)z$ -2300,$dS=-dF$ -2301,$s \to 1$ -2302,$\tilde M(a)=\bar M(a)-\tau a$ -2303,$\kappa_i(x)=\mathsf{E}[ X_i \mid X = x]$ -2304,"$(ccc.south |- mcc.south)+(0,-0.5)$" -2305,"$[0,1]\to[0,1]$" -2306,$p=\infty$ -2307,$\bar P(a) = \rho_g(X\wedge a)$ -2308,$0\rho_2(X)$ -2310,$s(t)$ -2311,$\rho(W_1\wedge a_0)$ -2312,$0.8 \le p < 0.9$ -2313,$\epsilon_2$ -2314,$k=0$ -2315,$1-2c\mathsf{Pr}(Z>\mathsf{E} Z)$ -2316,$\Delta X_j=X_{j+1} - X_j$ -2317,${X}_p=\mathsf{E}[|X|^p]^{1/p}$ -2318,$\iota:1$ -2319,"$x_{2,1}$" -2320,$Y_{d}=\sum_{s>d} X_{s}$ -2321,"$\phi(x_1,...,x_n)$" -2322,$Z\in\mathcal Q$ -2323,$\iota^\ast$ -2324,$X-P$ -2325,$g(s)q=0.1839$ -2326,$X_2=x-t$ -2327,"$X_{t+2,1}$" -2328,$\mathsf{MON}$ -2329,$G(x)= 1-g(1-F(x))$ -2330,$g'(s)\to\infty$ -2331,"$\mathbf {g(S)\, \Delta X}$" -2332,"$j \in \{5,\dots,8\}$" -2333,"$\mathbb{R}=(-\infty, \infty)$" -2334,$e^{-r_Dt}$ -2335,$\rho((X-a)^+)$ -2336,$Q_t$ -2337,$X_0 < \dots < X_{N-1}$ -2338,$\mathbf {Z_6}$ -2339,"$B_4 = [\epsilon_1, \epsilon_2]$" -2340,$a(w_1X_1+w_2X_2;X)=w_1a(X_1;X)+w_2a(X_2;X)$ -2341,$(P-L) / (A-P)=$ -2342,$AR\succ BR$ -2343,$a(x)=xa(1)$ -2344,$X(\mathbf{v})$ -2345,"$x_{1,1}$" -2346,$\mathsf{E}_{\mathsf{Q}}[X_i\mid X\le a](1-g(S(a))) + a\mathsf{E}_{\mathsf{Q}}[X_i/X\mid X >a]g(S(a))$ -2347,"$d, r>0$" -2348,"$\phi(s)= g'(1-s) = \frac{1-w}{1-p_0}1_{[p_0, 1)}(s) + \frac{w}{1-p_1}1_{[p_1, 1)}(s)$" -2349,"$S\subset \Omega=\{1,\dots,N\}$" -2350,$x\le 0$ -2351,$S_0=1$ -2352,$0=\mathsf{Pr}(X<1)<\mathsf{Pr}(X\le 1)=1/6$ -2353,$f(x)=|x|$ -2354,$\mathbf {\mathsf{P}(X)}$ -2355,$S_t \ge 0$ -2356,$p=F(a)$ -2357,$\Psi^{-1}(t)=\log(-\log(t))$ -2358,$\mathsf{E}[e^{X_t}]=e^{\mu t + \sigma^2t /2}$ -2359,$q(U_X) > m$ -2360,$\mathsf{var}(\sum C_i)=\sum (m_i v_i)^2 = n(mv)^2$ -2361,$Y_s=(Y\mid Y\le y_c)$ -2362,$\mathsf{P}(d\omega)$ -2363,$h(0)$ -2364,$P_i/v_i$ -2365,$\mathsf{E}[X_1\mid X_1+X_2=x]=mx/(m+n)$ -2366,$\mathsf{E}_\mathsf{Q}[X_1]$ -2367,$\lambda > 0$ -2368,"$c(1,2) - c(2)$" -2369,"$(0,1]$" -2370,$t<0$ -2371,$\mathsf{COMON}$ -2372,$\beta_i(x)/\alpha_i(x)> 1 > g(S(x)) / S(x)$ -2373,$\int_0^\infty (1-F(x))dx=\int_0^\infty xdF(x)$ -2374,$(dW_t)^2=dt$ -2375,"$\mathbf {\Delta\,g(S)}$" -2376,$\mathsf{TVaR}_{0.95}(X)=3699$ -2377,$g(0^+) = r/(1+r)$ -2378,$x\mapsto 1/x$ -2379,$m\in\mathbb{R}$ -2380,$\mathsf{VaR}_{0.7}(X_i)=-\log(0.3)=1.204$ -2381,$-S(a)+\tau=0$ -2382,$\rho(c)\ge c$ -2383,$\beta_i(X)$ -2384,$0.8\le p<0.9$ -2385,$\mathsf P(X \le q_X(p)) > p$ -2386,$1/X$ -2387,$\displaystyle\int_0^1 X(p)dp$ -2388,$\mathsf{E}[X\tilde Z]$ -2389,$\rho_c\leftrightarrow\mathcal Q$ -2390,$U(X)\ge U(Y)$ -2391,$\lambda X_1 +(1-\lambda) X_2$ -2392,$\mathbf {a_{1}'}$ -2393,$\mathsf{E}[X\mid t+d]$ -2394,$MV = \bar Q + \mathit{NPV}_{\infty}$ -2395,$g(s)=1-(1-s)^m$ -2396,$g(0.05)=0.05\nu + \delta=0.1364$ -2397,$\mathbf {pK}$ -2398,$\min_{\eta\in \mathbb{R}} \eta + \alpha \mathsf{E}[(X-\eta)^+] -\beta\mathsf{E}](X-\eta)^-]$ -2399,$g(S(x)) = 1 - h(F(x))$ -2400,$\mathsf{E}[X]=k/(k+\beta)$ -2401,$g(s)\le s$ -2402,$L_1$ -2403,$X_1=1000$ -2404,$S$ -2405,$x < y$ -2406,$\mathsf{Pr}(E)$ -2407,$p>0.5$ -2408,$\mathsf{E}[p] \le 1$ -2409,$\mathsf{E}[1_A]$ -2410,$x=(y-\mu)/\sigma$ -2411,$a\to\infty$ -2412,$X+tX_1$ -2413,$M = \beta g(S)-\alpha S$ -2414,$0 < \nu = 1-\delta < 1$ -2415,$d=(\log(a/S_0)-(r-\sigma^2/2)t)/\sigma\sqrt{t}$ -2416,$X(\omega)=1/\omega$ -2417,$1/n$ -2418,$H(X)>-H(-Y)$ -2419,$s/(1-p) \wedge 1$ -2420,$\mathbf {\beta_{1}}$ -2421,$\Phi$ -2422,$\lambda y=x$ -2423,$\mathsf{MON'}$ -2424,$g'(S_X(X))$ -2425,$b<1$ -2426,$X\mapsto \mathsf{E}[XZ]$ -2427,$w < s$ -2428,$m_2$ -2429,$\mathsf{Pr}(X\in A)=0$ -2430,$\le c$ -2431,$n-1$ -2432,$qX$ -2433,$\bar P_2$ -2434,"$(4,3)$" -2435,$(X_i)_i$ -2436,$20+10t$ -2437,$s=1-\alpha$ -2438,$Z=d\mathsf Q / d\mathsf P\ge 0$ -2439,$X_i(a) = aX_i/X$ -2440,"$c(1,2,3)-c(2,3)$" -2441,$\sum_i q_iX_i$ -2442,$\mathsf{Pr}({\omega})=1/6$ -2443,"$\mathbf {X'\,\Delta g(S)}$" -2444,$\kappa_j(x)/x > \alpha_j(x)$ -2445,$a_i'$ -2446,$-\int xdS=\int Sdx$ -2447,$c\ge 1$ -2448,$f(P)=\mathsf{E}[f(X)]$ -2449,$\mathbf{B}(1)=\mathbf{P_3}$ -2450,"$\bar Q_{0,0}:=a_{0,0}-\bar P_{0,0}$" -2451,$p_- < p_0 < p_+$ -2452,$\mathbf {\Delta gS}$ -2453,$g'(t)=1-r_0$ -2454,$q(p)=\mathsf{VaR}_p(X)$ -2455,$g(0+):=\lim_{s\downarrow 0}g(s)$ -2456,$z\ge 0$ -2457,$\mathsf{E}[W]$ -2458,$ \& $ -2459,$A\setminus B$ -2460,$(k_1!)(k_2!)\dots$ -2461,$Q(x)=1-P(x)$ -2462,$\sup(X)$ -2463,$\mathbf {F(x)=\mathsf{Pr}(X\le x)}$ -2464,$1=\delta+\nu$ -2465,$=1/\lambda-1=(1-\lambda)/\lambda$ -2466,$U_X$ -2467,$\mathsf{Pr}(X_n=0)=1-1/n$ -2468,$q_X$ -2469,$\mathit{EGL}_{ro}(a)$ -2470,$\mathsf{E}[\cdot\mid X]$ -2471,"$i=1,2,\dots,10000$" -2472,$Z=z(X)$ -2473,$\{X > x \}$ -2474,$X_{\mathsf j(a)+1}>a$ -2475,$g_j<1$ -2476,$\rho(X)=0$ -2477,$\sum_i x_iX_i$ -2478,$Xq$ -2479,$\phi(p)=g'(1-p)=b(1-p)^{b-1}$ -2480,$N=1000$ -2481,$A\subseteq \mathbb{R}^n$ -2482,$a=90$ -2483,$S_m=\mathsf{P}(X>X_m)=0$ -2484,"$g:[0,1]\to [0,1]$" -2485,$q(p)$ -2486,$g(s)=\nu s+\delta$ -2487,$m=$ -2488,"$q(p)\phi(p)\,dp$" -2489,$q^+(p)=\sup\ \{ x\mid \mathsf{Pr}(X < x) \le p \}$ -2490,$x>\mathsf{VaR}_p(X)$ -2491,$a(\mathbf{v})=\mathsf{TVaR}_p(\mathbf{v})=\mathsf{E}[X\mid X > q_{\mathbf{v}}(p)]$ -2492,$\hat x > x$ -2493,$\text{VaR}_{0.99}$ -2494,$P_X\{X=M\}=0$ -2495,$X=X_0+X_{-1}+X_{-2}+X_{-3}$ -2496,$x>0$ -2497,"$X_{i,j}$" -2498,$a_1=\int_0^1 (\partial a/\partial x_1)dt=\partial a/\partial x_1$ -2499,$1=\bar\nu+\bar\delta$ -2500,$(1-p)/p=1$ -2501,$s=S(a)$ -2502,$\partial\rho(Z)$ -2503,$\mathbf X$ -2504,$\rho(W_1\wedge a_1 \wedge (a_0-X_1))=\rho(W_1\wedge a_1)$ -2505,$\sum_i \kappa_i(x)=x$ -2506,$g(s)=s^{0.4}$ -2507,$(g(s_0)-g_0)/s_0 \ge g'(s_0)$ -2508,$X_n(0)=1$ -2509,"$X_{t,2}$" -2510,$W=Z$ -2511,$\phi(x):=(2\pi)^{-1/2}\exp(-x^2/2)$ -2512,$g(s)=\sqrt{s}$ -2513,$1-p=S(x)$ -2514,$p(\delta_p-il_p)$ -2515,$\alpha(X)$ -2516,$=1$ -2517,$g''$ -2518,$f=f_X$ -2519,$dW_t\approx W_{t+dt}-W_t$ -2520,$X(\omega_1) > Y(\omega_1)$ -2521,$H_g(X) \le H_g(Y)$ -2522,$M:=\max(X)$ -2523,"$0,10,20$" -2524,$1/9=0.11\dot 1$ -2525,$a=80$ -2526,$n-2$ -2527,"$((0, x), (1-p, p))$" -2528,$P=D=L/(1+R_L)$ -2529,$w(A)\le v(A)$ -2530,$2^{20}\approx 1$ -2531,$^{**}$ -2532,$\mathsf{LI}\iff\mathsf{SSD}$ -2533,$p_j$ -2534,$P$ -2535,$s_00$ -2608,$i= \alpha/(1-\alpha)$ -2609,$\rho(X_0) = \mathsf{E}[X_0Z]$ -2610,$X\ge x$ -2611,$Z(x)=g'(S(x))$ -2612,"$c = 1.0, 1.5$" -2613,$a_{d}=a(Y_{d})$ -2614,$\mathsf{SD}(X)$ -2615,$-A(-X)$ -2616,$t\ge 0$ -2617,"$\Omega=\{0,\dots,99\}$" -2618,$g'(S(x))\ge 0$ -2619,"$p~\text{Unif}[0,1]$" -2620,$R_A=R_f$ -2621,$\mathsf{VaR}_p(X)=q^-(p)$ -2622,$E(u(X)) \le E(u(Y))$ -2623,$\rho_g(V)= g(F(x^*)) \ge F(x^*)=\mathsf{E}[V]$ -2624,$\mathsf{E}[X]+\lambda\sigma(X)$ -2625,$\mathsf{Pr}(\{\omega \})= 1/100$ -2626,"$(2,-\x*0.75)$" -2627,$\mathsf{Pr}(A\le t)= 1/2 + \mathsf{Pr}(U\le t) /2 = 1/2 + t/2$ -2628,$a \ge 1$ -2629,"$\mathsf{biTVaR}_{0,p}^w(X)$" -2630,$\iota^\ast = (g(s^\ast)-s^\ast) / (1 - g(s^\ast))$ -2631,$\mathsf{E}_{\mathsf{Q}}[X_i]$ -2632,$g'(s)\ge 0$ -2633,$\mathsf{E}[X]+k\mathsf{Var}(X)=a(X)$ -2634,"$X:\Omega\to [0,\infty]$" -2635,"$\mathsf{TVaR}_{0.95}(X)=\int_0^{1000}g(S(x))\,dx$" -2636,$\rho(X_n)\to \rho(X)$ -2637,$\lambda_{obj}$ -2638,$W_0$ -2639,$cv=0.287$ -2640,$0=q(0)=q(Y+(-Y))\le q(Y) + q(-Y)$ -2641,$g_\tau(0)=0$ -2642,$\mathsf{P}(X=1)$ -2643,$\mathsf{Pr}(X\le y) < p$ -2644,$0.41$ -2645,$\mathsf P(X=q_X(p))>0$ -2646,$p=0.8$ -2647,$\kappa_1(10)$ -2648,$\mathsf{E}_\mathsf{Q}[X+tY]$ -2649,"$[0,p)$" -2650,$a_1'$ -2651,$S(x)=1-\Phi((x-\mu)/\sigma)=\Phi(-(x-\mu)/\sigma)$ -2652,$X_{t+1}$ -2653,$X=0$ -2654,$p\mapsto g(1-p)$ -2655,$\downarrow$ -2656,$X\wedge 20$ -2657,$\mathsf{TVaR}_1( X )$ -2658,"$x_1, x_2$" -2659,$\bar P_{2}$ -2660,$\Sigma$ -2661,$B\subset A$ -2662,$\bar P=\mathsf{TVaR}_{p^\ast}(X)$ -2663,$\bar P^a_g(X_i\subseteq X)$ -2664,"$\mathcal{M} = \{ f \mid \|f\|_q\le c, f\ge 0 \}$" -2665,$X \preceq_m Y$ -2666,$qX_i$ -2667,$\mathsf{E}[X_i\tilde Z]=\rho_g(X)/2$ -2668,$X \prec_n Y$ -2669,$\bar\iota=0.10$ -2670,$a=18000.0$ -2671,$\mathsf{TVaR}_{0.95}(X)=1000$ -2672,$s_0/2^{n+1}$ -2673,$\delta(x)$ -2674,$H[X]$ -2675,"$x_1,x_2$" -2676,$>100$ -2677,$\mathsf{Pr}(X\ge q(p))>1-p$ -2678,$dh - h_x dx = (r_h-\mu_L)(h-h_x x)dt$ -2679,$\alpha<1$ -2680,$2.576\times 6.258$ -2681,$\mathsf{TVaR}_{p*}(X)=a$ -2682,$\kappa_i(X) = X_i$ -2683,$a_i = a(X_i; X)$ -2684,$\rho(X_n(t))+t\pi$ -2685,$\mathsf{TVaR}_{p}$ -2686,$g(A)/p=59.142$ -2687,$Z(S_X(x))=-(x-\mu)/\sigma$ -2688,$\nu=1-\delta$ -2689,$\{\omega\in\Omega\mid X(\omega)\le x\}$ -2690,$X_n= X_g-X_c$ -2691,$s^*$ -2692,$\bar P_{0}$ -2693,$P = \mathsf{E}[X] + \pi \mathsf{SD}(X)$ -2694,$\rho(X_0+\epsilon Y)=\mathsf{E}[(X_0+\epsilon Y)Z_\epsilon ]$ -2695,$X\wedge 30$ -2696,$k+1/2$ -2697,$X'=\mathsf{E}[X\mid A]$ -2698,$\mathsf{Pr}(X_i>\bar q(s))=s$ -2699,$\lambda=5$ -2700,$D_3$ -2701,$\ge c$ -2702,$\kappa_i(X)$ -2703,"$\mathsf{PH,SA,CX}$" -2704,$\phi:=\rho\circ F$ -2705,$u_j(x) = 1 - exp(-\lambda_j x)$ -2706,$M_{1}$ -2707,$X-V$ -2708,"$\bar P_i = \sum_{j} X_{i,j}\Delta g(S_j)$" -2709,$f(t)$ -2710,"$[0, \epsilon_1]$" -2711,$\pi=1$ -2712,$a_l \le 1$ -2713,$\rho=\mathsf{E}$ -2714,$P_g\not\ll P_X$ -2715,$\delta_i=\delta$ -2716,$\mathbf {X_1pK}$ -2717,$p_Y>0.5$ -2718,$-g'(1-p)<0$ -2719,$\rho(X+Y) = \rho(X) + \rho(Y)$ -2720,"$(0.5,1]$" -2721,$F(x_0)=p_+$ -2722,"$(X_i, X)$" -2723,$\mathsf{E}[X_i\sum_j w_jZ_j]=\sum_iw_j\mathsf{E}[X_i Z_j]$ -2724,$\beta_i(x) =\mathsf{E}_{\mathsf{Q}}[X_i/X\mid X>x]=\mathsf{E}[(X_i/X)g'S(X))\mid X>x]$ -2725,$l(kX)=k\rho(X)$ -2726,$\int udv = uv - \int vdu$ -2727,$\mathbf {Q_2\Delta X}$ -2728,"$\mathcal F_0=\{\mathsf{var}nothing, \Omega\}$" -2729,$r_P-\mu_L$ -2730,$\bar P(a)=\displaystyle\int_0^a g(S(x))dx$ -2731,$g(x) = (x-\mu)^2$ -2732,$\mathsf{biTVaR}(Y)=\mathsf{TVaR}_{p^\ast}(Y)$ -2733,$\mathsf{E}[(X_i/X)g'(S(x)) \mid X > x]$ -2734,$0$ -2735,$p=1-g(1-F(x))$ -2736,$\bar S_i(3463)$ -2737,$X=X(\omega)$ -2738,$\int_0^1$ -2739,$D/L=\mathsf{E}[A\wedge L]/\mathsf{E}[L]$ -2740,$\esssup(X)g(0-)$ -2741,$\tilde p$ -2742,$\bar P'$ -2743,$\sum_i E[X_i|anything]\le _{cx} \sum X_i \le_{cx} F_{X_i}^{-1}(U)$ -2744,$\bar{\mathbf M}$ -2745,$\lambda = \lambda_0+\lambda_1$ -2746,$X_{-2}=C_1 + \cdots + C_n$ -2747,$X_{0}$ -2748,$\rho_g = \int g(S)$ -2749,$a={{break_even}}$ -2750,$x=q(1-g^{-1}(1-\tilde p))$ -2751,$\mathsf x\mathsf{TVaR}_p(X):= \mathsf{TVaR}_p(X)-\mathsf{E}[X]$ -2752,$0.5\le p^* \le 0.75$ -2753,$X(\omega)=1$ -2754,$P(a)=S(a)+\delta F(a)$ -2755,$\mathsf{TVaR}_{1-c\epsilon}(X) = \mathsf{VaR}_{1-\epsilon}(X)$ -2756,$q(p)=S^{-1}(1-p)$ -2757,$d_i= i/(1+i)$ -2758,$P=\sum_i P_i$ -2759,$B_i$ -2760,$a_1=a(W_1)$ -2761,$\rho=\esssup=\mathsf{TVaR}_1$ -2762,$\sigma(X)^2$ -2763,$g'(s)=1/(1-p)$ -2764,$0.4$ -2765,$f'_+(x)=\lim_{h\downarrow 0} (f(x+h)-f(x))/h$ -2766,$g'(1)=1$ -2767,$\mathcal Q_1$ -2768,$X\le m$ -2769,$dt^2$ -2770,$q=p$ -2771,$\sigma_d = \mu_d/5$ -2772,$Q_0$ -2773,$X_t=X_{t+1}$ -2774,$g(s)=0.1995$ -2775,$\log(0)=-\infty$ -2776,$\mathsf{VaR}_p(X_1)$ -2777,$W_t$ -2778,$Z_{\tilde X}$ -2779,$U0$ -2792,"$\{4,5\}$" -2793,$h(x)=f(x)/S(x)$ -2794,$S\Delta X\wedge a$ -2795,$r_U$ -2796,$(c(S\cup \{i\})-c(S))$ -2797,$\mu(dp)=f(p)dp$ -2798,$X\preceq_2 Y$ -2799,$\mathbf {\min a}$ -2800,$\mathsf{E}_{\mathsf{Q}}[X]=\infty$ -2801,$v(E)$ -2802,$\{Z\circ T\mid T:\Omega\to\Omega\text{\ PPT}\}$ -2803,$6/6$ -2804,$\Phi(\Phi^{-1}(s) + \lambda)$ -2805,"$[0, -k]$" -2806,$\rho(X)=\mathsf{E}[f_X X]$ -2807,$\mathsf P(X=\mathsf{VaR}_p(X))>0$ -2808,$\rho(W_0\wedge a_0)=\bar P_0 +\bar P'$ -2809,$U(t)$ -2810,"$\sum_i a(X_i, p^*)=a$" -2811,$p=1-s$ -2812,$q_{X_1}(p)+q_{X_2}(p)=q_{X_1+X_2}(p)$ -2813,$\mathsf{E}[X_i\mid X\le a]F(a) + a\mathsf{E}[X_i/X\mid X >a]S(a)$ -2814,$8.5$ -2815,$M_{1}\Delta X$ -2816,$\bar P_i(a)$ -2817,$T_i$ -2818,$L_0^y$ -2819,"$\mathbf {g(S)\,\Delta X'}$" -2820,$\mathsf{E}_Q\left[\dfrac{X_i}{X}(X\wedge A)\right] + \delta A \mathsf{E}_Q[X_i/X\mid X > a]$ -2821,$2$ -2822,$\rho(c)=\rho(0+c)=\rho(0)+c$ -2823,$U_X(\omega)=F(X(\omega)-) + V(\omega)(F(X(\omega)) - F(X(\omega)-))$ -2824,$s = 1-10^{-15}$ -2825,$s/g(s)$ -2826,$\bar F(a)=\int_0^a F(x)dx = a-\bar S(a) = \bar Q(a) + \bar M(a) = \mathsf{E}[(a-X)^+]$ -2827,$a(W)=\mathsf{E}[W] + 4\sigma(W)$ -2828,$\alpha f$ -2829,$\{\omega\in\Omega \mid X(\omega)=x\}$ -2830,"$[0,1]$" -2831,$1/(1-p)$ -2832,"$(0,0),\ (1,0),\ (1,1)$" -2833,$\omega_i\in B$ -2834,$g'(1)=\alpha$ -2835,$\mathsf{E}[Xe^{\pi X}]/\mathsf{E}[e^{\pi X}]$ -2836,$\le 1$ -2837,$-\rho(-X)$ -2838,$f_{opt} = 1-s/g$ -2839,$g(S(x_i-))-g(S(x_{i-1}))$ -2840,$\Delta \mathit{MV}_{ro}(a)$ -2841,$Q_i=a_i-P_i$ -2842,$0.0476/(1-0.0476)=0.05$ -2843,$q=0.9215$ -2844,$c\ge \mathsf{E}[cZ]$ -2845,$f(x)dx$ -2846,$\mathcal F'$ -2847,$p=\Phi((a-\mu)/\sigma)$ -2848,$\nu (1-s)$ -2849,$g'(1-s)$ -2850,$P(X_{-1}\wedge a_{ro})=9196.39$ -2851,$\omega_0$ -2852,$g_2(s) = 2s/3 + 1/3$ -2853,"$\mathbf {X_{i,j}}$" -2854,$x^\ast$ -2855,$2/3$ -2856,$\iota(s)=w/(1-w)$ -2857,$\kappa_i(x) = \mathsf{E}[X_i \mid X=x]=\mathsf{E}_{\mathsf Q}[X_i \mid X=x]$ -2858,$\phi(0)=0$ -2859,$\log(S) =\mu t$ -2860,$a\le (P(1+\iota)-S)/\iota$ -2861,$\mathsf{E}_\mathsf{Q}[\cdot]$ -2862,$g'(s_1) \ge (1-g(s_1))/(1-s_1)$ -2863,"$U, V$" -2864,$s^{0.642}$ -2865,$\kappa_i(x)=mx/(m+n)$ -2866,$\mathbf {n}$ -2867,$C\mathsf X$ -2868,$s_0=1$ -2869,"$\Omega=\{1,2\}$" -2870,$\kappa_i(x) = \mathsf{E}[X_i \mid X=x]$ -2871,$x' - \mathsf{E}_\mathsf{P}[X]$ -2872,$\rho(X_0)=\mathsf{E}[X_0Z]$ -2873,$X\mapsto\int X(\omega)Z(\omega)\mathsf(d\omega)$ -2874,$\rho(X_0+\epsilon Y)-\rho(X_0)$ -2875,"$\sigma_A,\sigma_L$" -2876,$P(a)=g(S_X(a))$ -2877,$Z_\epsilon\to Z$ -2878,$a\beta_1g(S)$ -2879,"$\mathsf{biTVaR}_{p,1}^w$" -2880,"$(s^\ast, g(s^\ast))$" -2881,$a^\star$ -2882,$\beta_i(x)$ -2883,$\mathbf {\alpha_2S\Delta X}$ -2884,$\rho(X)=1.169$ -2885,$U(\omega)=\omega$ -2886,${X}$ -2887,$D^f\rho_{X;\tilde X}(X_i)$ -2888,"$d(g(S(x)))/dx=-g'(S(x))\,dF/dx$" -2889,$S(x+a)$ -2890,$\rho(0)=0$ -2891,$\succeq^2$ -2892,$\rho(\lambda X)=\lambda \rho(X)$ -2893,$q(p) \times \phi(p)dp$ -2894,$E(X_{-1}(a))=\bar S_0(a)$ -2895,$q_2(t)=t^2$ -2896,$\sigma^2=\sigma_A^2 + \sigma_L^2 - 2\rho\sigma_A\sigma_L$ -2897,"$(0.5, 0.5)$" -2898,$a_lp}$ -2920,$\int X=0$ -2921,$\mathsf{j}(0)=0$ -2922,$0<\alpha\le 1$ -2923,$g'(S(x))>1$ -2924,$r_f /(1+ r_f)$ -2925,$X_c$ -2926,$-q(-Y)$ -2927,"$[1,\infty)$" -2928,$4.75$ -2929,$D_c$ -2930,"$X_{t-2,1}$" -2931,$L\mathsf{VaR}_p(X))>1-p$ -2940,$\mathsf{E}[X\wedge a] = (1-e^{-a\beta})/\beta$ -2941,$(r-i)\sum_t Q_t$ -2942,$\gamma(ds)$ -2943,$Z=20\cdot1_A$ -2944,$X_n(\omega)= 1$ -2945,$F_0 = P_{act}-\mathsf{E}_{rn}[U]$ -2946,$\mathsf{Var}(X)$ -2947,$f=(1-p)^{-1}1_{W}$ -2948,$\rho(X_n)=0$ -2949,$1_{X\le a}$ -2950,$af\le 1$ -2951,$ for estimates $ -2952,$X+W$ -2953,"$\mathsf{biTVaR}_{0,1}^w(X)=(1-w)\mathsf{E}[X]+w\sup(X)$" -2954,$\mathsf{TVaR}_p(X)=\mathsf{E}[X\mid X >\mathsf{VaR}_p(X)]$ -2955,$gS$ -2956,$-\rho(-X)\le \mathsf{E}[X] \le \rho(X)$ -2957,$\Delta X_7$ -2958,$Z=\tilde X_2$ -2959,$a\alpha_i(a)=\kappa_i(a)$ -2960,$\mathsf{Pr}(X < x)\le 0.99 \le \mathsf{Pr}(X\le x)$ -2961,$\mathsf{E}[(X-m)(1_{U_X\ge p}-B)]\ge 0$ -2962,$B-p(\nu(p) + il(p))$ -2963,"$(0,0,0,0,0,0,5,0,0,5)$" -2964,$\mathsf{VaR}_p(X)=q_X^{-}(p) = \sup \{ x\mid F_X(x) < p \}$ -2965,$\omega\in\Omega$ -2966,$g=0$ -2967,$\bar P(a)=\mathsf{E}_\mathsf{Q}(X\wedge a)$ -2968,$L_0^a$ -2969,$-5.91$ -2970,$\bar q_{X_1+X_2}(s)=q_{X_1+X_2}(1-s)$ -2971,$\Pi=B-p\nu(p)$ -2972,"$Y_{2,1}$" -2973,$U(a)=-s$ -2974,$\rho(X+Y) = \rho(\lambda(X/\lambda) + (1-\lambda)(Y/(1-\lambda))))$ -2975,$\mathsf{E}_G(X)$ -2976,$\mathbf {\beta_{2}}$ -2977,$P(x)$ -2978,$r=(1+\bar\iota)/(1+\tau)-1$ -2979,$\rho(-X)=-\rho(X)$ -2980,$R_L=-k R_f + \beta_L(R_M-R_f)$ -2981,$\mathbf {\Sigma}$ -2982,$g(t)$ -2983,$N := \lceil (1-p)M \rceil$ -2984,"$a_i=\mathsf{E}[X_i] + k\mathsf{cov}(X_i, X)$" -2985,$\mathsf{Pr}(X\le x)=0$ -2986,$\{2\}$ -2987,"$(\nu,\delta)$" -2988,$p\to\infty$ -2989,$1\le\lambda$ -2990,$P_1=\mathsf{E}[X_1g'(S_X(X))]$ -2991,$\rho E/(1-\tau) - rA$ -2992,$x\mapsto x^{1/2}$ -2993,"$j=0,\dots,m=8$" -2994,$\beta_i(a)/\alpha_i(a) > 1$ -2995,$=\displaystyle\int_0^\infty x f(x)dx$ -2996,$a_l-1<0$ -2997,$F_X(x):=\mathsf{Pr}(X\le x)$ -2998,$F_Y^{-1}(V)=q_Y(V)$ -2999,$Z_{\mathit{lin}}$ -3000,$\mathsf{E}[f(X-\pi P)] = f((1-\pi)P)$ -3001,$0\le p_0\le p^*\le p_1\le 1$ -3002,$log(x)$ -3003,$\mathsf{P}(A) = \mathsf{E}[1_A]$ -3004,$\rho(X)=\mathsf{E}_{\mathsf{Q}}[X]$ -3005,$\nu=\nu(a)<1$ -3006,$X_1$ -3007,$X(\cdot)$ -3008,"$Z=(0,0,0,0,0,0,0,0,5,5)$" -3009,$Z_{\mathit{lift}}$ -3010,"$\mathbf{B}:\left [0,1 \right ] \ni t \mapsto (x(t),y(t)) \in \mathbb{R}^2$" -3011,$\mathcal Q(X)=\{ \mathsf Q\in\mathcal Q\mid \rho(X)=\mathsf{E}_\mathsf{Q}[X] \}$ -3012,"$(\Omega, P)$" -3013,$0 \le p<1$ -3014,$\mathsf Q_X$ -3015,$\mathbf {M_{1}}$ -3016,$\mathsf{E}[X]=1/\beta$ -3017,$n < N-1$ -3018,$\bar P(x)=\int_0^x P(t)dt$ -3019,$\mathsf{E}[g'(S(X))]=1$ -3020,$F(x_0)\ge p$ -3021,$X-a$ -3022,$Z\not=0$ -3023,$\rho(\cdot)$ -3024,$p = (1-s)$ -3025,$p=0.417$ -3026,$(j)$ -3027,$\mathsf{E}[X_1\mid X=x]$ -3028,"$\int_{[0,1]}$" -3029,$y^{\ast}:=\min(y)$ -3030,$\mathsf{E}|X|<\infty$ -3031,$\mathsf{E}_{\mathsf{Q}}[X] = \rho(X)$ -3032,"$ (MA.south)+(0, -1) $" -3033,$q^-(U(\omega))$ -3034,$Q_2\Delta X$ -3035,"$\mu=0.1, \sigma=0.15$" -3036,$\mathsf{Pr}(X > q_{\mathbf{v}}(p))=1-p$ -3037,$P_X(dx)$ -3038,$Y_{2}$ -3039,$Q_1dX$ -3040,${}^nS_X(t)\le {}^nS_Y(t)$ -3041,$(0.5)(20)+(0.5)(30)=25$ -3042,$\rho(X)\not=\sum_i\rho(X_i)$ -3043,"$M\subset \{1,\dots, n\}\setminus \{i, j\}$" -3044,$u(x)=(1-e^{-\pi x})/\pi$ -3045,$55+0.675\times 3.807=57.572$ -3046,$D \rho(X_0)$ -3047,$\alpha(1-f)$ -3048,$90$ -3049,$L_{250}^{\infty}(x)$ -3050,$\mathsf{Pr}(X < x) \le 0.1 \le \mathsf{Pr}(X\le x)$ -3051,$d\mathsf{Q}/d\mathsf{P} = g'(S(X))$ -3052,$\mathsf{E}[XZ_\epsilon]\to \mathsf{E}[XZ]$ -3053,$q(1)=\infty$ -3054,"$(p,q(p))$" -3055,$\prec_2^*$ -3056,$1/r$ -3057,"$\mathbf{v}=(v_1,\ldots,v_n)$" -3058,$\rho(X+X_i)=\rho(X)+\rho(X_i)$ -3059,$\displaystyle\int$ -3060,$\mathsf{E}[X1_{U_X\ge p}]\ge \mathsf{E}[XB]$ -3061,$\alpha_i(x)<\kappa_i(x)/x$ -3062,$\mathbf {\mathsf{E}[X_i\wedge a_i]}$ -3063,"$\{1,2,3,4,5,6\}$" -3064,"$(X_1,\dots, X_n)'$" -3065,$1_A(x)=0$ -3066,$X_{-4}=x$ -3067,$\mu-\sigma^2/2$ -3068,$s(1)=s_3=1$ -3069,$g(x)=0$ -3070,$L_0^{a+y}=L_0^a+L_a^{a+y}$ -3071,$x_{#4}$ -3072,$\mathsf Q(A)=\mathsf{E}[Z1_A]$ -3073,$n\ge 1$ -3074,$\mathsf{E}[X_m\mid X_{m+n}=x]=mx/(m+n)$ -3075,$\mathsf{E}[X]=28$ -3076,$2.576$ -3077,$Q_j=1-g(S_j)$ -3078,"$B_2=[0,0]$" -3079,$\sum c_i^2$ -3080,$X_i(v_i)$ -3081,$\alpha_i(a)S(a)$ -3082,$X(\omega)=0$ -3083,$\lambda=(1-\alpha_p)^{-1}$ -3084,$g_i$ -3085,$U \ge U_s$ -3086,$\bar P = a - \bar Q$ -3087,$p(1-\nu(p)-il(p))$ -3088,$Z_{a}(x)=g(S_X(a))/S_X(a))$ -3089,$X(\omega_1)a'$ -3093,"$0.1, 0.4, 0.5,\dots, 0.9$" -3094,"$I(q,p) \ne I(p,q)$" -3095,$k=-\log(p)/u$ -3096,$S(x_{i-1})-S(x_{i})=S(x_i-(x_i-x_{i-1}))-S(x_i)=-S'(x'_i)(x_i-x_{i-1})=f(x'_i)(x_{i}-x_{i-1})$ -3097,"$x,y\in C$" -3098,$g^{-1}(x)\le s$ -3099,$\mathsf{Q}(A)=2\mathsf{P}(A\cap B)$ -3100,$f(x) < f(y)$ -3101,$\bar S(a):= \mathsf{E}[L_0^a(X)]=\mathsf{E}[X\wedge a]$ -3102,$\iota^{\star}$ -3103,$\zeta_{s} = \Phi^{- 1}(s)$ -3104,$Z(\omega)=\dfrac{1}{1+r}\dfrac{\mathsf Q(\omega)}{\mathsf{P}(\omega)}$ -3105,$\iota$ -3106,$\mathsf{E}[X_ih(X)]$ -3107,$d^* = D/L^*$ -3108,"$\rho(X) = \max\{\rho_c(X), \mathsf{TVaR}_{0.8}(X) \}$" -3109,$S(x)=1$ -3110,$g(s)=\Phi(\Phi^{-1}(s)+\lambda)$ -3111,$\rho_g(X)<\infty$ -3112,$M=\iota Q$ -3113,$q + 2pq + 3p^2q+\cdots=q(1+2p+3p^2+\cdots)=1/q$ -3114,$g'(1-p^* )=1$ -3115,$-1$ -3116,"$ In general, define $" -3117,"$(4,2)$" -3118,$\alpha=1$ -3119,$\mathsf{E}[X \mid X \ge x] = \mathsf{E}[X 1_{X \ge x}] / \mathsf{Pr}(X \ge x)$ -3120,$\alpha_{Cat} \le \beta_{Cat}$ -3121,$\rho(X)=\mathsf{E}_\mathsf{Q}[X]$ -3122,$R_S$ -3123,$dt$ -3124,$\mathsf{Pr}(X\ge x_0)=p_-$ -3125,$E_i\in\mathcal F$ -3126,"$\bar P_{0,0}:=\rho(Y_{0,0})$" -3127,$a_1 < a_0-X_1$ -3128,$ is different from the contact function $ -3129,$t < 2/3$ -3130,"$\omega\in[0,1]$" -3131,"$h(x):=H(x, 1, t)$" -3132,$g(s)=3s$ -3133,$\mathbf {X_1/X}$ -3134,$\mathsf{E}[(X-\mathsf{E} X)^+]$ -3135,$\{ Z\mid \rho(X)=\mathsf{E}[XZ] \}$ -3136,$\mathsf{E}[X_iZ]=\rho_g(X)/2$ -3137,$\rho(X) \ge \mathsf{E}[X]$ -3138,$p=0.5$ -3139,"$\lambda\rho(X) + (1-\lambda)\rho(Y) \le \max(\rho(X),\rho(Y))$" -3140,$\{n_s\}$ -3141,$S(X_0)$ -3142,$r_m$ -3143,$X_i=X_i(a)$ -3144,$\bar Q_{act} = \bar Q - F_0$ -3145,$\beta_i/\alpha_i$ -3146,$\mathsf{E}[X_i/X \mid X > x]$ -3147,$\bar P(a)= (1-e^{-a\alpha\beta})/(\alpha\beta)$ -3148,$S(x)=e^{-x/\mu}$ -3149,$m_i\ge0$ -3150,$M(x)/(1-S(x))$ -3151,$\mathsf{E}[X \mid \mathcal F_0]$ -3152,"$(r,c)$" -3153,$r\le 0$ -3154,$\mathsf Q\in\mathcal Q$ -3155,"$X\,\Delta S$" -3156,$Z=(1-p)^{-1}1_{X>q_X(p)}$ -3157,$\mathsf{E}[X]=\sum_{\omega\in\Omega} X(\omega)\mathsf{Pr}(\omega)$ -3158,$d(g(S(x))/dx=g'(S(x))f(x)$ -3159,$p={{p}}$ -3160,"$\mathsf{E}_{\mathsf{Q}}[Y]=\mathsf{E}[Y\,g'(S(X))]$" -3161,$v\in V$ -3162,$\rho(\tilde X_1)=\rho(X_1) + \mathsf{E}[X_2]$ -3163,$\mathsf{E}[B]=p$ -3164,$\iota=0.10$ -3165,$\hat p > p$ -3166,"$C(S_0, a, t)$" -3167,$c = 0.5(0.5)2.5$ -3168,$M = 0.603$ -3169,$A/(A-P)$ -3170,"$A,B,C,D$" -3171,$h=\sin(77 s)$ -3172,$g(s)=1-(1-s)^3$ -3173,$\mathsf{CTE}_{p_0}=\mathsf{E}[X \mid X \ge x_0]$ -3174,$\phi\in \mathcal E$ -3175,$F_1 \prec_1 F_0$ -3176,$\mathsf{E}[u(w-X)] = u(w-c)$ -3177,$\lim_{s \downarrow 0}1/g'(s)$ -3178,$\Delta_j =g'(s_j-)-g'(s_j+)=\phi((1-s_j)+)-\phi((1-s_j)-)$ -3179,$\Omega_0:=\{\omega\in \Omega\mid X(\omega)=\max(X)\}$ -3180,$f(s)\le s$ -3181,$\bar\iota(a)$ -3182,$h(X)=(X-\mathsf{E} X)$ -3183,$j>0$ -3184,$n=1$ -3185,$S_0$ -3186,$g(S(x))=g(S(x-))=1$ -3187,$\mathsf{E} X + c\mathsf{E}[((X-\mathsf{E} X)^+)^p]^{1/p}$ -3188,$\mathsf{E}[hY]$ -3189,$A\subset\mathbb{R}$ -3190,$f(p)=(1-p)\phi'(p)=-(1-p)g''(1-p)$ -3191,$\mathsf{TVaR}_0( X )=\mathsf{E}[X]$ -3192,$ then $ -3193,$\epsilon_1$ -3194,$i>0$ -3195,"$0, 1, 90$" -3196,$\beta_1<\alpha_1$ -3197,$\nu p$ -3198,"$n=1, p=1/{{p}}={{pf}}$" -3199,$q(U)=F^{-1}(U)$ -3200,$\sqrt{0.1}=0.316$ -3201,$L(X)=e^{kX}/\mathsf{E}[e^{kX}]$ -3202,$\ge 0.95$ -3203,$\mathsf{E}[X_i \mid X=x]$ -3204,$vL + da$ -3205,$g'(s)=\nu$ -3206,$b=0.5$ -3207,$a < b_h$ -3208,"$(-\x*.8, 2*2)$" -3209,$L>d$ -3210,"$a_{0,2}$" -3211,$\mathbf {s_0}$ -3212,$\mathsf{E}[Z\mid X>a]=g(S(a))/S(a)$ -3213,$a_i=a(X_i; X)$ -3214,$\dot f(t)=a(x)$ -3215,$A^c$ -3216,$P_i \ge \mathsf{E}[X_i]$ -3217,"$\mathsf P((a,b])=b-a$" -3218,$\mathbf {g_1(s)=s^{0.4}}$ -3219,$1-p$ -3220,$\lim_{s \downarrow 0} s/g(s) = \lim_{s \downarrow 0}1/g'(s)$ -3221,$\rho_\mu$ -3222,$\mathsf{EPD}_\pi(X)$ -3223,$\bar F(a)$ -3224,$\mathsf{E}[Z_i\mid X] \ne \mathsf{E}[Z_j \mid X]$ -3225,$\mathbf {Q_1\Delta X}$ -3226,$P(X_{0}(a_{gc}))$ -3227,$20+8t>20+10t$ -3228,$b=1$ -3229,$p_0 = p^\ast = p_1$ -3230,$Z\in L^1$ -3231,$Y$ -3232,$g(S(x))=u$ -3233,$\phi'(s)\ge 0$ -3234,$x\mapsto (x-d)_+^{n}$ -3235,$\{X \le x^*\}$ -3236,"$X_1=0,0,0,0,1,1,2,3,20, 400$" -3237,$m_1=m_2$ -3238,"$\dfrac{\partial\rho}{\partial P} = \dfrac{0.4^2 P}{\rho(P,R,a)}$" -3239,$\mathbf {t+3}$ -3240,$u = g(S(x))$ -3241,$\mathsf{E}[X_2Z]$ -3242,$\rho(X)=g(q)$ -3243,$\bar M=\bar P-\bar S$ -3244,$(1-p)^{-1} \min_x x(1-p) + \mathsf{E}[(X-x)^+]$ -3245,$q_Y(1-U)$ -3246,$h(s)$ -3247,$f^{-1}(A)\in\mathcal B$ -3248,$(1-p)^{-1}\mathsf{E}[X_i1_{X\ge x_p}(X)]$ -3249,$\beta_1g-\alpha_1S$ -3250,$X_2(a)$ -3251,$g'(s)=bs^{b-1}$ -3252,$\mathsf P(A)=1-p$ -3253,$dF(x)$ -3254,"$(0,g_0)$" -3255,$\kappa_1(X)$ -3256,$x \mapsto -x$ -3257,$A(1_{X>x_1} + 1_{X>x_2})= A(1_{X>x_1}) + A(1_{X>x_2})$ -3258,${Z}_p \le c$ -3259,$X:\Omega\to\mathbb{R}$ -3260,$C_1+\cdots + C_n$ -3261,$\mathsf{E}[Y_d]$ -3262,$\mathbf {\alpha_1}$ -3263,$\tilde X\wedge a$ -3264,$d+v=1$ -3265,"$\Omega=[0,1]$" -3266,$q_Y$ -3267,$D\rho_X(\cdot)$ -3268,$\mathbf {X_2}$ -3269,$g^{-1}(u)$ -3270,$\sum_{i}X_{i} = X$ -3271,$g_{ROE}$ -3272,$>1-p$ -3273,$\mathsf{Pr}(X < x)$ -3274,$a=\mathsf{VaR}_{1-\tau}(X)$ -3275,$\mathsf{Pr}(X < x) \le 1/6 \le \mathsf{Pr}(X\le x)$ -3276,$h(x):=f(x)/S(x)$ -3277,$X_n(\omega)=1$ -3278,$\mathbb{R}$ -3279,$S_Y$ -3280,$\chi^2$ -3281,$X=X' + X''$ -3282,$(X\wedge a)\Delta g$ -3283,$\rho(X)=\mathsf{E}[h(X)L(X)]$ -3284,$f(x)\approx 0$ -3285,$ but if $ -3286,$Q=(a-EL)/(1+r)$ -3287,$a\ge 0$ -3288,$N=5$ -3289,$\mathsf{Pr}(E')=1-\mathsf{Pr}(E)$ -3290,"$D_n,D_n^*$" -3291,$\{ X=x \}$ -3292,$X_d$ -3293,$P=g(s)$ -3294,$\int xdF(x)=\int xf(x)dx$ -3295,$X({\mathbf{v}})$ -3296,$g(s)=s^{0.9}$ -3297,"$X_{t,1}$" -3298,$x=X(p)$ -3299,$\mathsf{E}_{\mathsf Q}[X_i]=\mathsf{E}[X_ig'(S(X))]$ -3300,$\hat s$ -3301,$\sigma_i^2$ -3302,"$(1-s, 1-g(s))$" -3303,$\ge$ -3304,$h(p)$ -3305,$\kappa_1(x)=\mathsf{E}[N_1/(N_1+N_2)]x$ -3306,$\max(X)=1$ -3307,$R_f$ -3308,$\phi(s)=0$ -3309,$\mathsf{E} X + c{ X-MX }$ -3310,$\mathsf{Var}(X+c)=\mathsf{Var}(X)$ -3311,$\mathsf{TVaR}_{0.975}$ -3312,$l^\infty$ -3313,$x_p=\mathsf{VaR}_p(X)$ -3314,$\sum v_iX_i$ -3315,$\mathsf{E}[X] + c\mathsf{E}[(X-\mathsf{E} X)^21_{X>\mathsf{E}[X]}]$ -3316,$R$ -3317,$s=0.5$ -3318,"$(1-S(x),x)=(p,q(p))$" -3319,$0!=1$ -3320,$\rho(U)=1$ -3321,$x=1000$ -3322,$\mathsf{E}[YZ]\le 0$ -3323,$\mathsf{Pr}(X<2)=1/6<\mathsf{Pr}(X\le 2)=1/3$ -3324,$m(1)=0$ -3325,$a_{t} = a_{t-1}$ -3326,$\mathsf{E}[\phi] = 1$ -3327,$A=\{X(\omega) > x\}$ -3328,$\mathbf {X_2(a)}$ -3329,$0.1 < s < 0.2$ -3330,$p < 1$ -3331,$g(0+)\ge 0$ -3332,"$3.129=\lambda \sigma(Y_{0,0})$" -3333,$\beta_i(x)/\alpha_i(x)> 1 > S(x) / g(S(x))$ -3334,$S(x)\approx k x^\alpha$ -3335,"$\mathit{EGL}_{gc}(a)>\max(0, \mathit{EGL}_{ro}(a))$" -3336,$\alpha_1(99)=0.1$ -3337,$\mathsf{TVaR}_p(X)=80$ -3338,$m\ge n$ -3339,$(a-X)^+$ -3340,$M_1dX$ -3341,$(X\wedge l)(\omega)=X(\omega)\wedge l$ -3342,$a=a[X]$ -3343,$\mathsf Q_k(B_k)=\mathsf{P}(B_k)/\mathsf{P}(B_k)=1$ -3344,$\mathsf{E}_\mu[\phi(\mathsf{E}_\pi u\circ f)]$ -3345,$a^{\star}(X)-a(X)$ -3346,$\mathit{PV}_{r_X}(X) + \mathit{PV}_{r_f}(\text{UW profit tax})$ -3347,$A-A\Phi(d^*)=A\Phi(-d^*)$ -3348,"$j=0,\dots, m-1$" -3349,$(P-S)/(a-P)\ge \iota$ -3350,$S=\mathsf{Pr}\{X>x\}$ -3351,$\mathsf{E}[p]$ -3352,$r^*$ -3353,$F(x)=\P(X\le x)$ -3354,$\mathsf{P}_X$ -3355,"$\int |X_n(\omega) - X(\omega)| \,\mathsf{P}(d\omega)\to 0$" -3356,$\bar Q(x)$ -3357,"$(a,b] \subset [0,1]$" -3358,$\mathbf {\omega_i}$ -3359,$\mathsf{Pr}(X< q(p))\le p \le \mathsf{Pr}(X\le q(p))$ -3360,$\iff$ -3361,$\exp$ -3362,$D>L$ -3363,"$\mathsf{biTVaR}_{0,1}^{0.0476}$" -3364,$\iota(0.5)=\iota^{\star}$ -3365,$n \ge 1$ -3366,$\kappa_{2}$ -3367,$Y\in L^\infty$ -3368,$\{ X=\mathsf{E}[X] \}$ -3369,$\mathsf P(f^{-1}(A))=\mathsf{Pr}(A)$ -3370,$0 \ge \rho(-X+a)=\rho(-X) + a \ge -\rho(X) +a$ -3371,$\mathsf{E}[X^k]$ -3372,$a>0$ -3373,$1\le p \le \infty$ -3374,$Z>\mathsf{E} Z$ -3375,$\mathit{PFL}$ -3376,$X_i(a)=X_i\dfrac{X\wedge a}{X}$ -3377,$g'(1)$ -3378,$0\le \alpha\le 1$ -3379,$g(S(x))=0$ -3380,$\rho\ge 0$ -3381,$\nu(p)=1/(1+\iota(p))$ -3382,"$[0,\infty)$" -3383,$\uparrow$ -3384,$a_i + b_i\ \mathit{EL}$ -3385,$\mu t + \sigma dW_t -\sigma^2 dt /2 +o(dt)$ -3386,$F(x):=\mathsf{Pr}(X\le x)$ -3387,$h$ -3388,$4/6$ -3389,$X_2=c_2+2Y$ -3390,$-Y\ge 0$ -3391,$S(x_2)(x_3-x_2)$ -3392,$0\le\lambda \le 1$ -3393,$x \ge x^\ast$ -3394,$1/4 < s\le 1$ -3395,$A_X = 5.976$ -3396,$\rho(X+Y)\ge$ -3397,$M = r K$ -3398,$X_n(\omega)=n$ -3399,$r = 0.6565$ -3400,$\nu^{\star}$ -3401,$-\rho(-X) =b-\rho(b-X)$ -3402,$\mathsf{E}_{\mathsf Q}$ -3403,$\alpha_1SdX$ -3404,"$a(\cdot, p)$" -3405,$\tau \ge t+d$ -3406,$\mu(\{p\})=1$ -3407,$c\approx -\sigma^2u''(w)/u'(w)$ -3408,$\|Z\|_p = \mathsf{E}[| Z|^p]^{1/p}$ -3409,$X\wedge a=\sum X_i(a)$ -3410,$\kappa_i(x)=E[X_i \mid X=x]$ -3411,$\lambda_0$ -3412,$\epsilon /2^{n+1}$ -3413,$\nu(x)$ -3414,$S(x)=\exp(-\int_x^\infty h(t)dt)$ -3415,$g(P)$ -3416,$2x$ -3417,$P(a) = g(S(a))$ -3418,$[F(x)](\cdot)$ -3419,"$\Omega=\{\omega_1, \ldots, \omega_6\}$" -3420,$\mu-\sigma^2/2=0.0992$ -3421,$F(p)=0.6$ -3422,$\rho(X_j)$ -3423,$S(a)=\mathsf{E}[1_{X>a}]$ -3424,$\mathsf{E}[X_ie^{kX}]/\mathsf{E}[e^{kX}]$ -3425,$y=a$ -3426,"$\mu,\sigma$" -3427,$g_i=g^{-1}(u_i)$ -3428,$u=0.1$ -3429,$1_{U>s}$ -3430,"$\rho(X)=\int g(S(t))\,dt$" -3431,$\mathbf {t}$ -3432,$\{ x \mid F(x) \ge p \}$ -3433,$g(s)q$ -3434,$\mathsf{VaR}_1(X)$ -3435,$\sigma_L$ -3436,$\bar S_i(a)=\mathsf{E}[X_i(a)]$ -3437,$Q=1-g$ -3438,$L_a^{a+y}(X)$ -3439,$\rho(X)=\mathsf{SD}(X)$ -3440,"$\int_{[a,b]} h(x)dF(x)$" -3441,$\bar\nu(a)=1/(1+\bar\iota(a))$ -3442,$-g''(1-p) = \phi'(p) = (1-p)^{-1}f(p)$ -3443,$g(S_X(X))$ -3444,$d\mathsf{Q}/d\mathsf{P}$ -3445,"$(\Omega, \mathcal F, \mathsf P)$" -3446,$\mathsf{E}_{\mathsf{Q}}[Y \mid X] = \mathsf{E}[Y \mid X]$ -3447,$0\le p\le 1$ -3448,$1/(1+r_f) = \mathsf{E}[p]$ -3449,"$D^f\rho_{X\wedge a,X}(\cdot)$" -3450,$V^{\ast}(1)=p/(1+r-p)$ -3451,$H_k(X)=H_{g_k}(X)$ -3452,$f(x)/S(x)$ -3453,$\partial\bar P/ \partial a$ -3454,"$X_{t,d}$" -3455,$\Delta S=0$ -3456,$\mathsf{E}[X_i(1) \mid X(\mathbf{v}) = q_{\mathbf{v}}(p)]$ -3457,$s_3=1$ -3458,$0< a\le 1$ -3459,$B(1_{X\le x})$ -3460,$2^{-t+1}$ -3461,$\beta < \alpha$ -3462,"$\bar P_i(\mathbf{v},a)$" -3463,$\sum \Delta g(S)_jX_j$ -3464,$\rho(0) = \rho(0+0)\le \rho(0)+\rho(0)$ -3465,$a<\infty$ -3466,$X=Y/\lambda$ -3467,$a\alpha_i(a)$ -3468,$q(1-s)$ -3469,$a_2 = 2.157$ -3470,$\mathsf{TVaR}_p = 20(0.55x_{67}+x_{68}+x_{69}+x_{70})/71$ -3471,$\mathsf{TVaR}$ -3472,$q(\psi)$ -3473,$a_{ro}:=\mathit{VaR}_{p}(X_{-1})={{a_x0}}$ -3474,$( x_{(j)}-x_{(j-1)} )$ -3475,$l(\mathbf X)$ -3476,$p\nu(p)$ -3477,$w_{0.75}$ -3478,$0.7 \ge p < 0.8$ -3479,$\omega_1=1$ -3480,"$(1-g(S(x)),x)=(p,q(1-g^{-1}(1-p))$" -3481,$v=1/(1+\iota)$ -3482,$f$ -3483,$a(X_i)=2.665$ -3484,$\mathbf{B}'(0) = -3\mathbf{P_0}+3\mathbf{P_1}$ -3485,$g_3(s)=s^{0.7}$ -3486,$1-\hat p$ -3487,$P(A\cup B)\le P(A)+P(B)$ -3488,$\iff \rho$ -3489,$0\le s\le \epsilon$ -3490,$q(p)=25$ -3491,$\rho(X)\le c$ -3492,$X_n(\omega)\to 0$ -3493,"$(0,3)$" -3494,$g(s)=sv+d$ -3495,$a=P+S$ -3496,"$(x,y)\not=(0,0)$" -3497,$\bar P_0$ -3498,$S=1-F$ -3499,$-t$ -3500,$f(x) = \dfrac{dF}{dx}$ -3501,"$\mathbf {D^f\rho_{X\wedge 30,X}(X_2)}$" -3502,$\bar{\mathbf P}$ -3503,$-g''(s)=\alpha(1-\alpha)s^{\alpha-2}$ -3504,$p(x) = \mathsf{Pr}(\{\omega\mid X(\omega) = x\})=\mathsf{Pr}(X=x)$ -3505,$\sigma=1$ -3506,$P(a)=1-Q(a)=1-h(F(a))$ -3507,$\delta=\dfrac{\iota}{1+\iota}=\dfrac{M}{a}$ -3508,$s\le s^*$ -3509,$a' := (1-S)\Delta X$ -3510,"$\mathbf{j, p, S, \kappa_1, \Delta X, \Delta(X\wedge a)}$" -3511,$\mathbf {Q=1-g(S)}$ -3512,$w/s = g'(s-) - g'(s+)$ -3513,$e^{\mu_L}-1$ -3514,$X=m$ -3515,$k(s)$ -3516,$\mathsf Q(A)=\int_A f(\omega)\mathsf P(d\omega)$ -3517,$(g-S)dX$ -3518,"$k, b$" -3519,$\mathsf{E}_\mathsf{P}[X_j]$ -3520,$p^*$ -3521,$\int_0^\infty xf(x)dx$ -3522,$\Delta P$ -3523,$\alpha_i(x)=\mathsf{E}\left[\frac{X_i}{X}\mid X > x \right]$ -3524,$r$ -3525,$s+\delta p = 1-\nu p$ -3526,$\mathbf p$ -3527,$\mathsf{Var}(\lambda X)=\lambda^2\mathsf{Var}(X)$ -3528,"$m_0, s_1, m_1, s_2, m_2$" -3529,$=\displaystyle\int_0^\infty x \P_X(dx)$ -3530,$\mathit{NPV}_1 = \bar Q - \bar Q_{act} = F_0$ -3531,$\rho_g(X)=35.2$ -3532,$\max_\mathsf{Q} \mathsf{E}_\mathsf{Q}[X]$ -3533,$Z=Y-X$ -3534,$\mathbf r$ -3535,$\mathsf{TVaR}_{0.642}$ -3536,$g(S(x_B))-g(S(x_B-))$ -3537,$u = \alpha_i(x)S(x)$ -3538,$\alpha_1 < \alpha_2$ -3539,$Z(g(s))=Z(s)+\lambda$ -3540,$\mathbf {\rho(X)}$ -3541,$\mathit{NPV}_{\infty}=a_xF_0$ -3542,"$X_{1,0}=\cdots=X_{m,0}=X_0=0$" -3543,"$\Omega=\{\omega_1, \omega_2 \}$" -3544,$a(X_i+X_j) < a(X_i)+a(X_j)$ -3545,$m=0.25$ -3546,$\{Y\mid Y\preceq_2 Z\}$ -3547,"$(de.east |- lee.north)+(0.375,0.25)$" -3548,$\mathsf{E}[X] + \pi\mathsf{E}[X]$ -3549,$c(\{i\})=c(i)$ -3550,$\hat g(s)=1-g(1-s)$ -3551,$W_{s+t}-W_s$ -3552,$\mathsf{Pr}(X=x)=0$ -3553,"$(1,2)$" -3554,$1-s$ -3555,$D_2$ -3556,$x=200$ -3557,$\mathbf{v}$ -3558,"$(0,0,0,0,0,5,0,0,0,5)$" -3559,$P=l + \delta(a-l)$ -3560,$S/L$ -3561,"$\int_0^a F(t)\,dt$" -3562,$\\mathbf {\1}$ -3563,$\int_0^s \mu(dt)/(1-t)$ -3564,$Z\circ T$ -3565,$\mathbf {D^n\rho_{X\wedge 30}(X_2)}$ -3566,$=v_f \mathsf{E}_Q\left[\dfrac{X_i}{X}(X\wedge A)\right]$ -3567,$g(S(a))$ -3568,$\mathcal M_\rho$ -3569,$P=(1+r)\lambda\mathsf{E}[X]$ -3570,$F(a+)=\lim_{x\downarrow a} F(x)$ -3571,$f<1$ -3572,$\alpha>1$ -3573,$\mathcal F_0\times \mathcal F_1$ -3574,$\rho(Y)$ -3575,$\mathsf Q(\omega)\ge 0$ -3576,$\lim_{s \uparrow 1}g'(s)$ -3577,$k>2$ -3578,$S\to Y$ -3579,$\mathsf{E} X$ -3580,$s'(t)$ -3581,$g'\circ S_X$ -3582,$s=0.1$ -3583,$g = s/(1-f)$ -3584,$g(s)=A(1_{U < s})$ -3585,$\Delta g(S_j)=g(S_{j-1})-g(S_j)$ -3586,$A\wedge L$ -3587,$\mathbf {2\mathsf{VaR}_p(X_1)}$ -3588,$g'(1-s)+g(0+)\delta_1$ -3589,"$5^{-1},5^{-2},5^{-3},\dots$" -3590,$\mathsf{EPD}_p(X)$ -3591,$+$ -3592,$c(\alpha)x^\alpha g(x)$ -3593,$\mathit{NPV}_{\infty} = a_xF_0$ -3594,$\mathsf{E}[X]=\mathsf{E}[Y]$ -3595,$v/\sqrt{n}$ -3596,$X_h$ -3597,$\mathsf{E}[(a-X)^+]=\int_0^a F(x)dx$ -3598,$\mathsf{Pr}(X < x)\ge 1/6$ -3599,"$\mathsf{cov}(X_i,X)$" -3600,"$(p,t)$" -3601,$e^{-rt}S_t$ -3602,$9+1$ -3603,$(x-d)^+ \wedge l$ -3604,$\mathsf Q(B) = \mathsf P(A\cap B)/\mathsf P(A)=\mathsf P(A\cap B)/(1-p_0)$ -3605,$\mathsf{E}[S_t]=e^{\mu t}$ -3606,$Y_i$ -3607,$\sqrt{x}$ -3608,$\rho(X-X)=\rho(X)+\rho(-X)=0$ -3609,$dG/dF=g'(S(x))$ -3610,$D_m\subset D_n$ -3611,"$[0,1]\subset\mathbb R$" -3612,$r-1$ -3613,$d_f = r_f / (1+r_f)$ -3614,$\hat q(p)=q(1-g(1-p))$ -3615,$X=Y$ -3616,$\mathsf{Pr}(X_n=1)=1/n$ -3617,$U^{1/b}$ -3618,$X\preceq_1 Y$ -3619,$E(X-q(X))^+$ -3620,$X_{-2}$ -3621,$t=U_X(s)$ -3622,$\mathsf{E}_{\mathsf Q}[Y]$ -3623,$3^{30}=2.06\cdot 10^{14}$ -3624,$\rho(kX)\ge k\rho(X)$ -3625,$M(x)=P(x)-S(x)$ -3626,$H$ -3627,$a=\mathsf{VaR}$ -3628,$\int X_n=1$ -3629,"$\displaystyle\int_0^a \kappa_i(x)f(x)\,dx + a\alpha_i(a)S(a)$" -3630,$\kappa_i(x)\approx x -\sum_{j\not=i} \mathsf{E}[X_j]$ -3631,$\alpha_1(90) = (0.0816 \cdot 0.0625 + 0.1 \cdot 0.0625)/(0.0625+0.0625)=0.01135/0.125=0.0908$ -3632,$c_i$ -3633,$0 \le X_i(a) \le X_i$ -3634,$\sup_i f_i$ -3635,$D\rho_X(X_1)=6.2085$ -3636,$+\mathsf{NORIPOFF}$ -3637,"$(a,b)$" -3638,$t\downarrow 0$ -3639,$\rho_g(X)=\mathsf{E}_\mathsf{Q}[X]$ -3640,$\{\mathsf{P} \}$ -3641,$\mathsf{E}_{\mathsf Q}[Y] = \mathsf{E}[YZ]$ -3642,$\mathcal{G}=\sigma(X)$ -3643,$\pi$ -3644,$h(x)=-d/dx(\log(S(x)))$ -3645,$x=8$ -3646,"$\displaystyle\int_\Omega g(X(\omega), \omega)\mathsf{Pr}(d\omega)$" -3647,$X\_{2}$ -3648,$dS$ -3649,$\sum \alpha_i S\Delta (X\wedge a)$ -3650,"$g'(s) = \frac{1-w}{1-p_0}1_{[0, 1-p_0)}(s) + \frac{w}{1-p_1}1_{[0, 1-p_1)}(s)$" -3651,$\mathscr{E}$ -3652,$\mathsf{E}_{\mathsf Q}[X_i \mid X=x] = \mathsf{E}[X_iZ \mid X=x]/\mathsf{E}[Z \mid X=x] = \mathsf{E}[X_i \mid X=x]$ -3653,$pX$ -3654,$g(S(a))/S(a)$ -3655,$\sum X_i(a)\Delta g(S)$ -3656,"$(p,q(1-g^{-1}(1-p)))$" -3657,$0\le \pi\le 0.5$ -3658,$\bar\delta=\bar\iota/(1+\bar\iota)$ -3659,$q^-(F(x))=x$ -3660,$1-g(s)$ -3661,$P=L + d(a-L)$ -3662,$p\not=0.75$ -3663,"$a=0, b=\alpha$" -3664,$\mathsf{E}[X_i\mid X=q(p)]$ -3665,$\mathbf {\vert S\vert}$ -3666,"$\bar S(a)=\int_0^a S(x)\,dx$" -3667,$X=X\wedge a + (X-a)^+=\sum_i X_i(a) + (X-a)^+$ -3668,$r_f = 0.01$ -3669,$X_2=X-X_1$ -3670,$c_1$ -3671,$\max_\mathsf{Q} \mathsf{E}_\mathsf{Q}[X] - \alpha(\mathsf Q)$ -3672,$u^{(n-1)}$ -3673,$(r-\sigma^2/2)t$ -3674,$\tau_i=\tau$ -3675,$\tau=\tau_i=0$ -3676,$a=a(s)$ -3677,$\mathsf{E}[Z\mid X]=Z$ -3678,$\mathsf{E}[X_1\tilde Z]=\mathsf{E}[X_2\tilde Z]=500$ -3679,$f(L)=L$ -3680,$f(L) \le L$ -3681,$p=0.283$ -3682,$g'(s)=\alpha s^\alpha/s$ -3683,$n-4$ -3684,$xdF(x)$ -3685,$\mathsf{TVaR}_{0.8}(X)=25$ -3686,$X_0=X_1=0$ -3687,$Q_X$ -3688,$\mathsf{TVaR}_{p^\ast}(X)=\bar P$ -3689,$\mathsf{E}[(S_t-a)1_{\{S_t>a\}}$ -3690,$P_X(A)=0$ -3691,$L > a$ -3692,$f=0$ -3693,$f(x)dx=dp$ -3694,$P_X(A)=\mathsf P(X\in A)= F(b)-F(a)$ -3695,$Z(a')=g(S_X(a))/S_X(a))$ -3696,"$X_i(\omega), i=1,...,N$" -3697,$\alpha(\mathsf Q)$ -3698,$\phi(p)$ -3699,$\mu(\{p_1\})=w$ -3700,$G\mathsf X$ -3701,$\omega=0$ -3702,$\displaystyle\int_\Omega X(\omega)\mathsf{Pr}^*(d\omega)$ -3703,$P-D$ -3704,$X>a$ -3705,$\iota=$ -3706,$\lim_{t\to 0}a(X_1; X+tX_1)=a(X_1;X)$ -3707,$1+Z-\mathsf{E} Z$ -3708,$e = P/C$ -3709,$t^\star=1/2$ -3710,$t+1$ -3711,$1-B_p=B_{1-p}$ -3712,$\mathsf{Pr}(|X_n(\omega)-X(\omega)|>\epsilon)\to 0$ -3713,$\bar M(x)$ -3714,$X\not\preceq_n Y$ -3715,$0\le x < a$ -3716,"$Z_2:=\sum_{t+d=2} Y_{t,d}$" -3717,$ since the contact function $ -3718,"$c_1+c_2=(c(1) + c(1,2) - c(2) + c(2) + c(1,2) -c(1))/2=c(1,2)$" -3719,"$(-\infty, \infty)$" -3720,$a(X)\equiv a$ -3721,$\mathcal E(X)=c\mathsf{E}[X^2]$ -3722,$x^{**}$ -3723,"$D^f\rho_{X\wedge a,X}(X_i)$" -3724,$X(p)=q(T(p))$ -3725,"$(1-S(x), x)$" -3726,$\tilde X_1+\tilde X_2\succeq^2 \tilde X_1$ -3727,$\mathcal D(X)+\mathsf{E}[X]$ -3728,$S_{\mathbf{v}}$ -3729,$\mathsf{VaR}$ -3730,$\mathsf{E}[X]+\mathsf{SD}(X) \le \mathsf{E}[Y]+\mathsf{SD}(Y)$ -3731,$\bar S_i$ -3732,$\{X(\mathbf{v}) = q_{\mathbf{v}}(p)\}$ -3733,$\alpha_iSdX$ -3734,$c=0.5$ -3735,$K$ -3736,$g(p)/p-1$ -3737,$a(X_i; X)$ -3738,$\log(1+\mu t + \sigma dW_t)=\mu t + \sigma dW_t +o(dt)$ -3739,$\max(X)$ -3740,$\mathsf{E}[X_1\mid X < 2^{-m}]$ -3741,$x>\sup(X)$ -3742,$M=\inf\{ x\mid S(x)=0\}$ -3743,$\mathsf{VaR}_\pi(X)$ -3744,$0=\mathsf{Pr}(X<1)<1/6=\mathsf{Pr}(X\le 1)$ -3745,$-k<0$ -3746,$X_n=Y_1+\cdots +Y_n$ -3747,$^{}$ -3748,$\mathsf{CTE}_p(X)=(8+12+25)/3=15$ -3749,$p \ge 0.9$ -3750,$S_0=1000$ -3751,$\lim_{s \to 1}{\mathsf{E}[ r_{s} ] = - 1}$ -3752,$1_{U0}$" -3771,$\prod_{n\ge N}(1-\frac{1}{n})=0$ -3772,$X\le 0$ -3773,$g(s)=s^{0.8}$ -3774,$q \cdot X$ -3775,$p=0.1$ -3776,$\mathsf{E}[X_iX]$ -3777,"$(p, q(1-g^{-1}(1-p)))=(p, q(\hat p))=(p, \hat q(p))$" -3778,$\mathsf P(X\le q_X(p))=p$ -3779,"$\rho_1,\rho_2$" -3780,$P/(A-P)=P/Q$ -3781,$-\rho$ -3782,$\alpha_1(98)=0.1$ -3783,$\pi=1.2613$ -3784,$\gamma=0.421$ -3785,$8+11.1667=19.167$ -3786,$\mathsf{Pr}(Y_m > y) = 1 - (1 - \mathsf{Pr}(X > y))^n$ -3787,$\beta_i(x)/\alpha_i(x) < g(S(x))/S(x)$ -3788,$h(p)=s^3$ -3789,$\psi$ -3790,$\mathsf{VaR}_p(X)=\mu + \sigma \Phi^{-1}(p)$ -3791,$B_k$ -3792,$\bar P(\infty)=\mathsf{E}[q(U)\phi(U)]$ -3793,"$Binomial(s,N)$" -3794,$x=S^{-1}(g^{-1}(s))$ -3795,$e^{-rt}$ -3796,$\mathsf{VaR}_{p^*}$ -3797,$\mathsf{E}[X^2]$ -3798,$=\mathrm{MV}(y-T(X))^+$ -3799,$\mathsf{E}[YZ_\epsilon]\to\mathsf{E}[YZ]$ -3800,$p^{* }$ -3801,$\beta_Q=(a/Q)\beta_A + (P/Q)\beta_L$ -3802,$r\times n$ -3803,$F(2)=0.75$ -3804,$(80-11)\times 0.25$ -3805,"$S, S^{-1}$" -3806,$\mathsf{Q}'$ -3807,$q(0.1)=1$ -3808,"$k=0,1,\dots,n-1$" -3809,$q(1-g^{-1}(1-p))$ -3810,$\tilde X_j$ -3811,$\bar F$ -3812,$\pm\infty$ -3813,"$c\in[0,1]$" -3814,$dg$ -3815,$\rho_c(Y)=\mathsf{E}[Y]$ -3816,$p_Y<0.5$ -3817,"$\mathsf{E}[W]=\sum_{d\ge 0} \mathsf{E}[Y_{-d,d}]$" -3818,$\mathscr{P}$ -3819,"$(\mu,\sigma)$" -3820,"$(brR15 |- lee.south)+(-0.25,-0.25)$" -3821,$\pi=1.2497$ -3822,"$\mathsf{E}[(X-a)^+]= p\,\mathsf{E} X$" -3823,$\bar\iota$ -3824,$L(X)=(X-\mathsf{E} X)/\mathsf{SD}(X)$ -3825,$g(s) \ge 1$ -3826,$v(A\cup B) + v(A\cap B)\ge v(A) + v(B)$ -3827,$\bar P_\tau(a)=\bar P(a) + \tau(a-\bar P_\tau(a))$ -3828,$\nu>0$ -3829,$\mathsf{E}[X\mid\mathcal F_0]=\mathsf{E}[X]$ -3830,$P_g\{X=M\}=g(0+)>0$ -3831,$\mathsf{E}[X_i\mid X = x_p]$ -3832,$\Delta=a'-a$ -3833,$\alpha_i(x)S(x)$ -3834,$r_h-\mu_L=r-r_L$ -3835,$-0.0012$ -3836,$\rho(X)$ -3837,$\mathsf Q$ -3838,$+1$ -3839,$\implies\mathsf{FATOU}$ -3840,$\bar P(a)>\mathsf{E}[X\wedge a]$ -3841,$1-w$ -3842,$=1/(1-p)$ -3843,$Q_j = 1 - g(S_j)$ -3844,$A(X)$ -3845,$\mathsf Q(\omega)=Z(\omega)\mathsf{Pr}(\omega)$ -3846,$X\ge \mathsf{VaR}_p(X)$ -3847,$p_+-p_-$ -3848,$\mu(\{0\})=\phi(0)=g'(1)$ -3849,"$s\in (0,1]$" -3850,"$p\in (0,1)$" -3851,"$\lambda, \iota, \psi$" -3852,$h_{xx}$ -3853,$u=x$ -3854,$af + a(1-f)/q$ -3855,$\bar P_{act}$ -3856,$L_d^{d+l}(X)$ -3857,$r=0.06$ -3858,$x\mapsto x^{3/2}$ -3859,$L_0^{500}(x)$ -3860,"$A_3,B_3$" -3861,$\mathbf {\beta_{1}g(S)\Delta X}$ -3862,"$B_1=[0,0]$" -3863,$f(x)=x^2$ -3864,$E_\mathsf{Q}[1]$ -3865,$\mathsf{E}[(X-\mu)^2]$ -3866,$s\to 0$ -3867,$g_0=0$ -3868,"$\displaystyle\int_0^1 q(p)\,dp$" -3869,$P(X_{-1}(a_{gc}))=9094.25$ -3870,$\phi$ -3871,$S;g(S)$ -3872,$\kappa_{i^*}$ -3873,$\rho_a$ -3874,"$\bar P_{0,2}$" -3875,$x=e^{\mu + y\sigma}$ -3876,$f(w|s)$ -3877,$\mu_U = 15$ -3878,"$Y_{2,0}$" -3879,$r_{pq}:=\sqrt{p(1-p)}$ -3880,$A\subset B$ -3881,$g(s)=s^b$ -3882,$S_j=S_{j-1}-p_j$ -3883,$\{X = q_X(p) \}$ -3884,$X(\omega)=\omega$ -3885,$1_A(x)=1$ -3886,$g(s)=100s \wedge 1$ -3887,$(0.333...)(0.15)$ -3888,$\bar P_1$ -3889,$v(A\cup B)\le v(A)+v(B)$ -3890,$V(1)$ -3891,$x^2$ -3892,$a_{0}=a(Y_{0})$ -3893,$C=1-H$ -3894,$Y = NX$ -3895,$a_l < b_l$ -3896,"$1,9,10$" -3897,$g_0 \le 1-\alpha$ -3898,$(1+r)Z=\mathsf Q/\mathsf{P}$ -3899,"$\{0, 9, 10\}$" -3900,$\iota^*$ -3901,"$t=0,1,2,\dots$" -3902,$0.354 \cdot 8 = 2.83$ -3903,$x=z$ -3904,$(x+(X-x)^+)^n\not=x^n+((X-x)^+)^n$ -3905,$f(R) = \mathsf{E}[f(X)]$ -3906,$\delta_i$ -3907,$1-2/3=1/3$ -3908,$\mathsf{Amb}(X)$ -3909,$\rho^{ho}_c$ -3910,$X \le Y$ -3911,"$\mathsf{cov}(X,M)=\mathsf{cov}(X_i,M)$" -3912,$\mathbf {X'}$ -3913,$F(X(\omega))$ -3914,$\mathsf{VaR}_{p}( \cdot \mid \mathcal F_t)$ -3915,$d$ -3916,$\mathsf{E}[X\wedge a]= 2.4982$ -3917,$\mathsf{E} X + \inf_x \{\alpha_1\mathsf{E}[(x-X)^+] + \alpha_2\mathsf{E}[(X-x)^+] \}$ -3918,$M:=\esssup X$ -3919,$g(s)=1-\sqrt{1-s}$ -3920,$dF=-d(g\circ S)=$ -3921,$E_\mathsf{Q}[X_i]$ -3922,"$\mathsf{TVaR}_0,\mathsf{TVaR}_1$" -3923,$f_x$ -3924,$s^{-1}(\cdot)$ -3925,$X_{1c}$ -3926,"$(p, \mathsf{E}[X_i\mid X=q(1-g^{-1}(1-p))])$" -3927,$\mathsf{TVaR}_p(X)=\mathsf{TCE}_p(X)=\mathsf{E}[X\mid X \ge \mathsf{VaR}_p(X)]$ -3928,"$\omega\in [0,1]$" -3929,$\mathsf{PH}$ -3930,$N=X-L_{r_a}^{r_a+r_l}(X)$ -3931,$x_i-x_{i-1}=dx$ -3932,$P(\hat s)=\mathsf{E}[\hat s]=s$ -3933,$X_i(1)$ -3934,$0<\alpha_1<\alpha_2<1$ -3935,$v(B)$ -3936,$(dX_t)^2$ -3937,$L^2$ -3938,$g(s)=\Phi(Z(s)+\lambda)$ -3939,$|Y_n|\le 1$ -3940,$a=0$ -3941,$0.25 + U/4$ -3942,$F(q^-(p))\ge p$ -3943,"$[0,a)$" -3944,$0.909+0.273=1.182$ -3945,$\bar P=\bar S+\bar M$ -3946,$\rho(X)=\rho(X\wedge a) + \rho((X-a)^+)$ -3947,$X\wedge a=30$ -3948,$F$ -3949,"$(2,1)$" -3950,$X(\mathbf{v})=\sum_i X_i(v_i)$ -3951,$\Phi^{-1}(0)=-\infty$ -3952,$2/6$ -3953,$F_M$ -3954,$q_C(p)=\inf C$ -3955,$g(S)dX$ -3956,$\mu \cdot T_k$ -3957,$\rho:\mathcal{S}\to \mathbb{R}$ -3958,$F_I^{n*}$ -3959,$Q(x)$ -3960,$\mathsf{E}[X] + \pi\mathsf{E}[(X-\mathsf{E} X)^+]$ -3961,$X_1(v_1)$ -3962,"$g\in D_n^*=\{ g \mid (-1)^{k+1} g^{(k)} \ge 0, k=1,\dots,n-1, (-1)^n g^{(n-1)}\text{ non-increasing} \}$" -3963,$\mathbf {\mathsf{E}[X_i(a)]}$ -3964,$\bar Q_{1}$ -3965,$h=1+\lambda(f-\mathsf{E} f)$ -3966,$\Delta Q(a)$ -3967,$t\to 0$ -3968,"$i=1,2$" -3969,$\sigma=2.15$ -3970,$\mathsf Q\in \mathcal Q$ -3971,$\alpha \ge s_0 g'(s_0)/g(s_0)$ -3972,$g(s)=s^r$ -3973,"$t=1,2,...,\tau$" -3974,$Y(\omega)$ -3975,$Sdx$ -3976,$s=1/4$ -3977,$\int_0^\infty xg'(S(x))dF(x)=\int_0^\infty g(S(x))dx$ -3978,$\mathsf{E}[Xe^{\pi Z}]/\mathsf{E}[e^{\pi Z}]$ -3979,$X_2' = X_2+\cdots +X_n$ -3980,$1_{U>0.95}$ -3981,$g=u^2=0.01$ -3982,$100$ -3983,$X\wedge a \le X$ -3984,"$Y_{t,0}$" -3985,$s>s^\ast$ -3986,$g(s)-\hat g(s)$ -3987,$R:=\bar P_{act}-\bar S$ -3988,$Var[T]=s(1-s)/N$ -3989,$\sum w_i=1$ -3990,$\alpha_i(x) =\mathsf{E}[X_i/X\mid X>x]$ -3991,"$\{x_1,...,x_n\mid X < \max(X)-\epsilon\}$" -3992,$z=x$ -3993,$F_n(x)\to F(x)$ -3994,$c=2.5$ -3995,"$\rho(1000, 3000, 3500)$" -3996,$w(x)=e^{kx}$ -3997,$\mathsf{Pr}(X>0)$ -3998,$1_\omega(\omega')=1$ -3999,$g(s)=cs$ -4000,$f(t|s)$ -4001,$\mathbf {F}$ -4002,$\mathsf{E}[(a-X)^+]$ -4003,$\displaystyle\int_0^\infty u(x)dF_X(x)$ -4004,$\Lambda\dfrac{\mu_{U}}{\sigma_U} = \dfrac{E( r_{U} ) - r_{f}}{\sigma_{r_{U}}} \left(\dfrac{\mu_{U}}{\sigma_{U}}\right)$ -4005,$f'(x_0)$ -4006,$\mathsf{E}[X\mid X=x]\equiv x$ -4007,$y^{\ast}-x^{\ast} \ge \epsilon$ -4008,$(1+\rho)\mathsf{E}[C]$ -4009,$1-g$ -4010,$g(S(x_{i+1}-))-g(S(x_{i}))$ -4011,"$d\,F(X)$" -4012,$Q(a)$ -4013,$a_i=\mathsf{E}_\mathsf{Q}[X_i]$ -4014,$(1+c)\mu$ -4015,$\mathbf {\Delta(X\wedge a)}$ -4016,$\mathsf{E}[YZ]$ -4017,$\mathsf{VaR}_{0.99}$ -4018,$dG(x)=g'(S(x))dF(x)$ -4019,$n=100$ -4020,$\rho(c) = c$ -4021,$\delta_p$ -4022,$\sigma$ -4023,"$(s(t),m(t))$" -4024,$X>x$ -4025,$\mathsf{E}[X_i(a)]$ -4026,$\sigma_U = 1$ -4027,"$(p,q(p))=(1-S(x),x)$" -4028,$w_i\ge 0$ -4029,$\int X_n\to 0$ -4030,"$r_f\ge 0, r>0$" -4031,$Z$ -4032,"$X_1,X_2$" -4033,$r_L$ -4034,$\gamma=\mathsf{Pr}(X>\mathsf{E}[X])$ -4035,"$\Omega=[0,1]\times [0,1]$" -4036,$x_i$ -4037,$a(X)$ -4038,$g(s)+g'(s)(1-s)\ge 1$ -4039,$\mathbf {\alpha_1S\Delta X}$ -4040,"$\mathsf{cov}(X_i,\sum_j X_j)=\mathsf{cov}(X_i,X_i)=\mathsf{Var}(X_i)>0$" -4041,$\mathsf Q(A)=0$ -4042,$0-\rho(-H)$ -4064,"$Y_{0,1}$" -4065,$\mathsf{E}[X_ig'(S(X))]$ -4066,$a(X;X)=\rho(X)=\sum_i a_i$ -4067,$\displaystyle\int_0^\infty xg'(S(x))f(x)dx$ -4068,$A(X)\not= B(X)$ -4069,$\lim_{y\uparrow x} f(y)$ -4070,$\rho(X)= \mathsf{E}_{\mathsf{Q}_X}[X]$ -4071,$\psi^{-1}(p)$ -4072,$\mathcal Q\subset\mathcal M(\mathsf P)$ -4073,"$a(x_1,\dots,x_n):=a(X(x_1,\dots,x_n))$" -4074,$1-q$ -4075,$ds(t)/dt$ -4076,$X_{-3}=C'_1 + \cdots + C'_n$ -4077,$g(S(M-))/S(M-)$ -4078,$\mathsf{VaR}_{p_0}(X)=\sup X$ -4079,$p=1/2$ -4080,$y\not\in C$ -4081,$X_0=C_1 + \cdots + C_N$ -4082,"$2^0, 2^2, 2^4, ...$" -4083,$F_I$ -4084,$gdX$ -4085,$b_l \le 1 \le b_h=2-b_l$ -4086,$30+10t$ -4087,$m_1$ -4088,"$Y_{t,d}$" -4089,$F(x)=\sup\{ p\mid q(p) < x \}$ -4090,"$X_{i,j} \leftarrow \kappa_{i}(X_j)$" -4091,$1/g'(0)$ -4092,$1-g(1-p)$ -4093,$d\Pi = (r_h-\mu_L)\Pi dt$ -4094,$q(U_X) = m$ -4095,$\alpha_i(t)$ -4096,$\mathbf {gS}$ -4097,$U=X+Y$ -4098,$p^\ast$ -4099,"$0,0,0,1,2,5,8,12,23,40$" -4100,$0\le k < 2^m$ -4101,"$c=1,2,3$" -4102,$E[s]=0.1160$ -4103,"$\lambda([a,b]) = b-a$" -4104,$p^+$ -4105,$S_X(t)=S_{X\wedge a}(t)$ -4106,$h(X)=X$ -4107,$D_1\supset D_2\supset \cdots \supset D_\infty$ -4108,$g''(s)=-\phi'(1-s)\le 0$ -4109,$\prec_1^*$ -4110,$X=100$ -4111,$\mathsf{WCE}_p(X) = \mathsf{TVaR}_p(X)$ -4112,$X\wedge a(X)$ -4113,$\times$ -4114,$\bar M(a)$ -4115,$\mathsf{LI}$ -4116,$(p_0 < p^\ast < p_1)$ -4117,"$c = 0.5,1.0,\dots,2.5$" -4118,$\sup X_n=1\not=\sup X=0$ -4119,$IL$ -4120,$S(x)\leftrightarrow g(S(x))$ -4121,$\rho(X) = \mathsf{E}[X] + c\mathsf{E}[X-\mathsf{E}[X]]^+$ -4122,$\lambda=\sum_i \lambda_i$ -4123,$\mathsf{TVaR}_{0.8}$ -4124,$Q = M/\iota$ -4125,$\mathsf{Pr}(X>\mathsf{VaR}_p(X))=1-p$ -4126,$(a_i)_i$ -4127,$g(s)=d+sv$ -4128,$p\nu_p$ -4129,$f_i$ -4130,$\mathsf{P}(X=X_j)=\Delta S_j:=S(X_{j-1})-S(X_j)$ -4131,$P\approx \mathsf{E}[A(1)] + k\mathsf{Var}(A(1))/2$ -4132,"$X_{0,2}$" -4133,$aq_X(p) \}$ -4159,$X_i=\mathsf{E}[X_i\mid X]$ -4160,$S(x)>>0$ -4161,$q_B \le q_C$ -4162,$\mathsf{TVaR}_{0.75}$ -4163,$g'(s) < \infty$ -4164,$\hat p$ -4165,$\kappa_i(q(1-g^{-1}(1-\tilde p)))$ -4166,$q^-(p)$ -4167,$\rho(X-\rho(X))=\rho(X)-\rho(X)=0$ -4168,$g_0$ -4169,$\mathsf{TVaR}_p(X)=\mathsf{E}[X\mid X >\mathsf{VaR}_p(X)]=\sum_i\mathsf{E}[X_i\mid X>\mathsf{VaR}_p(X)]$ -4170,$dt\to 0$ -4171,$\{X\in L^\infty \mid \rho(X)\le c \}$ -4172,"$Y_{2,2}$" -4173,"$c_i=\displaystyle\int_0^1\dfrac{\partial c}{\partial x_i}(tx)\,dt$" -4174,$\rho(X_{-1}\wedge a_{ro})={{mvp_ro}}$ -4175,$\bar \iota = \dfrac{\bar M(a)}{\bar Q(a)}$ -4176,$\mathcal{N}_{X\wedge a}(X_i(a))$ -4177,$f'>0$ -4178,"$\bar M_{t,0}$" -4179,$E$ -4180,$p^\ast = 0.48732$ -4181,$r_P$ -4182,$\mathbf {t+1}$ -4183,$S=g(S)=1$ -4184,$\mu_d = (6-d)^2$ -4185,$g(s)=0.9s + 0.1$ -4186,$\left( g(S(x_{(j)}))-g(S(x_{(j-1)})) \right) / ( x_{(j)}-x_{(j-1)} )$ -4187,$t^\star$ -4188,$1_Z$ -4189,$\omega < p^-$ -4190,$q = s$ -4191,$\bar F(a):=\int_0^a F(x)dx=a-\mathsf{E}[X\wedge a]$ -4192,"$s\in (0,1)$" -4193,$\mathsf{E}[X] + \pi\mathsf{E}[((X-\mathsf{E}[X])^+)^2]^{1/2}$ -4194,"$\omega\in [0,0.1)\cup [0.25, 0.35) \cup [0.5, 0.6) \cup [0.75, 0.85)$" -4195,$80-11=69$ -4196,$g'$ -4197,$\rho(X)+c$ -4198,$S(x)=(1+x)^{-\alpha}$ -4199,$r_M$ -4200,$U(2)=0$ -4201,$\alpha_i(x)$ -4202,$\sup X\le \sup Y$ -4203,$\sigma(X)=\mathsf{E}[(X-\mathsf{E} X)^2]^{1/2}$ -4204,$S(x)=\Phi((-x+\mu)/\sigma)$ -4205,$\tilde X_1 + \tilde X_2 \succeq^2 \tilde X_1$ -4206,$p=F(a)=1-S(a)$ -4207,$v\mathrm{EL}+da\ge \mathrm{EL}$ -4208,$X=X_s + X_c$ -4209,"$\mathsf{VaR}_{0.995}=64,861$" -4210,$P = 3.1035$ -4211,$x=q(1-g^{-1}(1-p)))$ -4212,"$d=1,2,\dots$" -4213,"$(\x*1.2, 2)$" -4214,$h=1$ -4215,"$k_1, k_2$" -4216,$p=0.95$ -4217,"$s^{\ast}=1/2, \lambda^{\ast}=0$" -4218,$\esssup(X)=1$ -4219,$1-p \ge g^{-1}(1-p) \implies 1-g^{-1}(1-p) \ge p \implies q(1-g^{-1}(1-p))>q(p)$ -4220,"$x+y\wedge aX =\min(x+y,aX)$" -4221,$H(X)q(p) \}$ -4234,"$g'>0, g''<0$" -4235,$Y=c\in \mathbb R$ -4236,$h(u)=1$ -4237,$\lim_{\epsilon \downarrow 0} (f(x+\epsilon)-f(x))/\epsilon$ -4238,$\omega_1$ -4239,$r>0$ -4240,"$\alpha_i(\mathbf{v}, x)$" -4241,$\omega\ge 0.4$ -4242,$\mathsf{Pr}(B)=0$ -4243,$\bar q_{X_1+X_2}(s) \le 2\bar q(s)$ -4244,$\mathsf{E} X +\lambda_1 {(X-\lambda_2 \mathsf{E} X)^+}_1$ -4245,$f(t)=\rho(tX)$ -4246,$X_n\uparrow 1$ -4247,$\int S(x)dx$ -4248,$A\subset \Omega$ -4249,$(A-L)^+$ -4250,$P(x)/Q(x)$ -4251,$\mathsf{Pr}(X=x_i)=\mathsf{Pr}(X>x_{i-1})-\mathsf{Pr}(X>x_i)=S(x_{i-1})-S(x_i)$ -4252,$\mathsf{E}[WX] \le \rho(X)$ -4253,$r_U \Delta A - \Delta P$ -4254,"$\bar Q_{0,0}$" -4255,$s_0=0$ -4256,$g(S_{\mathsf{j}(a)})=0.5$ -4257,$-g''(s)=\alpha(\alpha-1)s^{\alpha-2}$ -4258,$\bar Q(a) =a-\bar P_g(a)$ -4259,$\exp(a)$ -4260,$s\mapsto g(s)$ -4261,$\alpha X$ -4262,$\mathsf{E}[XM]$ -4263,$c(S)\le c(T)$ -4264,$(1-\lambda)(1+\gamma)$ -4265,$\mathsf{E}[X] = \displaystyle\int_\Omega X(\omega)\mathsf{Pr}(d\omega)$ -4266,$1-\beta_i(t)g(S(t))$ -4267,$\mathsf{Pr}(X>x)$ -4268,$\mathsf{E}[X\mid X>2000]-2000=\mathsf{TVaR}_{F(2000)}(X)-2000=624$ -4269,"$(p, \mathsf{E}[X_i\mid X=q(p)])$" -4270,$L_a^{a+da}$ -4271,"$a_{0,t}' := a_{0,t-1}-X_{0,t}$" -4272,$-Y$ -4273,$P = \mathsf{E}[X] + \pi \mathsf{Var}(X)$ -4274,$\rho(X)=\mathsf{E}_\mathsf{Q}[X]-\alpha(\mathsf Q)$ -4275,$W_{t}$ -4276,$2^n$ -4277,$(1 - \nu F(a))$ -4278,$<$ -4279,$g'\left (S_X(X)\right )$ -4280,"$X_1=(0,0,0,0,0,0,2,4,8,0)$" -4281,$R_L=R_f + \beta_L(R_M-R_f)$ -4282,$cv=0.137$ -4283,$\mathsf{E}[(X-x)^+]$ -4284,$\mathsf{Pr}(X>a)$ -4285,"$(2,2)$" -4286,$1/x$ -4287,$A(1_{X_1>x_1}+1_{X_2>x_2}) \le A(1_{X_1>x_1}) + A(1_{X_2>x_2})$ -4288,$\Delta=\Phi(d^*)$ -4289,$F_g(x) = 1- g(S_X(x))$ -4290,$Z(1000)=(1-0)/(0.1-0)=10$ -4291,$\tilde X_1=X_1 + \mathsf{E}[X_2\mid X_1]$ -4292,$\tau=1$ -4293,$\rho(X_1) \ge D\rho_X(X_1)$ -4294,"$\mathcal Q =\{ \mathsf Q \mid \mathsf Q\ll \mathsf P,\ \alpha(\mathsf Q)=0 \}$" -4295,"$X_j=\sum_i X_{i,j}$" -4296,$\mathsf{SD}$ -4297,$n>1$ -4298,$-\phi(d^*)<0$ -4299,$\rho(\tilde X+X)=\rho(\tilde X)+\rho(X)$ -4300,$a(\mathbf{v})$ -4301,$P(dx)$ -4302,$\mathsf Q(X>a)/P_X(X>a)=g(S(a))/S(a)$ -4303,$\rho(X-P)=\rho(X)-P$ -4304,$a+da$ -4305,$r_pq$ -4306,"$m\ge 1, n\ge 0$" -4307,$(1-g)$ -4308,$x=2$ -4309,$T_k$ -4310,$X\le c$ -4311,$X \succeq Y$ -4312,$t$ -4313,$x=a$ -4314,"$\mathsf{biTVaR}_{p_0,p_1}^w(X)=\bar P$" -4315,$\log(X)$ -4316,$\mathsf{E}$ -4317,$\mathsf{P}(X=X_j)$ -4318,"$[0, 1-p)$" -4319,$\mathsf{VaR}_{0.95}(X)$ -4320,$S_t=a_0 + (1+c)\mu t - X_t$ -4321,$1/(1+r) = 0.893$ -4322,$D^n\rho_X(X_i)$ -4323,$A\subset \mathbb{R}$ -4324,$\mathsf{E}[X_1]=\mathsf{E}[Y_{0}]$ -4325,$\bar P_t = \rho(Y_{t})$ -4326,$W_1$ -4327,$s/(1-s)$ -4328,$E_1$ -4329,"$f:[0,1]\to[0,1]$" -4330,$A\cap B$ -4331,"$(p,q(1-g^{-1}(1-p)))=(1-g(S(x)),x)$" -4332,$X_1=0$ -4333,"$\beta = \mathsf{cov}[r,r_M]/ \sigma^2_{r_M}$" -4334,"$f(x, \cdot)\in L_p(\Omega, \mathcal{F}, \mathcal{P})$" -4335,$\bar P(a)=\rho_g(L_0^a(X))$ -4336,$S_t=\exp(\mu t + \sigma W_t)$ -4337,$P_i(x)=\beta_i(x)g(S(x))$ -4338,$1-L/P = (P-L)/P$ -4339,"$x_{1,2}$" -4340,$\mathsf{Q}\in\mathcal Q$ -4341,$w/(1-w)$ -4342,$\sup_\Omega |X_n - X| \to 0$ -4343,$\beta_i(a) g(S(a))$ -4344,$\prec_n^*$ -4345,$2^{-t}$ -4346,$\mathsf{Pr}(p(\omega)=0)=0$ -4347,$X_2/X$ -4348,$\Delta X=80-11=69$ -4349,$g(S(x))=\exp(-\alpha H(x))$ -4350,$\rho(\lambda X + (1-\lambda)\rho(X))$ -4351,$X\le Y+\Vert X-Y\Vert$ -4352,$(1-p)/(p\nu(p)^2)$ -4353,$\mathsf{E}[X_2\mid X_1]$ -4354,$I(F(x) < p)=\begin{cases} 1 & F(x)< p \\ 0 & F(x)\ge p\end{cases}$ -4355,$g(s)=1$ -4356,$\mathsf{Pr}(X = q(p)) > 0$ -4357,$x < x^\ast$ -4358,$X\wedge a(X)\le Y\wedge a(Y)$ -4359,$t\uparrow 0$ -4360,"$\eta_{p,\alpha_1}(X) < \eta_{p,\alpha_2}(X)$" -4361,$\mathsf{Pr}(\{\omega \mid X_n(\omega)\to X(\omega) \})=1$ -4362,$a_{\min}$ -4363,$\phi(t)=\int_0^t (1-p)^{-1}\mu(dp)$ -4364,$\mathsf{E}_\mathsf{Q_r}[X_j]$ -4365,$\mathsf{EPD}_s(X)$ -4366,$\mathsf{E} X + c{X-\mathsf{E} X}_p$ -4367,$x\ge a$ -4368,$N=r_a$ -4369,$\int S(x)dx = \int xdF(x)$ -4370,$\mathsf{VaR}_{0.7}(X)=$ -4371,$\mathsf P(X=\sup(X))>0$ -4372,$L(X)=(1-p)^{-1}1_{X\ge x_p}(X)$ -4373,$\mu = w \delta_{\alpha_1} + (1-w) \delta_{\alpha_2}$ -4374,$A(-X)$ -4375,$\mathsf{j}(90)=6$ -4376,$a - P$ -4377,$q^-_X(0.95)$ -4378,$1100 \le x \le 1250$ -4379,$\sigma\sqrt{t}$ -4380,$(L-A)^+$ -4381,"$\int Zd\mathsf P = \int d\mathsf Q/d\mathsf P\, d\mathsf P = \int d\mathsf Q =1$" -4382,$\mathcal A$ -4383,$d\mathsf Q/d\mathsf P$ -4384,$\rho(X) = \mathsf{E}_{\mathsf{Q}}[X] = \mathsf{E}_{\mathsf{Q}}[X\wedge a + (X-a)^+] = \mathsf{E}_{\mathsf{Q}}[X\wedge a] + \mathsf{E}_{\mathsf{Q}}[(X-a)^+] \le \rho(X\wedge a) + \rho((X-a)^+) = \rho(X)$ -4385,$t_f$ -4386,$\hat q(p)$ -4387,$\mathbf {a_{2}}'$ -4388,$p<1$ -4389,$r < n$ -4390,$\mathcal S(X)=\mathsf{VaR}_p(X)$ -4391,"$(\omega',\omega'')$" -4392,"$u\in[0, 1-p]$" -4393,$k_i=a_i/v_i$ -4394,"$(s_j, g_j)$" -4395,"$(3,3)$" -4396,$T(p)$ -4397,$D/C$ -4398,"$s\in[0, 1-p]$" -4399,$p=1$ -4400,$a_{d}' = a_{d-1}-X_{d}$ -4401,$g'\left (S(X)\right )$ -4402,$p=0.75$ -4403,$\mathbf {X_{1}/X}$ -4404,$\bar M(a) = \bar P(a) - \mathsf{E}[X\wedge a]$ -4405,$\{\omega\mid X(\omega) = x_1\}$ -4406,$\tau a_i$ -4407,$d\downarrow 0$ -4408,$p\ge 1$ -4409,"$g(s)=\min(s/(1-p),1)$" -4410,$\mathit{ROE}(s) = r_f + Ck(s)$ -4411,$\phi'(p)\ge 0$ -4412,$q(p)\phi(p)$ -4413,$\mu_0=\mu_1$ -4414,$w_l=1-c\gamma$ -4415,$0.7 \le p < 0.8$ -4416,$\int_\Omega X(\omega)\mathsf \mathsf{Pr}(d\omega)$ -4417,"$j=1,\dots, n$" -4418,$x=q_X(1-s)=\mathsf{VaR}_{1-s}(X)$ -4419,$\{p \ge p_-\}$ -4420,$(1-p)^{-1}$ -4421,$\alpha_i(x) = \mathsf{E}[X_i /X \mid X> x]\not=\mathsf{E}[X_i\mid X> x]/\mathsf{E}[X\mid X>x]$ -4422,$\omega<1/n$ -4423,$\tilde X = (x_{ij})$ -4424,$\mathbf {\mathsf{VaR}_p(X_1+X_2)}$ -4425,$q(p)=c$ -4426,$a(X_i;X)\le \rho(X_i)$ -4427,$\rho(0) \ge 0$ -4428,$g'(s-)\ge 0$ -4429,$\mathbf {\max a}$ -4430,$\exists$ -4431,$^1$ -4432,$x^{-\alpha}$ -4433,$k>1$ -4434,$D\rho_X(X_2)=45.1801$ -4435,$\mathsf{E}_\mathsf{Q_2}[X_j]$ -4436,$\mathsf{E}[X_i (X\wedge a)/X]$ -4437,$V(2)$ -4438,"$\rho_g(X)=\int_0^\infty g(S(x))\,dx$" -4439,$c(1)$ -4440,$\mathsf{E}[(X-x_l)^+]$ -4441,$X=X_{-1}+X_{0}$ -4442,$\mathsf{E}[Z_A]=1$ -4443,$xf(x)dx$ -4444,$t=0.06405$ -4445,$Y_{0}=\sum_{d>0} X_{d}$ -4446,$a=Q+P$ -4447,$Y\preceq Z$ -4448,"$a_{0,0}:=a(Y_{0,0})$" -4449,$X_1+X_2\sim 2X$ -4450,$l$ -4451,$r_h=r+\pi$ -4452,$\Delta Q_{ro}(a)$ -4453,$\bar Q_{0}$ -4454,$=1-\nu F(a)$ -4455,$X \le 0$ -4456,$X^{-1}(A)\in\mathcal F$ -4457,$\sup(X\wedge a)=a$ -4458,$\mathbf{P_i} \in \mathbb{R}^2$ -4459,$\mathsf{E}[\kappa_i(X)g'(S(X))]$ -4460,$\mathsf{E}[X_i(x)]$ -4461,$\bar\nu=1/(1+\bar\iota)$ -4462,$\rho(X/n)=\rho(n(X/n))/n=\rho(X)/n$ -4463,$1_{U_X\ge p}=0$ -4464,$\mathsf x\mathsf{TVaR}$ -4465,$F(X) - F_X(X-)=0$ -4466,$\tilde X$ -4467,$m'(0) = (m_1-m_0)/s_1$ -4468,$B \in\mathcal B_p$ -4469,$1/16$ -4470,$\tilde X_1 = X_1 + \mathsf{E}[X_2]$ -4471,$\bar\iota(a)=\bar\iota$ -4472,$\mathsf{TVaR}_p(X)-\mathsf{E}[X]$ -4473,$\mathsf P(X=X(\omega_0))>0$ -4474,$X=X_i + (X-X_i)$ -4475,$X(\omega)\mathsf{Pr}(\omega)$ -4476,$\rho(X_1+X_2) \le \rho(X_1)+\rho(X_2)$ -4477,$(0)$ -4478,$\mu_i$ -4479,$\mathsf{E}[kX]$ -4480,$\mathsf{VaR}_1=\esssup$ -4481,$#4$ -4482,$v=x$ -4483,$\phi(p)=g'(1-p)\ge 0$ -4484,"$(0,0)$" -4485,$s_2$ -4486,$\mathbf {a_1'}$ -4487,$F(x) < p \iff q^-(p) > x$ -4488,"$\mathbf {g(S)\,\Delta X}$" -4489,"$g(S)\,\Delta X$" -4490,$E'$ -4491,$\delta+\nu$ -4492,$\mathsf P(X\ge x_p)=1-p$ -4493,$\mu=7.8044$ -4494,$a_{2}'$ -4495,$p>0$ -4496,$z$ -4497,$\mathsf{j}(91)=7$ -4498,$\zeta_s = 8$ -4499,$ag(0+)$ -4500,$\rho-\iota g>0$ -4501,$R = P-L$ -4502,$\mathrm{sgn}(z)|z|^{1/(q-1)}/\|z\|_p^{q/p}$ -4503,$o(dt)$ -4504,$q^-$ -4505,$A_4 = [0; \epsilon_1 + \epsilon_2]$ -4506,$\mathsf{E}[q(U_X)1_{U_X\ge p}]$ -4507,$\mathsf{Var}(Y) \ge \mathsf{Var}(X)$ -4508,$0\le \omega\le 1$ -4509,$q(p)=e^{\mu+z_p\sigma}$ -4510,"$[f'_-(x_0), f'_+(x_0)]$" -4511,$p/\mathsf{E}[p]=p(1+r_f)$ -4512,$a(\cdot)$ -4513,$\mathsf{E}[X_i \mid X]$ -4514,$L_p$ -4515,"$X\ge 0,(\tilde X-X)\ge 0$" -4516,$\rho(\lambda X)$ -4517,$\mathbf {j}$ -4518,$Pr(X_{-1} > a)$ -4519,$X\wedge a / X$ -4520,$\tau=-1$ -4521,$\mathsf{TVaR}_{p^*}$ -4522,$X\ge x_p$ -4523,$A(1_{U>0.95})=A(1_{U\le 0.05})=g(0.05)=0.3017$ -4524,$\mathsf{TVaR}_{0.9}$ -4525,$2.576\sigma_d$ -4526,$\mathbf {Z_7}$ -4527,$\mathsf{TVaR}_1$ -4528,$\mathsf{E}[X\mid X>x]/\mathsf{Pr}(X>x)$ -4529,$X_1=\mathsf{E}[X\mid \mathcal F_1]$ -4530,$\bar P_n$ -4531,$\mathit{MV}_{gc}(a_{gc})=a_{gc}-\rho(X\wedge a_{gc})={{mv_gc}}$ -4532,$\mathsf{Pr}(X_n\in A)=1$ -4533,$1\wedge \cdot$ -4534,"$g(s)= \displaystyle\int_0^s \phi(1-p)dp = \min(s/(1-\alpha), 1)$" -4535,$\ll$ -4536,$0\le \alpha \le 1$ -4537,$\bar P=\mathsf{E}[W]+\lambda\sigma(W)$ -4538,$j=8$ -4539,$\rho(X)=\mathsf{E}[Xe^{kX}]/\mathsf{E}[e^{kX}]$ -4540,$\int_{\mathsf{E}[X]}^\infty (x-\mathsf{E}[X])^2 f(x)dx$ -4541,$S_0=1-p_0$ -4542,$\mathsf{E}[\cdot]$ -4543,$\mathbf {Z_\mathit{lift}}$ -4544,$g_{ROC}$ -4545,$\rho_1(X)$ -4546,$f(s) \ge s$ -4547,$Q(a)=h(F(a))$ -4548,$P = \mathsf{E}[X] + \pi \mathsf{E}[((X-\mathsf{E}[X])^+)^p]^{1/p}$ -4549,$\mathbf {a=1}$ -4550,"$\bar P_i(v_1, v_2, a) / v_i$" -4551,$q_C\le q_A$ -4552,$. Thus $ -4553,$k\mapsto k\rho(-X)$ -4554,$\mathsf{TVaR}_1=\sup$ -4555,$\lambda = \dfrac{E( r_{M} ) - r_{f}}{\sigma_{rM}}$ -4556,$g'(1)<1$ -4557,$u'''' \le 0$ -4558,$\mathbf {g_3(s)=s^{0.7}}$ -4559,$-X_i$ -4560,$ROE=(g-s)/(1-g)=m/(1-s-m)$ -4561,$X > a$ -4562,"$f(0,0)=0$" -4563,$\mathsf{Var}$ -4564,$l(kX)\le\rho(kX)$ -4565,$\lambda \ge 0$ -4566,"$0, 1/p$" -4567,$X\ge m$ -4568,$E(X_{0}(a))$ -4569,"$(0,1)$" -4570,$i=1\dots N$ -4571,$-(1-s)g''(1-s) + g(0+)\delta_1 + \sum_s s\Delta_s \delta_{1-s} + g'(1)\delta_0$ -4572,"$\mathsf{E}[X_{t,d}\mid \mathcal F_0]=\mathsf{E}[X_{t_d}]$" -4573,$\phi(0)=\mu(\{0\})$ -4574,$X_1=X_2=10$ -4575,$80=9.56 + 70.44$ -4576,"$\kappa_{i}(x) = \dfrac{\sum_{j:X_{j} = x} X_{i,j} p_j}{\sum_{j:X_{j} = x}p_j}$" -4577,$S(p)=1-p$ -4578,$x=q(\hat p)$ -4579,$g(s)\le 1$ -4580,$N\times d$ -4581,$X=a$ -4582,$P_{g}$ -4583,$x=q_{\mathbf{v}}(s)$ -4584,$dW_t$ -4585,$a_x=2$ -4586,$f(x)=\exp(-x/\mu)/\mu$ -4587,$\bar M_i(a)$ -4588,$Z\in \mathcal Q$ -4589,$U=4$ -4590,$f(x)=e^x$ -4591,$X_{-1}=C_1 + \cdots + C_N$ -4592,$M_i(x)+Q_i(x)$ -4593,$V=1_{X\le x^\ast}$ -4594,$\bar Q_{2}$ -4595,$\bar P_g(a)=\rho(X\wedge a)$ -4596,$g''(s)=0$ -4597,$\mathsf x\mathsf{VaR}_p(X):=\mathsf{VaR}_p(X)-\mathsf{E}[X]$ -4598,$K=3$ -4599,$a\mathsf{E}_{\mathsf{Q}}[...]$ -4600,$g(s)=\sqrt s$ -4601,"$\bar P_{0,0}$" -4602,"$(x,-x)$" -4603,$n=9$ -4604,$\hat q(p)=q(1-g^{-1}(1-p))$ -4605,$A(0)=0$ -4606,$\rho(X)\le\liminf \rho(X_n)$ -4607,$c$ -4608,$p^*=48.25/71=0.6796$ -4609,$\mathsf{E}[X]+k\mathsf{Var}(X)$ -4610,$d\tilde p=g'(1-p)dp=\phi(p)dp$ -4611,$BC$ -4612,$1 in a layer with loss probability $ -4613,$d=1$ -4614,$s/g(s)\le 1$ -4615,$1/\lambda$ -4616,$1-\alpha_i(x)S(x)$ -4617,$Z=Z_X$ -4618,$E[Z]$ -4619,$\rho(\tilde X_1)=\rho(X_1)+\rho(\mathsf{E}[X_2\mid X_1])$ -4620,$\sum_i X_i(a)=X\wedge a$ -4621,$\mathsf{P}(\{X\in A\})$ -4622,$M(x)/Q(x)$ -4623,$d\omega$ -4624,$\mathcal{Q}$ -4625,"$(x_B, g(S(x_B-))$" -4626,$X=q(U)$ -4627,$q_A(p) = \sup A$ -4628,$\lambda > 1$ -4629,$a \in \mathbb{A}$ -4630,$y\le q_C(p)$ -4631,$\rho(kX)$ -4632,$u=ug(1)=ug(1)+(1-u)g(0) \le g(u)$ -4633,$\Delta Q_{ro}(a) = a-a_{ro}$ -4634,$x_A=\partial x/\partial A$ -4635,$\mathsf{TVaR}_0$ -4636,$\lambda=0.73$ -4637,$Q^* > S$ -4638,$c\le 1$ -4639,$\omega=1$ -4640,$\tau=0.03$ -4641,"$\mathbf {S\,\Delta X}$" -4642,$p<0.9$ -4643,"$\beta, \kappa$" -4644,$a=a_0+(1+c)\mu$ -4645,$f_{\mathbf{v}}$ -4646,$(d\mathsf{Q}/dP)(x) = (1-p)^{-1}1_{x >\mathsf{VaR}_p(X)}$ -4647,$\frac{1}{1-p}\int_{1-p}^q \mathsf{VaR}_s(X)ds$ -4648,"$\bar L, \bar P, \bar M$" -4649,$\mathsf{E}(X)=$ -4650,$\nu$ -4651,$\tau$ -4652,$x_l < x=\mathsf{VaR}$ -4653,$0\le p < 1$ -4654,$Z\mid X$ -4655,"$X:\Omega\to[0,\infty)\subset \mathbb R$" -4656,"$[a, a+da]$" -4657,$f>0$ -4658,$S(x-)=0.1$ -4659,$\rho(X)\le b$ -4660,$s=0.45$ -4661,$(1-p)$ -4662,$Z(X(\omega))$ -4663,$\mathsf{E}[X_i]$ -4664,$\mathit{MV}_{ro}(a) = a-P(X_{-1}\wedge a)$ -4665,$9+1=10+0=10$ -4666,$\mathbf {\mathsf{VaR}_p(X_1)}$ -4667,$g \circ S$ -4668,$1+2c(1-\mathsf{Pr}(Z>\mathsf{E} Z))$ -4669,$\mathsf{P}(\{\omega_i\})=1/4$ -4670,"$\bar S_i(\mathbf{v}, a) := \mathsf{E}[X_i(\mathbf{v}, a)]$" -4671,$r_f>0$ -4672,$\sum_\omega Z(\omega)\mathsf{P}(\omega)=\mathsf{E}[Z]$ -4673,$X\wedge a$ -4674,$NT$ -4675,$p\ge p_0$ -4676,$-\rho(-H)=\rho(H)$ -4677,$\mathbf {X'\Delta S}$ -4678,$L_X(X)=\rho(X)$ -4679,"$\{(s_j, g_j)\} \cup \{(0,0), (1,1)\}$" -4680,$\displaystyle\int_0^\infty xdF(x)$ -4681,$g = s^{0.4}$ -4682,"$0.06 \times (64,861 - 7,500)=3,442$" -4683,$a_1'=a_0-X_{1}$ -4684,$N(t)$ -4685,$\rho(X)=\mathsf{E}_\mathsf{Q}[X]=\mathsf{E}[XZ]$ -4686,$v=S$ -4687,$\mathsf{VaR}_p(X)$ -4688,$\mathsf{E}_{\mathsf Q}[Y] = \mathsf{E}[Yg'(S_X(X))]$ -4689,$u^{iv}<0$ -4690,$\lambda_1$ -4691,$X_1=c_1-Y/2$ -4692,$\alpha < 1$ -4693,$Y+W$ -4694,$\bar q(s)=q(1-s)$ -4695,$P=\mathsf{E}[X]$ -4696,$L-f(L)$ -4697,$X=MX_2$ -4698,$a=a(X)$ -4699,$\alpha_i(a)$ -4700,$\bar\iota : 1$ -4701,$a < kP$ -4702,$\rho(X)=\sum_n X(n)\mathsf{P}(n)$ -4703,$X_i/X$ -4704,$\partial a/\partial v_i$ -4705,$U(-X)\ge U(-Y)$ -4706,$\rho(X)\le \lim\rho(X_n)$ -4707,$wq_Y(p)+(1-w)q_Z(p)$ -4708,$\mathsf{var}(\sum C'_i)=v_{res}^2 \sum c_i^2$ -4709,$p^-$ -4710,$h(0)=0$ -4711,$0\le p^\ast\le 1$ -4712,$\alpha\ge A(n)=\sum_s n_s(1-g(s))$ -4713,$af=1$ -4714,$N=n$ -4715,$q=1-p$ -4716,"$\{x_1,\dots,x_N\}$" -4717,"$(0,0,0,0,0,0,0,0,5,5)$" -4718,$X_n(\omega)=0$ -4719,$1-s_j$ -4720,$c(1)-c(\mathsf{var}nothing)=c(1)$ -4721,$\mathsf{TVaR}_p(X)-\mathsf{VaR}_p(X)=\sigma(\phi(\Phi^{-1}(p))/(1-p) - \Phi^{-1}(p))\to 1$ -4722,$\alpha_j'(x)<0$ -4723,$\mathsf{E}[X_1h(X)]$ -4724,$P=D$ -4725,$f(w) = \exp(-w)$ -4726,$1+r^*=(1+r)(1+\tau)$ -4727,$a=P+Q$ -4728,$X\wedge 10$ -4729,$\mathsf{E}_{\mathsf{Q}}[Y\mid X]\mathsf{E}[Z\mid X] = \mathsf{E}[YZ \mid X]$ -4730,"$u\in[0,1]$" -4731,$L_0^l(X)$ -4732,$j=1$ -4733,$g(s)=\mathsf{TVaR}_{.99}$ -4734,$m+1$ -4735,$\rho_h(X):=\mathsf{E}[X_h]$ -4736,$S(x):=\mathsf{P}(X>x)$ -4737,$9.67$ -4738,$\|\cdot \|_\rho=\rho(|\cdot |)$ -4739,$L^*$ -4740,"$(x_{2,1}, x_{2,2})$" -4741,"$(x,y)$" -4742,$p>1$ -4743,$\mathsf{VaR}_1$ -4744,$p=\Phi^{-1}(4)=3.17\times 10^{-5}$ -4745,$g(s)=s^a$ -4746,$X_i\Delta g(S)$ -4747,$x'$ -4748,$\mathsf{E}[g'(S(X))]=\int_0^\infty g'(S(x))dF(x)=\int_0^\infty -\frac{d}{dx}g(S(x))dx=g(S(0))-g(S(\infty))=g(1)-g(0)=1$ -4749,$\rho_g(X)=51.156$ -4750,"$(s,g(s))=(0.2, 0.36)$" -4751,$\delta^{\star}$ -4752,$\mathsf Q^t\cdot X$ -4753,$\mathsf{Pr}(\Omega)=1$ -4754,$s(0)=s_0=0$ -4755,$dS=-f(x)dx$ -4756,$1_{\{X>x\}}$ -4757,$\ge x$ -4758,$g'(1-p)$ -4759,$Z(x)$ -4760,$0.495(r-i)$ -4761,$\tau(a-\bar P_\tau(a))$ -4762,"$\bar Q_{0,2}$" -4763,"$u'>0, u''>0$" -4764,$Y(\omega)=0$ -4765,$g(S(x))=s$ -4766,$P/S$ -4767,"$p\in[0,1]$" -4768,$X=F^{-1}(U)$ -4769,$>1$ -4770,$r\times m$ -4771,$\mathsf{E}_{\mathsf{Q}}[(X-a)^+] \le \rho((X-a)^+)$ -4772,$\mathbf {D^n\rho_{X\wedge 30}(X_1)}$ -4773,$s=0$ -4774,$\hat q(p)=x$ -4775,$\mathscr{O}(f)$ -4776,"$1/2,1/4,1/4$" -4777,$n-5$ -4778,$q(1-g^{-1}(1-p))/q(p)$ -4779,$Z-X$ -4780,$s>0$ -4781,$\mathsf{E}_\mathsf{Q}[X_i \mid X=x]=\mathsf{E}[X_i g'(S(X))1_{\{X=x\}}] / \mathsf{E}[g'(S(X))1_{\{X=x\}}] = \mathsf{E}[X_i1_{\{X=x\}}]/\mathsf{E}[1_{\{X=x\}}]=\mathsf{E}[X_i\mid X=x]$ -4782,"$\pmb{j, p, S, \kappa_1, \Delta X, \Delta(X\wedge a)}$" -4783,$S(x)$ -4784,$0\le x < 1/6$ -4785,$K = \mathsf{E}[\exp (\lambda x)]^{-1}$ -4786,$Y=\mathsf{E}[Z\mid\mathcal G]$ -4787,"$\omega=0,1,\dots, 99$" -4788,${}^2S(t)=\mathsf{E}[(X-t)_+]$ -4789,"$0,0,1,2,3,6,10,18,36,52$" -4790,$\mathbf {X_{n}}$ -4791,$t=0$ -4792,$p=0.791$ -4793,$\ge \mathsf{E}[X]$ -4794,$f(x+)$ -4795,$X_{2c}$ -4796,$\mathsf{E}_{QQ'}[X_i(a)] \ne \mathsf{E}_{QQ}[X_i(a)]$ -4797,$\mathcal S$ -4798,$\mathbf {M_{2}}$ -4799,$q_{\mathbf{v}}(p)=\mathsf{VaR}_p(X(\mathbf{v}))$ -4800,$p(\omega)$ -4801,$0\le p^*\le 1$ -4802,$r_N$ -4803,$\rho(X_g)-\rho(X_n)=51.1560-49.8986=1.2574$ -4804,$\sum_i x_i\mathsf{Pr}(X=x_i)$ -4805,"$\mathsf{Q}_2(A)=2\mathsf{P}(A\cap (0.5, 1])$" -4806,$S(x)=0$ -4807,$5/6$ -4808,$\bar S(a)=\mathsf{E}[X\wedge a]$ -4809,$s^\ast=1/2$ -4810,"$a_{0,t}:=a(Y_{0,t})$" -4811,$0\le p_0 \le p_1\le 1$ -4812,$0.2$ -4813,"$X_1,X$" -4814,$(1-r_0)\delta_1$ -4815,"$[0,1-p)$" -4816,$\mathcal Q_2$ -4817,$\mathsf{E}[X\mid \mathcal F_{t+1}]$ -4818,$\mathsf{Pr}(\mathsf{var}nothing)=0$ -4819,$\rho\mapsto a^\rho(\ \cdot\ ;\ \cdot\ )$ -4820,$\lambda\rho(X)$ -4821,$\mathcal D(X)=c\mathsf{Var}(X)$ -4822,$\le$ -4823,$u''' \ge 0$ -4824,$\rho(X\wedge a)$ -4825,$Y_2$ -4826,$X=X_1+...+X_n$ -4827,$\mathsf{E}[X\mid A]$ -4828,$g_j$ -4829,$\mathsf{E}[X] + \pi \mathsf{SD}(X)$ -4830,$1 < \alpha < 2$ -4831,$\Delta \mathit{MV}_{ro}(a)$ -4832,$\phi(s)$ -4833,$p\cdot X$ -4834,$\mathsf{E}[U]=\mathsf{E}[X]$ -4835,$\mathbf {Z_\mathit{lin}}$ -4836,$\beta_i(x)=\mathsf{E}_\mathsf{Q}\left[ \dfrac{X_i}{X}\mid X > x\right]$ -4837,$p_0 \le p^\ast \le p_1$ -4838,$D\rho(\cdot)$ -4839,$\lambda=0.25$ -4840,$u'''>0$ -4841,${}^nS^{-1}_X(q)\le {}^nS_Y(q)$ -4842,$S_X$ -4843,"$(\Omega, \mathcal{F})$" -4844,$S\subset\Omega$ -4845,$\hat q(p) > q(p)$ -4846,$\mathsf j(a)$ -4847,$X+c$ -4848,$S(y_j-)-S(y_j)$ -4849,$\mathbf {Z_3}$ -4850,$\Phi^{-1}(0.995)=2.576$ -4851,$g(p)/p$ -4852,$\alpha_2(99)=0.9$ -4853,$\alpha_iS\Delta X$ -4854,$L_0^a(X)=X\wedge a$ -4855,$\mathsf{E}[v^T] \ge v^{\mathsf{E}[T]}$ -4856,$g(s)=s^{0.7}$ -4857,${}^2S^{-1}(t)=q\mathsf{TVaR}_q(X)$ -4858,$\bar P_{0}=\rho(Y_{0})$ -4859,$\mathsf{E}_\mathsf{Q}[\mathsf{E}[X_i \mid X]]$ -4860,$g'(s)=\phi(1-s)\ge 0$ -4861,$\mathsf{MONO}$ -4862,$\bar M_i(a)>0$ -4863,$g(S(0-))=1$ -4864,$\rho(0)=\rho(0+0)=\rho(0)+\rho(0)$ -4865,$\rho(X_0)$ -4866,$\delta_p/\nu_p = \iota_p$ -4867,$\mathcal{Q}=\mathcal{M}$ -4868,$\rho \ge \mathsf{E}[X]$ -4869,"$d=1,\dots,N$" -4870,$x=0$ -4871,$\mathsf{j}$ -4872,$\mathsf{E}[X] + c\mathsf{E}[(X-\mathsf{E} X)_+^2]$ -4873,"$E_1,\dots,E_N$" -4874,$\mathsf{E}[Z\mid X]=0$ -4875,$(1-p)x_0$ -4876,$U\le p$ -4877,"$(x_1-\epsilon,x_1]$" -4878,$\sigma=0.15$ -4879,$pl(p)$ -4880,$g'(0)$ -4881,$P = \mathsf{VaR}_\pi(X)$ -4882,$C_i$ -4883,$x\mapsto (x-a)^+$ -4884,$\beta_L$ -4885,$D\rho_X(X_i)$ -4886,"$\alpha_1,\alpha_2$" -4887,$\{X>x\}$ -4888,"$x_{2,2}$" -4889,$w=1$ -4890,$\mathbf n$ -4891,$\mathsf{E}[Z\mid X]$ -4892,$F_2\prec_2 F_1$ -4893,$Z_8$ -4894,$T^{-1}(A)$ -4895,$\mathsf{TVaR}_{0.95}(Y)=0.8\mathsf{E}[X]=2000$ -4896,$g'(s) = rs^{r-1}$ -4897,$\mathsf{Q}\in\mathscr{M}$ -4898,$\mathsf{P}_X(A) :=\mathsf{Pr}(X\in A)$ -4899,$\rho(X)/2$ -4900,$ro$ -4901,$\alpha(\mathsf Q) < \infty$ -4902,$x_p$ -4903,$X\Delta S$ -4904,$S=e^{\mu t}$ -4905,$\Delta gS$ -4906,$s^{th}$ -4907,$\mathsf{E}[(X-\mu)^n]$ -4908,$X=X_0+Y$ -4909,$20$ -4910,$\mathsf{TI}$ -4911,$b-a$ -4912,$\sigma(Z)=\sqrt{\mathsf{var}(Z)}$ -4913,"$\tau(a-\rho_{a,\tau}(X))$" -4914,$\rho(X) < \infty$ -4915,$Y=c$ -4916,$\rho(W_0\wedge a_0)$ -4917,$g(0.01)=0.1$ -4918,$f(x)=(x-d)^+1_{\{x \le m \}}$ -4919,$X\circ T$ -4920,$\mathsf{E}[X\wedge 0]=0$ -4921,"$(\x*.75, -2)$" -4922,"$\mu=8.7, \sigma=2.5$" -4923,$p=0.05$ -4924,$\mathit{RV}$ -4925,$n=7$ -4926,$X_4=X_5=10$ -4927,$\mathsf{E}_{\mathsf Q}[.]$ -4928,$n\to \infty$ -4929,$i\not=j$ -4930,$H(x)$ -4931,$\mathsf{E}_{\mathsf Q}[X_i \mid X]$ -4932,$a\le \rho(X)\le b$ -4933,$EL$ -4934,$\alpha_i'(x)<0$ -4935,$q_Z$ -4936,$dp=f(x)dx$ -4937,$v_{res}\sqrt{(1+v^2)/n}\approx v_{res}v/\sqrt{n}$ -4938,$\rho(X_i)\le 0$ -4939,$s=s_1+s_2$ -4940,$\displaystyle\int_0^1 X(1-g^{-1}(1-\tilde p))d\tilde p$ -4941,$F(x_0)= p_+>p_0$ -4942,$g''(s)<0$ -4943,"$(s,m)$" -4944,$U = A = 8.149$ -4945,$P(a)da$ -4946,$B(p)$ -4947,$Q=a-P$ -4948,"$2^1, 2^3, ...$" -4949,$c(S)= \rho\left( \sum_{i\in S} X_i \right)$ -4950,$\partial f_{\bar x}/\partial x_i$ -4951,$\log(x)$ -4952,$L_d^{d+l}$ -4953,$\alpha(\mathsf Q)\not=0$ -4954,$X\le a$ -4955,$\kappa_i(x)/x$ -4956,$\bar P_{d}=\rho(Y_{d})$ -4957,$D^n\rho_X(X_2)=45.1838$ -4958,$f(x)=1$ -4959,$X_0+\epsilon Y$ -4960,$g_\tau$ -4961,$\phi'(s)ds$ -4962,$S\approx \mathsf{E}[X]$ -4963,$g'(1)>0$ -4964,$A=8.13$ -4965,$\mathbf {\kappa_1}$ -4966,$X_n$ -4967,$a=P+Q=EL+M+Q$ -4968,$\mathit{MV}_{ro}(a_{ro})$ -4969,$g(0-)f(\esssup(X))$ -4970,$S(x)=d/dx(\mathsf{E}[X \wedge x])$ -4971,$g_1$ -4972,$g'(1-p)=\nu$ -4973,$\mathbf {\rho(X\wedge a)}$ -4974,$X_1+X_2=X$ -4975,$|t|$ -4976,$\prec_n$ -4977,$P(X\wedge a)=\bar P(a)$ -4978,$\bar x$ -4979,$x_h>x=\mathsf{VaR}$ -4980,$1+\gamma$ -4981,$S/P$ -4982,$X_0$ -4983,$b_h$ -4984,$\mathsf P(X>a)>0$ -4985,$(1+\gamma)^{t-x}$ -4986,$n > 2$ -4987,$=\displaystyle\int_0^\infty x \P(\{X \in dx \})$ -4988,$\mathbf {a_2'}$ -4989,$\phi'(p)=f(p)/(1-p)\ge 0$ -4990,$\mathsf{VaR}_{0.98}$ -4991,$\sup X$ -4992,$h_f$ -4993,$\lambda>0$ -4994,${10\choose 5} = 252$ -4995,$T$ -4996,"$i,v$" -4997,$a_i':=\sum \alpha_i(1-S)\Delta (X\wedge a)$ -4998,$F:\mathbb{R}^n \to \mathcal{X}$ -4999,$u_i$ -5000,$N=40$ -5001,$Pr(X > a)$ -5002,$X_i(a')$ -5003,$t\mapsto s(t)$ -5004,$a_{1}$ -5005,$\int_0^1 f(p)dp = 1 - \alpha < 1$ -5006,$X=q(U_X)$ -5007,$t=w$ -5008,$E[X_2 | X]$ -5009,$B=\Omega$ -5010,"$1 million auto accident, a $" -5011,$3^{20}$ -5012,"$(-\x*0.75, -2)$" -5013,$\bar S_i(x)$ -5014,$dX$ -5015,$D\rho_X(X_1)$ -5016,"$\int_0^\infty z(x)\,dF(x)=1$" -5017,"$X_{t+1,1}$" -5018,$\log$ -5019,$(1-g(s))q$ -5020,"$(0,0,\dots,0,10)$" -5021,"$\iota, \iota(p)$" -5022,$\mathsf{Var}^+(X) = \int_{\mathsf{E}[X]}^\infty (x-\mathsf{E}[X])^2 f(x)dx$ -5023,$\mathsf{TVaR}_{0.5}(X_2)=45.5$ -5024,$t-2$ -5025,$Z_2$ -5026,$\prec_2$ -5027,$0\le x < X_1$ -5028,$\mathsf{E}_{\mathsf{Q}}[X] \le \rho(X)$ -5029,$a=\sum_i a_i$ -5030,$s<0.1$ -5031,"$a(x_1,x_2)=\sqrt{3x_1^2 + 4x_2^2}$" -5032,$\mathsf{E}[XZ_j] = (5)(1/10)(8)+(5)(1/10)(9)=8.5=\mathsf{TVaR}_{0.8}(X)$ -5033,$E[u_j(W_j - X_j + Y_j - H[Y_j])]$ -5034,$0\le \mathsf{Pr}(E)\le 1$ -5035,$1-p=0.9$ -5036,$h(s)=1-g(1-s)$ -5037,$(P-L)/A$ -5038,$X_1(10)$ -5039,$AR\succ BY$ -5040,$w_0$ -5041,$q_X\le q_Y$ -5042,$0 < \alpha\le 1$ -5043,"$\mathsf{biTVaR}_{0,1}^w$" -5044,"$a_i=\rho(X_i, p^*)$" -5045,$\mathsf{E}[1_{U_X\ge p}]=\mathsf{E}[B]$ -5046,$\phi(p)=g'(1-p)$ -5047,$1/(1+r)$ -5048,$\dfrac{1}{1+\iota} p$ -5049,$p(1-p)$ -5050,$\rho(X) = \int_0^\infty g(S(x))dx$ -5051,$\sum S\Delta(X\wedge a)$ -5052,$V^*$ -5053,$\partial a/\partial v_1$ -5054,"$A_1=[-k,-k]$" -5055,$p=0.25$ -5056,$a^{\star}(X)$ -5057,$0.8 \ge p < 0.9$ -5058,$\mathcal{G}$ -5059,$g'(s-)$ -5060,$k$ -5061,$\rho(X_n) \downarrow \rho(X)$ -5062,$q_X(U)$ -5063,$wq_X(p)+(1-w)q_Z(p)$ -5064,"$p\in [0,1]$" -5065,$g(s) \approx m_0+(1+m'(0))s$ -5066,"$Y_{0,0}:=\sum_{d>0} X_{0,d}$" -5067,"$Y_m=\max(X_1,\dots,X_m)$" -5068,$\mathsf{VaR}_{0.99}(X_1)=150$ -5069,$0.01$ -5070,"$t^\star \in [0,1]$" -5071,"$\{1,2,\dots, n\}$" -5072,$a < \max(X)$ -5073,$\mathcal N_X(X_i)$ -5074,$x^{\ast}:=\min(x)$ -5075,$0.5L_{250}^{500}(x)+0.75L_{500}^{750}+L_{750}^{1000}$ -5076,"$x_0, x_1, x_2$" -5077,$\sum (1-S)\Delta (X\wedge a)$ -5078,"$[0,\infty)\subset\mathbb{R}$" -5079,$\mathsf{Pr}(X=\mathsf{VaR}_p(X))=0$ -5080,$\mathsf{E}[(X-a)^+]/\mathsf{E}[X]$ -5081,$\bar Z = F(\bar x)$ -5082,$^2$ -5083,$q_{X}(p)=\sqrt{2}\Phi^{-1}(p)$ -5084,$a = a(\mathbf{v}) = a(X(\mathbf{v}))$ -5085,$s=1$ -5086,$S\cdot dX$ -5087,$s$ -5088,$\mathsf{E}[X\mid \mathcal F']$ -5089,$S(x)=u$ -5090,$\sup_{\omega\in\Omega} (f(\omega)+g(\omega)) \le \sup_{\omega\in\Omega} f(\omega) + \sup_{\omega\in\Omega} g(\omega)$ -5091,$\mathsf{E}_\mathsf{Q}[0]=0$ -5092,"$0,1,1,1,2,3, 4,8, 12, 25$" -5093,$\triangleright$ -5094,$\mathsf{TVaR}_p(X)=51.156$ -5095,"$A\subset [0, \infty)$" -5096,"$\Delta\,g(S)$" -5097,$f(x)\le f(y)$ -5098,$\rho(X) = \mathsf{E}[X] + \lambda \mathsf{E}[(X-\mathsf{E}[X])^+]$ -5099,$da$ -5100,$\mathsf{E}[X]=0.6$ -5101,$S=1$ -5102,$L_{250}^{\infty}$ -5103,$\mathcal S(X)=\mathsf{E}[X]$ -5104,$0 = x_0< x_1<\cdots < x_n < \cdots$ -5105,"$\nu p\,da=\nu F(a)\,da$" -5106,$X=\mathsf{E}[Y \mid \mathcal F']$ -5107,$\hat p:=1-g^{-1}(1-p)$ -5108,"$X(x_1, x_2)=(x_1+x_2)Y$" -5109,$1-F(x)=1-p$ -5110,$\mathcal F_t$ -5111,$\rho(X)=\rho(X-Y+Y)\le \rho(X-Y) + \rho(Y)$ -5112,$c \le 0$ -5113,$S(x_{(j)})(x_{(j+1)}-x_{(j)})$ -5114,$p=0.9$ -5115,$\rho(X+Y) \le \rho(X) + \rho(Y)$ -5116,$e^{X_t}$ -5117,$n\times r$ -5118,"$f'_\omega (\bar x, h)$" -5119,"$Y_{t,d+1}$" -5120,$F(b)-F(a)$ -5121,$\rho(X)\ge\rho(X+Y)\ge \rho(X)+\mathsf{E}[YZ]$ -5122,$a_{ro}:=\mathit{VaR}_{p}(X_{-1})=10743.5$ -5123,$\rho(X) - (-\rho(-X))=\rho(X)+\rho(-X)$ -5124,$Z_\epsilon$ -5125,$(\beta_i g(S))'(x)=-\mathsf{E}[X_i\mid X=x]g'(S(x))f(x)/x=-\kappa_i(x)g'(S(x))f(x) / x$ -5126,$\{3\}$ -5127,$\lim_{\epsilon \downarrow 0} (f(x-\epsilon)-f(x))/\epsilon$ -5128,$\max_{\mathsf{Q}} \mathsf{E}_\mathsf{Q}[0] -\alpha(\mathsf Q) =\max_{\mathsf{Q}} -\alpha(\mathsf Q)= -\min_{\mathsf{Q}} \alpha(\mathsf Q) = 0$ -5129,"$1,9,4,4,2,$" -5130,$g(S)$ -5131,$\mathsf{WCE}_p(X) := \sup\ \{ \mathsf{E}[X \mid A] \mid \mathsf{Pr}(A) > 1-p \}$ -5132,$\mathsf{TVaR}_p( X )$ -5133,$\mathsf{MON}'$ -5134,$\mathsf{TVaR}_{p_1}(X)$ -5135,$1_{X < q(1-s)}-(1-g)$ -5136,$g(x)=e^{2\pi i x\theta}$ -5137,"$\mathsf{E}[Y_{0,0}]+\lambda\sigma(Y_{0,0})=58.129$" -5138,$f=f(s)$ -5139,$l=a$ -5140,"$H(A, L, t)$" -5141,$\mathsf{TVaR}_{0.75}=4\left( \frac{90}{8}+\frac{98}{16}+\frac{100}{16}\right)=94.5$ -5142,$\mathit{NPV}$ -5143,$E_k$ -5144,$g(s)=s^\rho$ -5145,$X\ge 0$ -5146,$1.2\times 10^9$ -5147,$p=F(x)=\mathsf{Pr}(X\le x)$ -5148,$\mathsf{Pr}(X > \mathsf{VaR}_p(X))$ -5149,$f'(a)$ -5150,$y\in A$ -5151,$0 < \lambda \le 1$ -5152,"$\mathsf{cov}(X_i,X)/\sigma_X$" -5153,$s=S(x)=\mathsf{Pr}(X>x)$ -5154,$t_1$ -5155,$\lambda>1$ -5156,$g(S(x))=g(0)=0$ -5157,$D^n\rho_{X\wedge a}(X_i)$ -5158,$\mathsf{E}[X\mid \mathcal F_t](\omega)$ -5159,$\tau < t+d$ -5160,$s_2=1$ -5161,$\mathsf{E}[X_i\mid \{X=X(\omega)\}]$ -5162,$\mathsf j(a)=\max \{ j:X_j < a \}$ -5163,$g'(S(x))\ge 1$ -5164,$1-\tilde p=g(S(x))$ -5165,$F_m\succ_m F_0$ -5166,"$X_{t,d+1}$" -5167,$A(-X)=-B(X)\not=-A(X)$ -5168,$g=1$ -5169,$0.99$ -5170,$f_t$ -5171,$\mathsf{Var}^+(X)$ -5172,$E[YZ]$ -5173,$1-r_0$ -5174,$\lambda=0$ -5175,$\mathsf{E}[X_i\mid X]$ -5176,$\beta_2g-\alpha_2S$ -5177,$\rho(X)=\mathsf{E}[Xg'(S(X))]=\mathsf{E}[\sum_i X_i g'(S(X)))]=\sum_i \mathsf{E}[X_ig'(S(X))]$ -5178,$\mathsf{E}[X_i\wedge a_i]$ -5179,$x^*$ -5180,$\lambda t$ -5181,$\{X > \mathsf{VaR}_p(X)\}$ -5182,$r_f = 0.02$ -5183,$x=1$ -5184,"$[s_0, s_1]$" -5185,$(\beta g(S))'(x)=-\kappa_i(x)g'(S(x))f(x)/x$ -5186,"$a_{0,1}$" -5187,$X_{d}$ -5188,$q(p)=\inf\{x \mid F(x)\ge p \}$ -5189,"$([0,1], \mathcal B, \mathsf P)$" -5190,$\alpha(\mathsf{Q})=\infty$ -5191,$\rho_a(0) = \rho(0 \wedge a(0)) = \rho(0 \wedge 0) = \rho(0) = 0$ -5192,$\rho(X)\le \rho(\lambda X)/\lambda$ -5193,$c(\sum_{i\in S} X_i)$ -5194,$g(0)=0$ -5195,$\alpha_{1}$ -5196,$0 < b \le 1$ -5197,$pX + (1-p)Z$ -5198,$\pi(X)$ -5199,${}^nS^{-1}(q)$ -5200,$\sup X=\inf$ -5201,$Q^*$ -5202,$v-\nu^{\star}=(\iota^{\star}-i)/v\nu^{\star}$ -5203,$_{ro}$ -5204,$\iota=\delta/\nu$ -5205,$m'(1) = -m_2/(1-s_2)$ -5206,$D^n\rho_{X\wedge a}(\cdot)$ -5207,$P_i=\mathsf{E}_\mathsf{Q}[X_i]$ -5208,$(M-N)\times d$ -5209,$S(x_0)=1$ -5210,$\mathsf{E}_\mathsf{Q}[X1_A] / \mathsf{E}_\mathsf{Q}[1_A]$ -5211,$10/11$ -5212,$f(L)=(L-a)^+$ -5213,$\tilde Z_X:=\mathsf{E}[Z\mid X]$ -5214,$\mathsf{j}(a) = \max\{ j:X_j < a \}$ -5215,"$3.807=\lambda \sigma(W_{0,0})$" -5216,$h(x)=\sqrt x$ -5217,$ for $ -5218,$S(x-)=1$ -5219,$\{ Z\not=0 \}$ -5220,$\iota=(g(s)-s)/(1-g(s))$ -5221,$\tau=0$ -5222,$\mathsf{E}[X_iZ]$ -5223,$(r-i)Q_t$ -5224,$\delta p$ -5225,$\mathsf{TVaR}_p = q(p)$ -5226,$\mathsf{E}[Z]=g(1)-g(0)=1$ -5227,$\sigma=0.1980$ -5228,$X_1> x_1$ -5229,$\mathsf{E}[X_1\mid X=20]= 14$ -5230,$1/(1-p)>1$ -5231,$\mathbf {\mathcal Q}$ -5232,$\lambda\mathsf{E}[X]$ -5233,$q_V(p)=0$ -5234,$(1-s)^{-1/2}/4$ -5235,$g(0-)$ -5236,$(s+\iota) / (1+\iota)$ -5237,$k = 1.4 + 1.8s$ -5238,$\Psi(x)=1-\exp(-e^x)$ -5239,$=\displaystyle\int_0^\infty S(x)dx$ -5240,$dp$ -5241,$da\to 0$ -5242,"$(lee.east |- lee.north)+(0.25,0.25)$" -5243,$G$ -5244,$X'=0$ -5245,$\rho_g$ -5246,$\mathsf{E}[X_i/X\mid X>x]$ -5247,$s > 0.5$ -5248,$\rho=0.12$ -5249,$\beta_1g(S)dx$ -5250,$X(x)=x$ -5251,$g(S(x)) = S(x) + \delta(F(x))F(x)$ -5252,$L_X \in \mathcal L_\rho$ -5253,$g-S$ -5254,$x_0$ -5255,$0=\rho(0)$ -5256,$Xm1=X_{-1}$ -5257,$1-g^{-1}(1-p')$ -5258,$B(1_{U>0.95})=B(1_{U\le 0.05})=h(0.05)=1-g(1-0.95)=0.0203$ -5259,$\mathcal D(X)=\rho(X)-\mathsf{E}[X]$ -5260,$\phi(p)\ge 0$ -5261,$E(X_{-1}\wedge a)$ -5262,$n=8$ -5263,$R/Q$ -5264,$q < p$ -5265,$x=wy + (1-w)z$ -5266,"$B_3=[-k, \epsilon]$" -5267,$Q = 5.0449$ -5268,$n'=7$ -5269,$g'(t)>0$ -5270,"$j=0,\dots, N-1$" -5271,$0\ < p < 1$ -5272,$(S_t-a)^+$ -5273,$\alpha+\beta = \iota^\ast/(1+\iota^\ast)$ -5274,$\sin(x)$ -5275,$\mathbf{P_i}$ -5276,$a_{gc}:=\mathit{VaR}_{p}(X)={{a_x}}$ -5277,$\mathsf{VaR}_{0.995}$ -5278,$P(X_{-1}(a_{gc}))={{mvp_gc}}$ -5279,$\rho''(x)=-U''(x)>0$ -5280,$\{\omega\mid X(\omega)=x\}$ -5281,$\kappa$ -5282,$e$ -5283,$\omega'=\omega$ -5284,$0.3 < s <0.4$ -5285,$\mathbf {d=2}$ -5286,$g(s)=s^\alpha$ -5287,$X_1-X_2$ -5288,$\mathbf {g(S)\Delta X}$ -5289,$a = \sum_i a_i$ -5290,$\rho(X)=1$ -5291,$H(X)\le H(Y)$ -5292,$Y=X$ -5293,$\{\omega\in \Omega \mid (X\wedge a)=a \}$ -5294,$X\ge x_0$ -5295,$r=1$ -5296,"$\bar Q_{0,1}$" -5297,$Y\preceq_2 X$ -5298,$\rho(X)=k\mathsf{Var}(X)$ -5299,$\delta = \iota\nu$ -5300,$g'(1-s)=\phi(s)$ -5301,$q(U_X) < m$ -5302,$\alpha_1$ -5303,$A(X+Y)\le A(X)+A(Y)$ -5304,"$a_{0,t}' = a_{0,t}$" -5305,"$j=5,6$" -5306,$\mathsf Q_k$ -5307,$\lambda < 1$ -5308,$\mathcal E:=\{Y \circ T \mid T \text{ PPT} \}$ -5309,$Xp$ -5310,$F(x)=\mathsf{P}(\{X\le x\})$ -5311,"$(lee.east |- lee.south)+(0.375,-0.25)$" -5312,$p_j=\mathsf{P}(X=X_j)$ -5313,$dF=-dS=$ -5314,$m(s) := (1-s)\wedge m(s)$ -5315,$\mu_{rU} = M/K = 0.133$ -5316,$y \wedge (x-a)^+$ -5317,$\mathcal A=\{X\mid \rho(X)\le 0 \}$ -5318,"$Y_{0,0}$" -5319,$\bar P_{1}$ -5320,$\alpha_1+\alpha_2=\beta_1+\beta_2=1$ -5321,$a_l>b_l$ -5322,$X_0=0$ -5323,$\Delta Q_{gc}(a)$ -5324,$P_j=\sum_{i=0}^j p_i$ -5325,$\{y_j\}$ -5326,$X=3$ -5327,$\mathsf{Pr}(q^-(F(X))\not=X)=0$ -5328,$\rho(X)=\bar P$ -5329,$\alpha(\mathsf Q)\ge 0$ -5330,$a_l$ -5331,$A$ -5332,$v(AB) + v(ABCD) = 3/2 > v(ABC) + v(BCD) = 4/3$ -5333,$\sum p_jX_j$ -5334,$0.5+U/4$ -5335,$n=3$ -5336,$\bar\nu$ -5337,$p^*=1$ -5338,$r_K = \exp (\lambda) - 1$ -5339,$v(\mathsf{var}nothing) =0$ -5340,$n\mathsf{Pr}(Y > y_c)$ -5341,$x<1$ -5342,$a(X)=a(\sum_i X_i) = \sum_i a_i$ -5343,$P(X_{-1}(a))=\bar P^a_0$ -5344,$\kappa_{1}$ -5345,$\{\omega\in\Omega \mid X(\omega) \le x\}\in\mathcal F$ -5346,$\mathsf{TVaR}_{0.6975}$ -5347,$F(q^-(p))=p$ -5348,$\mathsf{E}[X]+\mathsf{var}(X)/\mathsf{E}[X]$ -5349,$B_2 \succ A_2$ -5350,$\hat{s}$ -5351,$\rho(X+\rho(X))=\rho(X)-\rho(X)=0$ -5352,$\mathsf{NORM}$ -5353,$Y\succeq X$ -5354,$\lim_{x\to\infty} xg(S(x))=0$ -5355,$\int xdF$ -5356,$\mathbf {M_1\Delta X}$ -5357,$t > 2/3$ -5358,$\mathsf{E}_\mathsf{Q}[X]=\mathsf{E}[XZ]$ -5359,$p=1-s_j$ -5360,$\mathsf{E}[\mathsf{E}[X_iZ\mid X]]\not=\mathsf{E}[\mathsf{E}[X_i\mid X]\mathsf{E}[Z\mid X]]$ -5361,"$d,v\ge 0$" -5362,$X_1\le X_2$ -5363,$r_D$ -5364,$x=\max(X)$ -5365,$c=0$ -5366,$1/\lambda = \sum_j 1/\lambda_j$ -5367,$>0$ -5368,$\rho_a(X)>2\rho_a(X_1)$ -5369,$Z(200)=0$ -5370,$A=\{X>x\}$ -5371,$n\ge 0$ -5372,$\bar P(a)\le a$ -5373,$\mathsf{Pr}(X < x)=p=\mathsf{Pr}(X\le x)$ -5374,$\displaystyle\int_0^\infty g(S(x))dx$ -5375,$M(x)$ -5376,$\mathbf {M\Delta X}$ -5377,$\rho(\tilde X)=\mathsf{E}_{\mathsf{Q}}[\tilde X]$ -5378,$\int_0^1 F^{-1}(p)dp$ -5379,$e_x=\sum_t {}_tp_{x}$ -5380,$g'\left (S_{X\wedge a}(X\wedge a)\right )$ -5381,$0 < g' \le 1$ -5382,$\mathit{NPV}_1$ -5383,$w(Z)/\mathsf{E}[w(Z)]$ -5384,$0.75+U/4$ -5385,$g_2$ -5386,$r_D=0$ -5387,$\displaystyle\int_\Omega X(\omega)\P(\omega)$ -5388,$p:=1-s$ -5389,$\bar\delta=\bar\iota\bar\nu$ -5390,$\rho(aX)=a\rho(X)$ -5391,$f(x-)$ -5392,$\mathsf{E}_\mathsf{Q}[X_i(a)]$ -5393,$A_i\cup A_i^c$ -5394,"$(s_0,g(s_0))$" -5395,$Q_0=0.25$ -5396,$3$ -5397,$X=\sum_t B_t/2^i$ -5398,$\iota(s)=(1-s)/(1-1)=\infty$ -5399,$Z_A=(1-p)^{-1}1_A$ -5400,$Q\circ T\in\mathcal{Q}$ -5401,$\mathsf{Pr}(B\le t) = 1/2 + 1_{t>1/2}(1/2)$ -5402,$\mathcal Q$ -5403,$\Delta X_j=X_{j+1}-X_j$ -5404,$w$ -5405,$t>\tau$ -5406,$1-g(S(t))$ -5407,$ to be the set of all sample points where the insurance event $ -5408,$1-1_{X>a}=1_{X\le a}$ -5409,$s=1-p$ -5410,$f(x)=x$ -5411,$s \approx 0$ -5412,$j=9$ -5413,$\mathsf{E}[Z(X)]=1$ -5414,$k\le m$ -5415,$\{\mathsf{E}_{\mathsf Q}[X_i] \mid \mathsf Q\in\mathcal Q(X)\}$ -5416,$\mathsf{E}[|X|]<\infty$ -5417,$\epsilon$ -5418,$\mathsf{E}[X_i (X\wedge a)/X \mid X=x] = \mathsf{E}[X_i\mid X=x] (x\wedge a)/x$ -5419,$\bar Q(a)=a-\bar P(a)$ -5420,$#2$ -5421,$\rho(X) = \mathcal{N}_{\tilde X}(X)$ -5422,$p$ -5423,$\mathbf {a=0.93}$ -5424,$3/4 \pm 1/4$ -5425,$10^{-2}$ -5426,$\mathsf{E}[X\wedge a] + d(a - \mathsf{E}[X\wedge a])$ -5427,$\mathcal B$ -5428,"$(\Omega,\mathcal F, \mathsf{P})$" -5429,$\epsilon>0$ -5430,"$g(s) = \nu s + \delta, s>0$" -5431,$X(\omega)\ge a'$ -5432,$\mathsf{E}[Xe^{hX}]/\mathsf{E}[e^{hX}]$ -5433,$r=0.025$ -5434,$\mathsf{E}[1_{X>a}]=\mathsf{P}(1_{X>a}=1)$ -5435,$\{X=q_X(p)\}$ -5436,$m$ -5437,$\mathcal F_0$ -5438,$L_0$ -5439,$m\le 4$ -5440,$\mathsf{TVaR}_1(X)=\sup(X)$ -5441,$\mathbf {d=0}$ -5442,$q(p)=\mathsf{VaR}_{p}(X)$ -5443,$\rho(X-Y)\le 0$ -5444,$P_{i}(a)$ -5445,$\rho(X)=\mathsf{TVaR}_p(X)$ -5446,"$\mathbf{v}=(v_1,v_2)$" -5447,$\kappa_i(t)=E[X_i \mid X=t]$ -5448,"$(s, g(s))$" -5449,"$(-1,1)$" -5450,$n\times 1$ -5451,$g'(S(x))<1$ -5452,$X_{1}$ -5453,$\rho(X)\le\lim \rho(X_n)$ -5454,$q^+(p) := \sup\ \{x \mid F(x) \le p \} = \inf\ \{ x \mid F(x) > p \}$ -5455,$M-N$ -5456,"$i=2,3,4,5$" -5457,$\mathsf{Pr}(X_n=Y)=\mathsf{Pr}(X=Y)=0$ -5458,$X_i(v_i)=v_iX_i(1)$ -5459,$X\le Y$ -5460,$S\Delta X'$ -5461,$t\mapsto \rho(X) + t\mathsf{E}_{\mathsf Q_X}[Y]$ -5462,$\rho(X\wedge a)=0.909$ -5463,$(1+\gamma)F_0$ -5464,$\sigma=\sqrt{s(1-s)/N}$ -5465,$\iota(s)$ -5466,$a-\bar P(a)$ -5467,$\mathbf {\mathsf{P}(X)=\Delta S}$ -5468,$F^{-1}$ -5469,$\rho(X) = \max_{\mathsf Q\in \mathcal Q} \ \mathsf{E}_\mathsf{Q}[X]$ -5470,$\mathbf {X'p}$ -5471,$\kappa_2(X)$ -5472,$U$ -5473,"$Y_{t,1}$" -5474,"$k=1,2,\dots,n-1$" -5475,$g(S(x-))=1$ -5476,$X_0 + \epsilon Y$ -5477,"$\displaystyle\int_0^a \kappa_i(x)g'(S(x))f(x)\,dx + a\beta_i(a)g(S(a))$" -5478,$m(s)$ -5479,$x_0 \ge q^-(p)$ -5480,$X(\mathbf{v}) = \sum_i X_i(v_i)$ -5481,$a=9532.0$ -5482,$L_{250}^{1000}(x)$ -5483,"$\sigma=13,108$" -5484,$\mathsf{E}[r] = \mu_r = M/K = 0.132$ -5485,$T_2 := ((n+1)-pN)x_n$ -5486,$\{ X>x \}$ -5487,$\rho(X)=\mathsf{E}[X] + c\sigma(X)$ -5488,$\iota = \dfrac{g(s)-s}{1-g(s)}$ -5489,$\mathsf{E}[|X_1|]<\infty$ -5490,$S_{\mathbf{v}}(t)=\text{Pr}(X({\mathbf{v}})>t)$ -5491,$g(s) = s^r$ -5492,$\Delta X$ -5493,$=$ -5494,$R^2$ -5495,$\mathsf{E}[X(1_{U_X\ge p}-B)]=\mathsf{E}[(X-m)(1_{U_X\ge p}-B)]\ge 0$ -5496,$S(x_4)$ -5497,$\kappa\ge K(n)=\sum_s n_s(1-g(s))k(s)$ -5498,$X+100$ -5499,"$\Omega=\{0,1,2,\dots \}$" -5500,$S_X(x) \ge S_{X_1}(x)$ -5501,$R(X)$ -5502,$g(S_6)\Delta X'_6$ -5503,$\rho(X-\rho(X))=0$ -5504,$\alpha_i(x) = \mathsf{E}[X_i /X \mid X> t]\not=\mathsf{E}[X_i\mid X> t]/\mathsf{E}[X\mid X>t]$ -5505,$P = \mathsf{E}[X] + \pi \mathsf{E}[(X-\mathsf{E}[X])^+]$ -5506,$g(0+) > 0$ -5507,"$X_i,X$" -5508,$p=0$ -5509,$r_h=\mu_L=0$ -5510,$g(0^+)>0$ -5511,$\mathrm{Pr}_{rn}\{P_{act}>P\}$ -5512,"$(I, \mathcal B, \mathsf P)$" -5513,$\mathbb{R}^3$ -5514,$ is not continuous and $ -5515,$\mathsf{Pr}(X=1)=s$ -5516,$E'=\Omega\setminus E\in\mathcal F$ -5517,$a_x$ -5518,"$\{1,2,\dots,10000\}$" -5519,$\Pi$ -5520,$ipl(p)$ -5521,$a'(x)=a(1)$ -5522,$-g''(t) = w \delta_{\alpha_1}/\alpha_1 + (1-w) \delta_{\alpha_2}/\alpha_2$ -5523,$a=\infty$ -5524,$\bar P_g$ -5525,$\sum_i a_i=\sum_i a(X_i;X)=\rho(X)$ -5526,$a^\rho$ -5527,"$(\Omega,\mathcal F,\mathsf{P})$" -5528,$p_{\mathit{cl}}$ -5529,$h(1)=1$ -5530,$\Delta_{1}$ -5531,$p^+=\mathsf P(X\le q_X(p))$ -5532,$\rho=\dfrac{M}{l} = \dfrac{1-\lambda}{\lambda}$ -5533,$\rho_1$ -5534,$S_{\mathbf{v}}(a)$ -5535,$^\circledR$ -5536,"$g(s)=\min(g_1(s), g_2(s))$" -5537,$a(X)=\mu+4\sigma$ -5538,$\mathsf{Pr}(X < x) \le 0.4 \le \mathsf{Pr}(X\le x)$ -5539,$pq$ -5540,$\rho(X+Y)=\rho(X) + \rho(Y)$ -5541,$g'=2/3$ -5542,$\mathsf{VaR}_p(X) = \mathsf{E}[X] + \pi(X)\mathsf{SD}(X)$ -5543,$X\wedge a\Delta g$ -5544,$P=80$ -5545,$\rho(-H)=\rho(C)-1=-0.05$ -5546,$i$ -5547,$S_i(x)=\alpha_i(x)S(x)$ -5548,$X'(\omega) \le Y'(\omega)$ -5549,$B_t(\omega)=\omega_t$ -5550,$S(x)/P(x)$ -5551,"$\int_0^s g'(t)\,dt=\nu s$" -5552,$L_X(v)=l(v)$ -5553,$\mu=\log(\theta)$ -5554,$\mathsf{E}[X] + \pi\mathsf{Var}^+(X)$ -5555,$T_{(1)}=W$ -5556,$t\in\mathbb{R}$ -5557,"$(x_{1,i}, x_{2,k(i)})$" -5558,$\rho_g(X\wedge a)=\bar P(a)$ -5559,$g'>0$ -5560,$X\wedge a = \sum_i X_i(a)$ -5561,$t=-\log(1-p)$ -5562,$\mathsf{E}[X]$ -5563,$S_X(y)$ -5564,$n=2^m+k$ -5565,$\mu t$ -5566,"$1/2, 1/4$" -5567,$\mathsf{CX}$ -5568,$\sigma^2 = \sum \sigma_i^2$ -5569,$\iota=M/Q$ -5570,$\sup_\mathsf{Q} (\mathsf{E}_\mathsf{Q}[X] - \alpha(Q))$ -5571,$AB$ -5572,"$\displaystyle\int_0^a \beta_i(x)g(S(x))\,dx$" -5573,$\bullet$ -5574,$366.4$ -5575,"$\tilde X:[0,\infty)\to[0,\infty)$" -5576,$1-\alpha_i(t)S(t)$ -5577,$F_1$ -5578,$a=\mathsf{VaR}_p$ -5579,$(a'-X)^+$ -5580,$\mathbf {X_3}$ -5581,$\mathsf{FSD}$ -5582,$a={{a_x}}$ -5583,"$(0.2, 0.304)$" -5584,$e^{\mu_A}-1$ -5585,$\mathsf{E}[(X-\mathsf{E} X)^+]={(X-\mathsf{E} X)^+}_1$ -5586,$-\rho(X-Y)\le \rho(Y)-\rho(X)$ -5587,$\mathsf{Pr}(X > a) \le \epsilon$ -5588,$\mathsf{E}[Z_A\mid X]$ -5589,$B^c_k$ -5590,$-$ -5591,$d+l$ -5592,$0.1005$ -5593,$\mathsf{E}[X\wedge a]$ -5594,$r_i$ -5595,$=v_f \mathsf{E}_\mathsf{Q}\left[\dfrac{X_i}{X}(X\wedge a)\right]$ -5596,$\bar\delta a$ -5597,$c > 1/2$ -5598,"$\mathsf{PML}_{n, \lambda}(X)=\mathsf{PML}_{n, \lambda}$" -5599,$f(x)$ -5600,$h(1-p)=1-g(p)=1-\sqrt{0.9}=0.051$ -5601,$\mathsf{E}[X_i\mid X](x)$ -5602,$\mathsf{P}(B)=0.5$ -5603,$Gn$ -5604,$\mathcal F$ -5605,$g_2(s)=\sqrt{s}$ -5606,$v_f=1/(1+r_f)$ -5607,$B\subset \Omega$ -5608,$\bar S(x)$ -5609,"$s_j,g_j\in[0,1]$" -5610,$\mu=21.315$ -5611,$a_{gc}=P(X_{-1}(a_{gc}))+P(X_{0}(a_{gc}))+\mathit{MV}_{gc}(a_{gc})$ -5612,$X0=X_{0}$ -5613,$X=(X\wedge a) + (X-a)^+$ -5614,$(\mathsf{TVaR}_p - q(p))/(1-p)$ -5615,$X \preceq_n Y$ -5616,$\lambda_i$ -5617,$\mathsf{Pr}(X_n>\epsilon)\to 0$ -5618,$\mathsf{VaR}_{0.95}(X)=3395$ -5619,"$W_2=\sum_{t+d=2} Y_{t,d}$" -5620,$\mathsf{E}[X\mid X\ge \mathsf{VaR}_p(X)]$ -5621,$a\ge \sup(X)$ -5622,$\mathsf{E}[\log(X)]$ -5623,$a=Q+R$ -5624,$p/q-1=(p-q)/q>0$ -5625,$\alpha_1\ge \beta_1$ -5626,"$c_1=(c(1) + c(1,2)-c(2))/2$" -5627,$Z\in D\rho(X_0)$ -5628,$\cdots$ -5629,$\mathsf{E}[Xw(X)]/\mathsf{E}[w(X)]$ -5630,$d\bar S(a)/da$ -5631,$\mathsf{P}(X=X_j)=S_{j-1}-S_j$ -5632,$\omega'=0$ -5633,$\rho(Y)=g(pq)$ -5634,"$\phi(s) = (1-p)^{-1}1_{[p, 1]}(s)$" -5635,$S(x_i-)-S(x_i) =\mathsf{Pr}(X=x_i)$ -5636,$dg/ds$ -5637,$T_1 := X_{n+1} + \cdots + X_{N-1}$ -5638,$\displaystyle\int_0^\infty xd(g\circ F)(x)$ -5639,$\mathsf{E}[X \mid X \ge q^+(p)]$ -5640,$\mathsf{POS\ LOAD}$ -5641,$\mathsf{E}[X_iZ]=500$ -5642,$R_x$ -5643,$t\mapsto W_t$ -5644,$\mu+\lambda\sigma$ -5645,$\rho(X)\le\rho(0)=0$ -5646,$\kappa_2$ -5647,$k(i)$ -5648,$\mathsf{E}[X^k]\le \mathsf{E}[Y^k]$ -5649,$\chi( s ) = p - \log(s)$ -5650,$C$ -5651,$0\le x\le 1000$ -5652,"$\Omega=(0,1)$" -5653,$D(t)$ -5654,$\mathsf{Pr}[X > a]$ -5655,$w(x)=x$ -5656,$\mathsf{E}_{\mathsf Q}[\kappa_i(X)]$ -5657,$Z(X)$ -5658,$1 < x < 2$ -5659,$P/A$ -5660,$\mathsf{TVaR}_{p^*}(X_1)+\mathsf{TVaR}_{p^*}(X_2)=80$ -5661,$g(S(x))$ -5662,$s<0.20$ -5663,$M_i = \beta_ig-\alpha_iS$ -5664,"$[0,1,\dots,n]$" -5665,$X\Delta g(S)$ -5666,$\mathsf{E}[X^k] \le \mathsf{E}[Y^k]$ -5667,$\mathbf {M}$ -5668,$\mathsf Q\not\ll \mathsf P$ -5669,$q(p')=q(p)$ -5670,$100G$ -5671,$g(x)$ -5672,$c-1$ -5673,$\lambda$ -5674,$S(x)=\mathsf{Pr}(X>x)=1-F(x)$ -5675,$C^1$ -5676,$q^-(F(x))\le x$ -5677,$\mathsf{E} X + c\mathsf{E}[\vert X-\mathsf{E} X \vert^p]^{1/p}$ -5678,$\alpha_i(a)S(a)=\mathsf{E}[(X_i/X)1_{X>a}]$ -5679,$h(p)p$ -5682,$a=f=1$ -5683,$R_L=(L-P)/P$ -5684,$\omega\mapsto \psi=F(X(\omega))$ -5685,$\tilde M_i(a) = \bar P_i(a) - \mathsf{E}[X_i(a)]$ -5686,$r-i$ -5687,$\sigma=0.4$ -5688,$y$ -5689,$d>0$ -5690,$\mathsf{TVaR}_p(X)= \sum_i X_iZ_i / 10$ -5691,$F_0=2$ -5692,$\rho(X+c) = \rho(X)+c$ -5693,$X\ge X+Y$ -5694,$X > x$ -5695,$c(X(\mathbf v))$ -5696,$\beta-\alpha$ -5697,"$(1+t)(1), (1+t)(2),\dots,(1+t)(10)$" -5698,$q = 1-p$ -5699,$\rho_g(X)=g(s)$ -5700,$\kappa_1$ -5701,$\Delta_d=a_{d}'-a_{d}$ -5702,$_{gc}$ -5703,$\mathbf {p}$ -5704,$q(p')$ -5705,$f_i(x+y)=f_i(x)+f_i(y)$ -5706,$=\mathrm{MV}(T(X))$ -5707,$F(a-)=\lim_{x\uparrow a} F(x)$ -5708,$g(S(x))>S(x)$ -5709,$s_0/2^{n}$ -5710,$\alpha f/(1-g)$ -5711,"$a_i=a(X_i, p^*)$" -5712,$\mathsf{E}[X_1]=\mu$ -5713,$\Delta X=X_1$ -5714,$V(U)$ -5715,"$f(x)=\int_0^1 f'(tx)\,dt$" -5716,$9$ -5717,$\rho(X_0)\ge \mathsf{E}[X_0 Z_\epsilon]$ -5718,$S_{X_{-1}}(a)$ -5719,$g(S_4)=0.5$ -5720,$S(x)>0$ -5721,$\mathsf{E}[YZ\mid X]=Z\mathsf{E}[Y\mid X]$ -5722,$q(1)$ -5723,$x_{max}$ -5724,$a \ge 0$ -5725,$E[s|t]=0.08353$ -5726,$ag(S_{\mathsf{j}(a)})=(80)(0.5)=40$ -5727,$\rho(\tilde X\wedge a)\le a$ -5728,$\mathsf{E}[X](1+\pi)$ -5729,$\preceq$ -5730,$X'$ -5731,"$\mathsf{NORM,TI}$" -5732,"$X^+=\max(X,0)$" -5733,$h(s) < s$ -5734,$g(s)>s$ -5735,$1_{U a)=0$ -5750,$f(p)=(1-p)\phi'(p)$ -5751,$gc$ -5752,$\mathcal F_1$ -5753,$\kappa/x$ -5754,$r_0$ -5755,$=\exp(8.7103 + \Phi^{-1}(0.995)\times 1)$ -5756,$\rho(X)=\mathsf{E}_{\mathsf Q}[X]$ -5757,$(1-f)$ -5758,$1+V^{\ast}(1) > V(2)$ -5759,$\mathsf{E} X + c{(X-\tau)^+}_p$ -5760,$D = L^* - L$ -5761,$Z_\mathit{lift}$ -5762,$\pi_1$ -5763,$p<0.01$ -5764,$f(s)$ -5765,$\lambda X_1$ -5766,$\mathbf {X_{2}/X}$ -5767,$h(X)$ -5768,$\mathsf{E}[X\wedge a] = \dfrac{k}{\beta-1}F(a)-\dfrac{a}{\beta-1}S(a)$ -5769,"$\int |X_n(\omega) - X(\omega)|^p\, \mathsf{P}(d\omega)\to 0$" -5770,$\rho_2(X_i)=0.5$ -5771,$W0$ -5800,$\mu(\{\alpha \})=1$ -5801,$\mathsf{Var}(U)>\mathsf{Var}(X)$ -5802,"$Y_{t,d=0}$" -5803,$(l-X)^+$ -5804,"$\rho(X)=\max(\rho_1(X), \rho_2(X))$" -5805,$9/6$ -5806,$j=2$ -5807,$\rho_1(X_i)=1$ -5808,$D^n\rho(\cdot)$ -5809,$\mathsf{FATOU}$ -5810,$p_0$ -5811,$\bar P=\bar P_1+\bar P_2$ -5812,$\mathsf{CTE}_p(X)=(12+25)/2=18.5$ -5813,$f=1$ -5814,$U_X = F(X-) + V(F(X) - F(X-))$ -5815,$ROL = EL + \lambda (\mathit{EL} (1 - \mathit{EL})/w)^{1/2}$ -5816,$q$ -5817,$v_{res}$ -5818,"$\{1,\dots,n \}$" -5819,$\mathit{MV}_{gc}(a_{gc})=a_{gc}-P(X\wedge a_{gc})=5583.9$ -5820,$q_Y(U)$ -5821,$x^{\ast}$ -5822,$P = \mathsf{E}[X] + \pi \max(X)$ -5823,$g''(s)=-s^{-3/2}/4$ -5824,$d\tilde p=g'(S(x))f(x)dx$ -5825,$\mathsf{E}[(-Y)Z]\ge 0$ -5826,$N\times 1$ -5827,$F_X$ -5828,$\mathcal{G}\subset\mathcal{F}$ -5829,$\preceq_n$ -5830,$s \to 0$ -5831,$A\subseteq \Omega$ -5832,$r =$ -5833,$t=1$ -5834,"$(s_i,m_i)$" -5835,$F_X(x)\ge F_Y(x)$ -5836,$g'''>0$ -5837,$T=1$ -5838,$\mathsf{E}[X_ih(X)]=\mathsf{E}[\mathsf{E}[X_ih(X)\mid X]]=\mathsf{E}[\mathsf{E}[X_i\mid X]h(X)]=\mathsf{E}[\kappa_i(X)h(X)]$ -5839,$\mathsf x\mathsf{VaR}$ -5840,$\mathcal F_{\tau}$ -5841,$\mathsf{E}[X]=\int_0^\infty S(x)dx$ -5842,"$\mathbf X = (X_1, \dots, X_n)$" -5843,$\bar P_{act} = \bar P + F_0 > \bar P$ -5844,$x=\sum_i \mathsf{E}[X_i\mid X=x]$ -5845,$(f)$ -5846,$y^2 - 2\sigma y=(y -\sigma)^2 -\sigma^2$ -5847,"$[0,t]$" +0,$Xmi=X_{-i}$ +1,$0.00 | \$ +2,$\text{VaR}_{0.99}$ +3,$\mathsf{E}[XB]$ +4,$\alpha=2$ +5,$>1$ +6,$\rho^o_h$ +7,$S(M)=0$ +8,"$\rho_m(Y)=w_{p_1,p_2}\delta_{p_1} + (1-w_{p_1,p_2})\delta_{p_2}$" +9,$\int_0^1 \phi(p)dp=1$ +10,"$\mathsf{cov}(X_i,X)/\sigma_X$" +11,$0 < \alpha_1=\alpha^{-1} < 1$ +12,"$M(X_1, a)+M(X_2, a)=M(X_1+X_2, a)$" +13,$d=0.1/1.1$ +14,$V_2 = V_1 / 12$ +15,"$[-\epsilon, \epsilon]$" +16,$\mathcal{A}_y$ +17,$\rho(X) + c = \rho(X+c)\ge \rho(X) + \mathsf{E}[cZ]$ +18,"$250,000 with a layer \$" +19,$U_X > p$ +20,$\Delta \mathit{MV}_{gc}(a)$ +21,$\mathsf{E}=\mathsf{F}'\mathsf{F}$ +22,"$i=1,\dots,n_d$" +23,$\sigma\sqrt{T}$ +24,$t_* > t^*$ +25,$\mathbf Y$ +26,$m_Y(s)=\mathsf E[Y\mid S=s]$ +27,$\mathsf{E}[X_1\mid X_1+X_2=x]=mx/(m+n)$ +28,$g(S_t(a(t)))$ +29,$2^{-72}=1/4722366482869645213696=1/4.7\times 10^{21}$ +30,$\rho_L^E(t_1)$ +31,$\mathsf{C}$ +32,"$\psi=\psi_{X,r}(u)$" +33,$\beta_1g(S)dX$ +34,$\mu(\Omega)\not=1$ +35,$Z$ +36,"$(X_i, a_i)$" +37,$\sqrt{0.9}=0.95$ +38,$\rho(X_n)\uparrow 0$ +39,$a=30$ +40,$A - \mathsf E[A] \succeq_2 A_0$ +41,$\mathbf {\max a}$ +42,$q<\infty$ +43,$\beta=1$ +44,$X\mapsto \mathsf{E}[XZ]$ +45,$u^{(n-1)}$ +46,$0.5\mathsf{TVaR}_0+0.3\mathsf{TVaR}_{0.5}+0.2\mathsf{TVaR}_0.9$ +47,$N_a$ +48,$0\le N\le G$ +49,$11 million occurs a loss of $ +50,$\mathsf{E}(X^2)=$ +51,$X_1+X_2\sim 2X$ +52,$\bar\nu$ +53,$Z_a=q_a/p_a>1$ +54,$X=X(\omega)$ +55,$e_i/s$ +56,$\mathbf T=\mathbf M'\mathbf C^t$ +57,$X\le 0$ +58,$X_T$ +59,"$\bar P_{0,2}$" +60,$\mathsf{E}[\iota Q] = \mathsf{E}[\iota]\mathsf{E}[Q]$ +61,$P=(1+r)\lambda\mathsf{E}[X]$ +62,$\not\Rightarrow$ +63,"$\Theta_i^Y := D^n_{\mathsf{TVaR}_{1-s_i},X}(Y)$" +64,$S(x)=1-F(x)$ +65,$i^{\star}$ +66,$m=mg^{ak}/g^{ak}$ +67,$d=0.13$ +68,$Z=Z(X)$ +69,$\mathsf{E}[X\mid T=t]=\mathsf{var}phi(t)$ +70,$-2<\alpha<-1$ +71,$v=(1+i)^{-1}$ +72,"$(Alice)+(0,-2.5)$" +73,$26$ +74,$ \& $ +75,$\mathbf {\beta_{1}}$ +76,$p'= e^{l+t}/(1+e^{l+t})$ +77,$\ll 1/a_0$ +78,$R_2=C_2$ +79,$Z_t$ +80,"$F(\omega, x)$" +81,$y^*=\min(y)$ +82,"$(p, \mathsf{E}[X_i\mid X=q(p)])$" +83,$Z=W_1 + W_2$ +84,$R_i = P_i - U_i$ +85,$r_h=0$ +86,$E[X_1 | X]$ +87,$s_{j} = \sum_{k = j}^{M}p_{k}$ +88,$Z = \sum_j X_j$ +89,$X \mapsto X+k$ +90,$\rho(Z_2)$ +91,$U=F_X(X)$ +92,"$(Alice)+(0,-3.5)$" +93,$rm_0$ +94,$\check g(1-t)^2=(1-kt)^2=1-2kt+k^2t^2$ +95,"$\forall A\in\mathscr{F},\ \omega\to P(A, \omega)$" +96,"$X,X_1,X_2$" +97,$256$ +98,$\phi\equiv 1$ +99,$\mathit{RV}$ +100,$c_i=\displaystyle\sum_{i\not\in S\subset\Omega}\dfrac{|S|!(N-|S|-1)!}{N!}\times$ +101,$\mathsf{E}[X] = \int_0^1 q(p)dp$ +102,$(a-X_{\mathsf{j}(a)})$ +103,$\mathsf P(T^{-1}(A))=\mathsf P(A)$ +104,$A_i$ +105,$R(x)=pd_i+(v-\nu^*)\sqrt{pq}$ +106,"$k=0,1,\dots$" +107,$\int_0^\infty (1-F(x))dx=\int_0^\infty xdF(x)$ +108,$n \to \infty$ +109,$st=k$ +110,$\mathscr Z$ +111,$x_u$ +112,$\beta_i(k) = \mathsf{E}_q[\kappa_i(X)/X\mid X > X_k]$ +113,"$L_{p,p+\delta}$" +114,$\rho(X) = sup_Q \mathsf{E}_Q(X)$ +115,$z_i >\zeta$ +116,$\alpha / \beta$ +117,$\{A_i\} \subseteq \mathcal{M}$ +118,$\mathsf{MV}$ +119,$C_1+\cdots + C_n$ +120,${}^{[<75]}$ +121,$0\not\in\Theta_p$ +122,$(\beta_igS-\alpha_i S)/(gS-S)$ +123,$\int S$ +124,$\mathsf{Pr}hi^{-1}(0)=-\infty$ +125,$s_0/2^{n}$ +126,$\epsilon_1$ +127,$r_D$ +128,$\alpha_k^i$ +129,"$p \in [1,\infty]$" +130,$2^{-t+1}$ +131,$\mathbf {t+3}$ +132,$A\in\mathscr{F}$ +133,$a_1 = a(Y_{1})$ +134,"$b'_{X,r}(Y)=\sup \mathcal E'_{X,r}(Y)$" +135,$\rho(X)=-U(-X)$ +136,"$\mu,\nu$" +137,$p(1-\nu(p)-il(p))$ +138,$X_n=Y_1+\cdots +Y_n$ +139,"$\nu(dy)=\delta_0(dy)+1_{(0,\infty)}dy$" +140,$(m+1)$ +141,$\Delta Q_{gc}(a) = a_{gc}-P(X_{0}(a_{gc}))-a$ +142,"$\boldsymbol{j, p, S, \kappa_1, \Delta X, \Delta(X\wedge a)}$" +143,$X\le_{cx} Y$ +144,$\mathsf{E}(X\wedge a)$ +145,$<\mathsf{E}[X_1]$ +146,$X_i(X\wedge a)/X$ +147,$P=\nu(\bar S + \iota a)$ +148,$\eta\gg\zeta$ +149,"$\mathbb{Q}(F)=\R(X\times F),\ \forall F\in\mathcal{B}$" +150,$g(S_k)$ +151,$ to be the set of all sample points where the insurance event $ +152,$B = g^{b} \pmod{p}$ +153,$k\le m$ +154,$\tau =0$ +155,$\mathsf{Pr}(A\mid\mathscr{G})_{\omega_0}\ge 0$ +156,$X_1$ +157,$\sigma=0$ +158,$0 = x_0< x_1<\cdots < x_n < \cdots$ +159,$= (g-s)/(1-g)$ +160,$g(s)=\nu s+\delta$ +161,$\lambda X$ +162,$\rho_t$ +163,$\mathcal F_0\times \mathcal F_1$ +164,$\rho_c\leftrightarrow\mathcal Q$ +165,$q_C(p)=\inf C$ +166,$g(T(X)\mid \lambda)$ +167,$P^i = \mathsf E[Z\cdot X^i]$ +168,$q=0$ +169,$g^x(s^\star)$ +170,$N:=(1-\alpha)M$ +171,"$100,000~ has had a per-occurrence limit of \$" +172,$\tilde \rho$ +173,$X^m=X+X^a$ +174,$55+0.675\times 3.807=57.572$ +175,$\mathsf{E}_{\mathsf Q}[X_i(a)]=\mathsf{E}[X_i(a)g'(S(X))]$ +176,$m\to\infty$ +177,"$(fun3a.south -| fun3a.south east)+(\smlspc,-\smlspc)$" +178,$P=L+M$ +179,$g'S(X)$ +180,$g'(1-p) dp$ +181,"$(\Omega, \mathscr{F})$" +182,$(3) \rightarrow (9 = 9) \rightarrow (27 = 4) \rightarrow (12 = 12) \rightarrow (36 = 13) \rightarrow (39 = 16) \rightarrow (48 = 2) \rightarrow (6 = 6) \rightarrow (18 = 18) \rightarrow (54 = 8) \rightarrow (24 = 1)$ +183,"$\delta^{\star}, d$" +184,$a < kP$ +185,$\mathsf E[XY] \not=\mathsf E[X]\mathsf E[Y]$ +186,$X_1(z_1)$ +187,"$u : (a, b) \to \mathbf R$" +188,$Z\circ T_Z=Z$ +189,$\inf_t\ \{ t+(1-\alpha)^{-1}\mathsf{E}(Z-t)_+ \}$ +190,$r_N$ +191,$p=0.831588$ +192,$\kappa_i(x)=O(x)$ +193,$X_u$ +194,"$=E_{id}=E_{y,n}$" +195,$1-\beta_i(t)g(S(t))$ +196,$f(L)=(L-a)^+$ +197,$T_x\wedge n$ +198,$f(s)$ +199,$\dfrac{px^{x+p-1}e^{-1}}{\mathscr{G}amma(x+p+1)}$ +200,$X\ge m$ +201,$\omega_I < s$ +202,$x_00$ +206,$a_h=2-a_l<2-b_l=b_h$ +207,$\mu(dp)$ +208,$g(S(a))$ +209,$M^-$ +210,$a=5$ +211,$\int_0^\alpha$ +212,$pd_i=F(x)d_i$ +213,$-21.6$ +214,$Y_t=X_1+\cdots + X_{N(t)}$ +215,$R^2=0.87$ +216,$F_i = X_i(1 - (X\wedge a)/X)$ +217,"$a(\cdot, p)$" +218,"$178.7 billion of expenses. Commissions and brokerage accounted for 25.1 percent and claim adjustment services for 13.5 percent of the total. Taxes licenses and fees were 6.3 percent. However, their remaining expense items are broken out by expense category, such as employee salaries and benefits or advertising, rather than insurer value-add function. They also reported a cost of capital of 13 percent, applied to equity capital of $" +219,$\zeta=\zeta(G)$ +220,$\kappa'(\theta)=\tau(\theta)=\mu$ +221,$(X\wedge a)/X$ +222,$V=\sum_i V_i$ +223,$0\le p_0 \le p_1\le 1$ +224,$=dP(a)/da = g(S(a))$ +225,$S_0=1$ +226,$\kappa_i(x)= \mathsf{E}[X^i\mid X=x]$ +227,$X\mid\theta$ +228,"$X_{(1)},\dots,X_{(n)}$" +229,$\kappa(s)=\log\mathsf{E}[e^{sX_1}]$ +230,$\mathsf P X = \mu P_I X$ +231,$X=4$ +232,$1_{\{X>x\}}$ +233,$\mathsf{E}_{\mathsf Q}[Y]=\mathsf{E}[YZ]$ +234,$\zeta_t=0$ +235,$F_0$ +236,$\phi(s)=g'(s)=s^{1/\rho}/(s\rho)$ +237,$\Delta_j s_j \ge 0$ +238,$\mathbf {M_{1}\Delta X}$ +239,$YL$ +240,$g''_\tau(s)=g''(s)/(1+\tau)\le 0$ +241,"$u_1,\dots, u_n$" +242,$T(p)=r$ +243,$E\setminus F\in R$ +244,$\mathcal Q$ +245,$F_t$ +246,$m\in \mathcal M$ +247,$\phi_t$ +248,$V=V_1$ +249,$H_k(X) \le H_k(Y)$ +250,$p=F(\mathsf{E}(X))$ +251,$t\mapsto \rho(X) + t\mathsf{E}_{\mathsf Q_X}[Y]$ +252,$\tau a_i$ +253,$\tau(\theta)$ +254,$\nabla (\zeta NF) = \zeta\nabla NF$ +255,$p=p_a$ +256,$\mathsf{E}_\mathsf{Q_r}[X_j]$ +257,$\beta\to 1/\mu$ +258,$\dfrac{\partial \mu}{\partial \eta}=\dfrac{1}{g'(\mu)}$ +259,$L_X(s)/L_Y(s)\to\infty$ +260,$m(1+\frac{m^2}{a^2})$ +261,$\rho_g(X)=\mu+\lambda$ +262,$s^{th}$ +263,$L_0^a(X)$ +264,$t a)$ +331,$\rho(X+tY)$ +332,$X_1 =\mathsf{E}[X_1 \mid C]$ +333,$\{\mathscr{F}_t\}$ +334,$q(p)=25$ +335,$\sum e_i^2 / (n-k-1)$ +336,$(1-p)x_0$ +337,$\beta < \alpha$ +338,$(g(s)-s)/(1-g(s))=\iota$ +339,$L^\infty$ +340,$p(\nu_p-l_p)$ +341,$\kappa$ +342,$u_1 > 0$ +343,$ for estimates $ +344,$1/\sqrt{\lambda}$ +345,"$(s_n,g_n)$" +346,$U''(x)<0$ +347,$A\subset\mathbb{R}$ +348,$W_1$ +349,"$\{90,\dots,99\}$" +350,"$\nu = (1,1,\dots ,1)$" +351,"$-l, -l+1, \dots, 0, \dots, l-1, l$" +352,$U_i$ +353,$t_*=0.206< 0.5 < 0.544=t^*$ +354,$W \equiv T_{(1)}=min_k{T_k}$ +355,$q\to\infty$ +356,$\beta_i(a)/\alpha_i(a) < 1$ +357,$\mathsf{SD}(G')=\nu$ +358,$l/P$ +359,$\rho_g(X)=35.2$ +360,$q^-(p)=\inf \{ x \mid F(x) \ge p \}$ +361,$\mu^3$ +362,$\nabla_i(X) = \mathsf E[X^i\mid p>0.99]$ +363,${rl:.3f}+{rex:.3f}={rl+rex:.3f}$ +364,$\mathsf{Var}(s) =\mathsf{E}[s^2]$ +365,$s = 1-10^{-15}$ +366,$p\not=0.75$ +367,$Q_1=\rho(V_1)$ +368,$v_f(\mathsf{E}_Q[X_i] - \dfrac{\mathsf{E}_Q[X_i]}{\mathsf{E}_Q[X]}\mathsf{E}_Q[(X-A)^+])$ +369,$P=a - v(a-\mathit{EL})$ +370,$\omega\not=\omega'$ +371,$\alpha=0.7205$ +372,$|Z|$ +373,$\rho(X)=\int_0^1 q(1-g^{-1}(1-t))dt$ +374,$\bar P^a_i$ +375,$d(y;\mu) =2\left(\dfrac{y}{\mu} - \log\left(\dfrac{y}{\mu} \right) - 1\right)$ +376,$=\mathsf{E}[X_i / X]$ +377,$q=X$ +378,$P = 53.565 = v EL + d \max(L) = 46.6 / 1.15 + (0.15 / 1.15) \times 100$ +379,$\mathsf{E}[X_1\tilde Z]=\mathsf{E}[X_2\tilde Z]=500$ +380,"$17,500) than project A has (\$" +381,$X=\sum X_i$ +382,$p^+=\mathsf P(X\le q_X(p))$ +383,"$(Z_1,\dots,Z_k)$" +384,$\rho(-X_n)\downarrow 0$ +385,"$(t_1,t_2)$" +386,$Q=\nu a'$ +387,$\mathsf{TVaR}_p(X)=25$ +388,$\tilde X_1$ +389,"$(s,g(s))=(0.2,0.333)$" +390,$t=i+1$ +391,$V(\mu)=\kappa''(\theta)=\tau'(\tau^{-1}(\mu)=1/(\tau^{-1})'(\mu)$ +392,$X_t - X_{t-1}$ +393,$\mathsf{E}(Y(a))=\mathsf{E}(Y\wedge a)=\int_0^a S_Y(t)dt$ +394,$A_x\cap A_y\not=\emptyset$ +395,$\omega^l=(\omega^n)^j=1$ +396,$p={p.dists['tvar'].shape:.3f}$ +397,$\sum_{j=0}^{a} q_j = 1$ +398,"$\Omega_i^X := (1-T_{R-1})\rho_e^U(X,s_i)$" +399,$\sum_i h^i= 0$ +400,"$T:(M,\mathcal{B})\to ??$" +401,$P=\mathsf E_q$ +402,$\{ v_i \}$ +403,$\iota(?)=\dfrac{g(s)-s}{1-g(s)}$ +404,$\sigma^2=\sigma_A^2 + \sigma_L^2 - 2\rho\sigma_A\sigma_L$ +405,$N(t)$ +406,"$481,689,597 | 13.2% | $" +407,$\mathsf E[X] =\displaystyle\int_\Omega S(x)dx$ +408,$X_{n \wedge \tau}$ +409,$Z=\lambda X$ +410,$\bar P_n$ +411,$(g(s_0)-g_0)/s_0 = g'(s_0)$ +412,"$\lambda, \iota, \psi$" +413,$n^{-1}\sum_i a_i^2=1$ +414,$r(X+Y)=r(X) + r(Y)$ +415,$(1-g)$ +416,"$X(x_1, x_2)=(x_1+x_2)Y$" +417,$B(-X)=-A(X)$ +418,$(90 + 100) / 2$ +419,$\mathsf{Pr}hi^{-1}(0.995)=2.576$ +420,$M_{1}$ +421,"$767 billion of capital, part of $" +422,$\beta_i(x)/\alpha_i(x)0$ +509,$x < X(\omega)$ +510,$E[X_i/x | X = x]$ +511,$\mathsf E X= a_1s_1 + a_2 s_2$ +512,$p_i = \mathsf P(X=x_i)$ +513,$1-s_L \le p_1 \le p_2 \le 1$ +514,$\forall x$ +515,$t_1^* = 0.544$ +516,$g(1-p)>1-p$ +517,"$\sigma=13,108$" +518,$\mathsf{E}[X \mid X > q(p^*)]=103.333$ +519,$\mathbf {s_1}$ +520,$(T\lambda)\{\lambda_t\Omega=\infty\}=0$ +521,$00$ +560,$\alpha(p)$ +561,"$Z_i\sim \mathrm{DM}^*(\theta, \nu_i)$" +562,$S(x_{max})=0$ +563,$B=g^k\pmod p$ +564,$X0=X_{0}$ +565,$s_L>0$ +566,$X > k$ +567,"$(R,S)$" +568,$\lambda=1/\sigma^2$ +569,$g'(p)=\phi(1-p)$ +570,$ be the compound of $ +571,$\mathsf{Q}\sim \mathsf{P}$ +572,$\circ$ +573,$\rho=\rho_\phi$ +574,$v(E)$ +575,$\rho =$ +576,$\gamma>0$ +577,$a-L_0^a$ +578,$Y_2$ +579,$g_\tau(0)=0$ +580,"$(s_1, s_0]$" +581,$k=1.333$ +582,$\nu(p)=p$ +583,$r_y$ +584,$\alpha_\epsilon-\alpha$ +585,$F^{(2)}=[F^{(-2)}]^*$ +586,$(1+r)\lambda \mathsf{E}[X]$ +587,$\mathsf{Pr}r({\omega})=1/6$ +588,$m / s^2$ +589,"$100,000 investment income, \$" +590,$\{ a_n\}$ +591,$\omega=\exp(2\pi i/n)$ +592,$g(t)=h(t) / (1+l(t)) > 0$ +593,$-(1-s)g''(1-s) + g(0+)\delta_1 + \sum_s s(g'(s-)-g'(s+))\delta_{1-s} + g'(1)\delta_0$ +594,"$t=0,1,\dots, T$" +595,$\mathbf {s_0}$ +596,"$\mathscr{F}\times\Omega\to [0,1]$" +597,$g'(S(x)))=0$ +598,$1/t$ +599,$0 < p < p^\star < 1$ +600,$sgn(z)|z|^{1/(q-1)}/\|z\|_p^{q/p}$ +601,$\mu dt$ +602,$\mathsf{Pr}(A\mid\mathscr{G})_{\omega_0}$ +603,$\mu t$ +604,$\exp$ +605,$dS = \mathbf{n}dudv$ +606,$X_i\Delta g(S)$ +607,$\delta=\rho/\nu$ +608,"$\mathrm{ED}^*(\theta, \lambda)/\lambda$" +609,"$uv \in [0, s_1], u, v \in (s_1, s_0]$" +610,$l=\sum_i l_i$ +611,$g_0=g(0+)$ +612,$P_{\text{req}}$ +613,$g(s)=\sqrt{s}$ +614,$X_1 = fI + g + N$ +615,"$\alpha_i(\mathbf{x}, x)$" +616,$\mathcal{A} \subseteq \mathcal{C}$ +617,$g(s)=\displaystyle\int_{1-s}^1 \phi(t)dt = \displaystyle\int_0^s \phi(1-t)dt$ +618,$q_{\mathbf{x}}(p)=\mathsf{VaR}_p(X(\mathbf{x}))$ +619,$X_{-2}$ +620,$0\in\Theta$ +621,$|\hat F(f)|$ +622,"$\mathbb{R}=(-\infty, \infty)$" +623,$S_X$ +624,$\hat\rho_{\mathscr F_1}$ +625,$\iota a$ +626,$\beta(\beta-1)s^{\beta-2}(1-s)$ +627,$g_o:=h_R + (s_{R+1}-s_R)(1-h_R)/(1-s_R) \le g_{R+1} \le g_e:=h_R + (s_{R+1}-s_R)(1-h_R)/(s^\star-s_R)$ +628,$\alpha(\alpha+1)\beta$ +629,$k> 0$ +630,$(1-T_{R-1})\rho$ +631,$g_3(s)=s^{0.7}$ +632,$F_X(t)> F_Y(t)$ +633,"$[0, a]$" +634,$\rho(X^{\oplus n}) = n(v\mathsf E[X] + d\max(X))$ +635,$Z(\omega)$ +636,$>a$ +637,$B_s$ +638,"$i=1,\dots,N$" +639,${\lambda\alpha}/{\beta}$ +640,$L=\mathsf E[X\wedge a]$ +641,$\mathbf{r}\ge 0$ +642,$s_x^2 = (n-1)^{-1}\sum_i (x_i - \bar x)^2$ +643,$\mathcal{A}\otimes \mathcal{B}$ +644,$\rho(\lambda X + (1-\lambda)Y)\le \lambda \rho(X) + (1-\lambda)\rho(Y)$ +645,$\mathsf{cv}(X_i)$ +646,$\mathsf{TVaR}_{p^*}$ +647,$\sum \Delta g(S)_jX_j$ +648,$K_\theta$ +649,$\rho(1_A) = \rho(1) = 1$ +650,$g(s)=s/(1-p)\wedge 1$ +651,$\mathbb{Q}(A)=0$ +652,$v_1X_1(1)$ +653,$ρ$ +654,$-stable distribution with Lévy density $ +655,$\sup_\mathsf{Q} (\mathsf{E}_\mathsf{Q}[X] - \alpha(Q))$ +656,$[xf(x)] \times dx$ +657,"$50,000 | \$" +658,$R(x)=pd+(\delta^*-d)\sqrt{pq}$ +659,"$t\in[0,1)$" +660,$\sup_n E[|X_n|]<\infty$ +661,$\rho(X+\rho(X))=\rho(X)-\rho(X)=0$ +662,$\sigma=0.5$ +663,$x=2.15$ +664,${s:.3g}$ +665,"$j=n+1,\dots,m$" +666,$E[X\_{1}(a)]$ +667,$(\mathscr{F})$ +668,"$\phi_{\bar x}(Z)=\langle Z,\zeta_{\bar x} \rangle$" +669,$3.2 \times 10^{15}$ +670,$(-0.2092) \cdot (-0.5) = 0.1046$ +671,$\mathsf{E}[X]+\mathsf{var}(X)/\mathsf{E}[X]$ +672,"$\mathrm{Ga}(\mu, \sigma^2)$" +673,$10+0$ +674,$2\left( y\log\frac{y}{m} - (y-m) \right)$ +675,$\hat\rho(A_k)$ +676,$P(a) = S(a) + \delta F(a)$ +677,$\Delta_j s_j$ +678,$X(\mathbf{v})=\sum_i X_i(v_i)$ +679,$I_t$ +680,$S=1-F$ +681,$2.439 > 2 \times 1.204 = 2.408$ +682,$e^{\mu_L}$ +683,$\mathsf{E}[Y\mid X] = X$ +684,$\beta > 0$ +685,$A(-X)=-B(X)$ +686,$\mathbf {2\mathsf{VaR}_p(X_1)}$ +687,$k_1 uv$ +688,"$[0, 1]$" +689,"$150,000 rather than \$" +690,$\mathsf{Pr}r(\mathsf{CP}(\lambda)=0)=e^{-\lambda}$ +691,$\zeta_1=0.5$ +692,$\alpha > 1$ +693,"$(rep.south) + (0.5, -1.0)$" +694,$f_0$ +695,$\mathsf{P}_X$ +696,$K_{k+X}(t)=kt+K_X(t)$ +697,$\{\omega\in \Omega \mid (X\wedge a)=a \}$ +698,$P_g\not\ll P_X$ +699,$F_X(x):=\mathsf{Pr}r(X\le x)$ +700,"$X^n_t=1_{[1+T_n, \infty)}$" +701,$\bar\delta=\bar\iota/(1+\bar\iota)$ +702,$\partial a/ \partial x_i$ +703,$100.03 \$ +704,$80-11=69$ +705,$\rho(Y_n)=0=\rho(0)$ +706,$\tilde Z=\mathsf P(X=\sup(X))^{-1}1_{X=\sup(X)}$ +707,$g(S(a))-S(a)$ +708,$2X$ +709,"$\mathbf {g(S)\, \Delta X}$" +710,$n$ +711,$24/6=4$ +712,$\mathsf{TVaR}_{0.98}$ +713,$\rho(X)=\mathsf{E}_Q(X)=\mathsf{E}_Q(Y)+\mathsf{E}_Q(Z)$ +714,$F_n(x) := 1 - J_n(x)/J_n(0)$ +715,$K=3$ +716,$\zeta NF$ +717,$a = q_X(0.995)$ +718,$\mathsf{E}_{\mathsf{Q}}[Y \mid X] = \mathsf{E}[Y \mid X]$ +719,$X=X_h$ +720,"$s \in [s_n,1]$" +721,$\alpha/\beta^2$ +722,$X_t=\mu t + \sigma W_t$ +723,"$(\Omega, \mathcal{F}, \mathbb{P}, \{\mathcal{F}_t\}_{t \geq 0})$" +724,$\mathsf{E}[Z \mid X]$ +725,$\beta_1<\alpha_1$ +726,$X^{\oplus N}$ +727,"$650,000 − \$" +728,$\mathsf{E}[X_i/X \mid X=a]$ +729,$\lambda = \mathsf{Pr} \times \mathbb{Q}$ +730,$f_i(x+y)=f_i(x)+f_i(y)$ +731,$g(0+)=\delta$ +732,"$[0, -k]$" +733,$\mathsf{TVaR}_{0.95}(X)=\mathsf{E}[XZ]$ +734,"$[0,1,\dots,n]$" +735,"$g_n = \max \{f_1,\dots,f_n\}$" +736,"$\rho(X) = \max(\mathsf{E}_{\mathsf{Q}_1}[X], \mathsf{E}_{\mathsf{Q}_2}[X])$" +737,$\nu V_{G_1}(m/\nu)$ +738,$\rho \in \mathcal R$ +739,"$\displaystyle\int_0^{F(a)} \kappa_i(q(p))\,dp + a\alpha_i(a)S(a)$" +740,$\rho_m(X) = \mathsf{E}(X) + (\rho_m(X)-\mathsf{E}(X))$ +741,$xf(x)dx$ +742,$\beta_i(x)/\alpha_i(x)> 1 > S(x) / g(S(x))$ +743,"$Z=(0,0,0,0,0,0,0,0,5,5)$" +744,"$u\in D_n=\{ u \mid u^{(k)} \ge 0, k=1,\dots,n-1, u^{(n-1)}\text{ nondecreasing} \}$" +745,$4 \times 10^{-7} - 10^{-8}$ +746,$\mathsf{E}[X] + \pi \mathsf{SD}(X)$ +747,$X_i-X_{i-1}$ +748,$q^-(p) = \inf\ \{ x\mid F(x) \ge p\}$ +749,$m + ra = ks$ +750,$e_x=\sum_t {}_tp_{x}$ +751,$n-1$ +752,$-1$ +753,$\mathsf{TVaR}_{0.642}$ +754,$k_0>\ge 2$ +755,$\rho_i$ +756,$\delta_p/\nu_p = \rho_p$ +757,$\rho(-X+a)=\rho(-X) + a \le 0$ +758,$r = g^k$ +759,$MV = \bar Q + \mathit{NPV}_{\infty}$ +760,$\mathscr{G}$ +761,$Z_1=Z\circ T_A$ +762,"$(fun4a.south -| fun3a.west)+(-\medspc,-\medspc)$" +763,$dx=x_{i+1}-x_i$ +764,$\Delta_s = \phi_s$ +765,"$723,103,276 | -22.8% | $" +766,$\lambda(e^t-1)$ +767,$ from policyholder as premium and capital $ +768,$tE[\text{jumps}]$ +769,$\hat X_i=\hat x_i$ +770,"$\mathsf{Tw}_1(\mu, \sigma^2)$" +771,$\lambda\to (\alpha+1)/\alpha$ +772,$\rho^u_g$ +773,$\liminf \rho(X_n) \ge \rho(X)$ +774,$X=X^+-X^-$ +775,$H\le h$ +776,$X\le m$ +777,$\theta = \tau^{-1}(\mu)$ +778,$g(S(x)) - S(x)\ge 0$ +779,$Q\in \mathscr{P}$ +780,"$a,b$" +781,$\mathsf{E}[X_i\mid X=q(p)]$ +782,$x^\alpha$ +783,$\rho(X)=\max_i \rho_i(X)$ +784,"$f(s,t)=\lim_n f_n(s,t)$" +785,$0 \leq s < t$ +786,$S=X$ +787,$M_X(k)\le M_Y(k)$ +788,$\tF$ +789,$\eta_i$ +790,$f(y;\theta)=c(y) e^{\theta y +(-2\theta)^{1/2}}$ +791,$M(u) = c_1 / |u|^\alpha$ +792,$|X_t| \leq M$ +793,$|X_\alpha| > c$ +794,$\sum_i X_i(v_i)$ +795,$g(s) = t_{df}(t_{df}^{-1}(s)+\lambda)$ +796,${}_nE_x$ +797,$y\pm 2\sqrt{V(\mu)}$ +798,$\nu=\nu(p)$ +799,$e^{-\beta x}$ +800,$r-i$ +801,$1.1M - \$ +802,$R>C$ +803,"$(Alice) + (0,-2)$" +804,$v\mathsf E[X] + d\max(X)=\rho(X)$ +805,"$f(x_i;\theta) = g(\theta, \sum_i x_i)h(x_i)$" +806,$\alpha_1 < \alpha_2$ +807,$\mathcal{B}\subset \mathscr{F}$ +808,$\rho_{(g)}=\max\{\mathsf{E}(ZX) \mid Z\in \mathcal{A}\}$ +809,"$G(x,\omega)=c_k(x)$" +810,$\mathsf{CTE}_p(X)$ +811,$P=\sum_i P^i$ +812,$\kappa_j(x)/t>\alpha_j(x)$ +813,$\zeta=\Omega$ +814,$X = X_0 + M + A$ +815,$m_Y(s)\to\infty$ +816,$s <1$ +817,$\mathsf E[X_iZ]$ +818,"$d,v\ge 0,\ d+v=1$" +819,$\px=\sum_i \mathsf{Pr}(B\cap A_i)$ +820,"$\mathcal F_0=\{\mathsf{var}nothing, \Omega\}$" +821,$dx_i$ +822,$p^{* }$ +823,"$32,942 after four years, when the loss of \$" +824,$a=P+Q=\max(L)=100$ +825,$P_Q-\mathsf{P}[Q(a)]$ +826,$V_\lambda(m)=\mathsf{Var}(X_\lambda)=\lambda\mathsf{Var}(X_1)=\lambda V_1(m/\lambda)$ +827,$X_i = \mathsf E[X\mid I_i]$ +828,$p_a$ +829,"$\bar P_{t,0}$" +830,$4/3$ +831,$\rho(X_n)\to\rho(X)$ +832,$f(x)=e^x$ +833,$v''(x) > 0$ +834,$r_m$ +835,$\mathsf{E} X + c{X-\mathsf{E} X}_p$ +836,$\theta_d=0.60$ +837,$\rho(X)=\mathsf{SD}(X)$ +838,$\mathsf{E}[Z]=g(1)-g(0)=1$ +839,$4 \times 10^3 - 10^4$ +840,"$x\in\Omega=[0,1]^N$" +841,$kROL$ +842,$h=1+\lambda(f-\mathsf{E} f)$ +843,$\rho(X)\ge \mathsf{E}[X]$ +844,$E[X_i/X | X > x]$ +845,$Z\circ T_B=Z$ +846,$\displaystyle\int_0^\infty xd(g\circ F)(x)$ +847,$\mathscr{O}(\zeta)=\{\zeta \circ T \mid T\in \text{MPT}\}$ +848,$λ$ +849,$\mathsf{E}[f(X)] \le \mathsf{E}[f(Y)]$ +850,$s(M+1) := 0$ +851,"$10,000 per-occurrence deductible, a \$" +852,$E=hc/\lambda = 10^{-6}/\lambda$ +853,$S_n=X_1+\cdots + X_n$ +854,$q(u_i)$ +855,$e^{-x\beta}$ +856,${}_b\bar V=1-\bar a_{x+b}/\bar a_x$ +857,$\mathbf {Z_8}$ +858,$Z_{\mathit{lin}}$ +859,$\mathsf{E}[X_i\mid X](\omega)$ +860,$B\in \mathcal{B}$ +861,$E'$ +862,$d+l$ +863,$f(y;\mu)=\dfrac{1}{\sqrt{2\pi}} e^{-(x-\mu)^2/2}$ +864,$1 \times 10^{19}$ +865,"$(r,c)$" +866,$p>0$ +867,$^{}$ +868,$f(x) \mapsto kje^{\theta x} f(x)$ +869,$\mathsf{E}[X_0]=\infty$ +870,$ for different values of $ +871,$p\approx 0.99$ +872,"$\text{LOSS}=\text{LR}\,\text{PREM}$" +873,$\mathcal F_1=\sigma(I)$ +874,$1_{X>x}$ +875,$k_i=a_i/v_i$ +876,$p=0.9999$ +877,$r_O$ +878,$C=1-H$ +879,"$50 of the amount allowed on each claim in the classes under subsections 3, 4, 4-B, 5 and 6 must be deducted from the claim and included in the class under subsection 8. Claims may not be cumulated by assignment to avoid application on the $" +880,$\mathsf{Var}(Z)=\sigma^2\mu^p$ +881,$I/a + U/R > 0$ +882,$X-100$ +883,$\sigma(\mathcal{A})$ +884,$r_{qp}=\sqrt{pq}$ +885,$\rho(kX)\ge k\rho(X)$ +886,"$(s,g(s))$" +887,$t^\star=1/2$ +888,$Y\le X=0$ +889,$u_j(x)$ +890,$g\in\mathscr{P}$ +891,"$0,0,1,2,3,6,10,18,36,52$" +892,$p_j > p_i \ge p^*$ +893,$\mathsf{E}[B]=p$ +894,$\epsilon /2^{n+1}$ +895,"$12 billion in high quality short-term assets, and \$" +896,$c_a\mapsto c_\lambda$ +897,$m(1+m)(1+\frac{a+1}{a}m)$ +898,$q_{\mathbf{v}}(p)$ +899,$P=\mathsf{EPV}(L; r_I) + RA + SM$ +900,"$\mathrm{DM}^*(\theta, \sum_i \nu_i)$" +901,$X_1(10)$ +902,$x>\sup(X)$ +903,$\mathcal B_p$ +904,$R$ +905,$\int_0^1 e^{-x/\mu}dx=\mu(1-e^{-1/\mu})<\infty$ +906,$H + C = H + (1 - H)\equiv 1$ +907,$g\in\mathscr P$ +908,$1-g(S)$ +909,"$A_3,B_3$" +910,"$i=2,3,4,5$" +911,$\sin{}$ +912,$g(s)\le s$ +913,$\mathbf {X(a)}$ +914,"$(X_1,X_2)$" +915,$a_i = \mathsf E[X_i \mid X \ge a]$ +916,$E[X_0] = 0$ +917,$\sqrt{x}$ +918,$\pi(X) := \rho(X\wedge \alpha(X))$ +919,$A(1_{U>0.95})=A(1_{U\le 0.05})=g(0.05)=0.3017$ +920,$10^5 - 10^{12}$ +921,"$X:\Omega\to[0,\infty)\subset \mathbb R$" +922,$da > 0$ +923,"$\mathsf{cov}(Z,X)=\mathsf E[ZX] - \mathsf E[Z]\mathsf E[X]=\mathsf E[ZX] -\mathsf E[X]$" +924,$(1-t)^{th}$ +925,$K = 5.029$ +926,$\mathscr{G}amma\in\mathscr{G}$ +927,$\nu\ll T\mathsf{Pr}$ +928,$\hat y(m)=m-m\log(m)$ +929,"$^{\,2}$" +930,$\mathsf{E}[(X-\mathsf{E}[X])^+]$ +931,$d\mathsf{Q}=g'(1-p)dp$ +932,$x_6$ +933,$p=0.9$ +934,$x_0 \ge q^-(p)$ +935,$g_{R+1}$ +936,$10^7$ +937,"$G=\mathrm{cl}\{\, (\mathsf{E}_\mathsf{Q}[X_i], \mathsf{E}_\mathsf{Q}[X]) \mid \mathsf Q\in\mathcal Q \, \}$" +938,$\hat p>p$ +939,$\bar M(a)$ +940,$H[X]$ +941,$p^+$ +942,"$p, q \in G$" +943,"$\forall \gamma>0: \int_{|y|>\gamma} K_\delta(y)\,dy=1$" +944,$\|Z\| = \mathsf{E}(| Z|^p)^{1/p}$ +945,"$\mathsf{CP}(\lambda,X)$" +946,$t=0$ +947,$\rho\in \mathcal R$ +948,$Q_i = \sum_j Q_j^i$ +949,$X=\mathsf{E}[Y \mid \mathcal F']$ +950,$\mathrm{PV}_i = 0$ +951,$1\wedge \cdot$ +952,$\rho(X)=\mathsf{E}(q(U)\phi(U))=\mathsf{E}_Q(q(U))$ +953,$\mathsf{E}[X_i/X \mid X > a]$ +954,"$\psi(S,T)=1$" +955,$M_r$ +956,$\mathsf{E}_Q[\text{ceded loss}]$ +957,$Q_2dX$ +958,$\mathsf{E}[X\mid\mathscr{G}]$ +959,$S\mathsf{Pr}$ +960,$j(x)=1/x^{\alpha + 1}$ +961,$r_H$ +962,"$(\Omega, \cal F)$" +963,$H(\omega)$ +964,$f:\Lambda\to\Omega$ +965,$\alpha_Y \le \alpha_X < \alpha_Y + 1$ +966,$k = 1.4 + 1.8s$ +967,$V(\mu)\propto \mu^2$ +968,$Y = X + Z$ +969,$\rho(p)=\rho(F(x))$ +970,$\mathsf{TVaR}_{1-s_i}$ +971,$-\log(1-e^\theta)$ +972,$X(x) = \sum_i x_iX_i$ +973,$2n^2$ +974,$\forall y\in Y$ +975,$\mathsf{E}_\mathsf{P}[X_j]$ +976,$\Omega\to\Omega$ +977,$dP_g/dP_X$ +978,$x \times [f(x)dx]$ +979,$\zeta_1=\cos(\theta\pi/2)$ +980,$X=\sum_i \Delta_iX$ +981,$(r-i)Q_t$ +982,${}^{[>190]}$ +983,$3.1 - 100$ +984,"$1,376.27 \$" +985,$c > C$ +986,$L/(1-\pi)/(1-v)$ +987,$z_{p/2}\le 2z_p$ +988,$R_1(t)\ge P(1)$ +989,$\check S/\check M$ +990,$|\phi(-2\pi f)|$ +991,$\mathbf R_{\le 0}$ +992,$X={abs(x.X1+x.X2):.0f}$ +993,$\bar a_{n\!\urcorner}$ +994,$AVaR_\lambda(X)=\frac{1}{\lambda} \int_0^\lambda VaR_\alpha(X)d\alpha$ +995,$\theta=c=\nu^2$ +996,$\tilde F$ +997,$\tilde W$ +998,$(\partial \alpha/\partial x_i)q_X(\alpha)q_\zeta(1-\alpha)$ +999,$X(\omega) = 1/\omega$ +1000,$\theta=-e^{-\mu}$ +1001,"$25,734.38 - \$" +1002,$\mathsf{Var}(r(X))\ge 1/\mi(\mu)$ +1003,$a(\mathbf{v})=\mathsf{TVaR}_p(\mathbf{v})=\mathsf{E}[X\mid X > q_{\mathbf{v}}(p)]$ +1004,$(x)$ +1005,$t=s$ +1006,$D ⊃ C N$ +1007,$S\cdot dX$ +1008,$\mathsf{Pr}(A\mid\mathscr{G})(\cdot)$ +1009,$(g(S(a)) - S(a)) / (1 - g(S(a)))$ +1010,$L=1\times s=s$ +1011,$\subseteq$ +1012,$S(x)\leftrightarrow g(S(x))$ +1013,$S(a)$ +1014,$p'$ +1015,$LR_{\mathsf{PH}}$ +1016,$f(\square)\mapsto f(\square)-1$ +1017,$E[X_a(a)]$ +1018,$(X-a)^+\wedge y$ +1019,$f(\lambda)$ +1020,$10^{15} - 10^{19}$ +1021,$\rho(X) \le \rho(Y)$ +1022,"$N\sim\text{Mixed Poisson}(\lambda=0.08 \times (\text{vehicles insured}), cv=0.075)$" +1023,$P=\mathrm{EL} + r(a -P)=v\mathrm{EL} + da$ +1024,$1-p=q$ +1025,$\rho(X-Y)\le 0$ +1026,$\text{E}(G^3)=g$ +1027,"$a'_{X,r}(Y)=\inf \mathcal E'_{X,r}(Y)$" +1028,$P_1(t)=tR_1(t)$ +1029,"$D_n, D_n^*$" +1030,$A=\mathsf{VaR}_{1-\delta}(X)$ +1031,$t\mapsto W_t$ +1032,"$p_0,p_1\in(0,1)$" +1033,$r = 0.12$ +1034,$\alpha/\beta=\alpha\mu^{p-1}/(\alpha+1)\to 1$ +1035,$\mathcal D(X+c)=\mathcal D(X)$ +1036,$Y=1-X$ +1037,$X\le x$ +1038,$\{y_j\}$ +1039,$0\ge \lambda \le 1$ +1040,$a\le X\le b$ +1041,$M^{\tau_n}_t = M_{t \wedge \tau_n}$ +1042,$a=\max X$ +1043,$s_l = f / (n+1)$ +1044,$x\downarrow 0$ +1045,$k=1/(1-p)=(d+vs)/s$ +1046,$\rho(\cdot\mid \mathcal F_1)$ +1047,$se(b_i)$ +1048,$\mathsf{E}[U(Z)] = \mathsf{E}[U(Z) \mid A] = \mathsf{E}[U(X)]p + \mathsf{E}[U(Y)](1-p)$ +1049,$H_g(X) \le H_g(Y)$ +1050,"$\Sigma_{ij}=\rho_{ij}=\rho(Z_i,Z_j)$" +1051,"$\mathbf{w}=(w_1,\dots,w_n)$" +1052,$h(s)=\check g(s) = 1-g(1-s)$ +1053,$f(t|s)$ +1054,$g'(s)=\phi(1-s)$ +1055,$J_n(0) < \infty$ +1056,$t_0^* t]\not=\mathsf{E}[X_i\mid X> t]/\mathsf{E}[X\mid X>t]$ +1060,$\check g(1)=1$ +1061,"$(s_{R+1}, g_{R+1})$" +1062,$\exists x\ [\forall z\ (z=\emptyset)\rightarrow z\in x \wedge \forall x\in x\forall z\ (z=S(x)\rightarrow z\in x)]$ +1063,"$x_1, x_2$" +1064,$g=0$ +1065,"$\sigma_A,\sigma_L$" +1066,$3\mu(U)/2$ +1067,$2y(\mathrm{atan}(y) - \mathrm{atan}(m))+\log\left( \frac{1+m^2}{1+y^2} \right)$ +1068,$\liminf \rho(X_n) \ge X$ +1069,$\rho_t(X) = \rho_t(-\rho_{t+1}(X))$ +1070,$P(t)=P_0(t)+P_1(t)$ +1071,$e^{-\beta x}\approx 1$ +1072,"$a_1=1, y_1=3$" +1073,$\mathsf{E}(\theta)=1$ +1074,$Q^* > S$ +1075,$V(0)>0$ +1076,"$j=1,\dots,n=10$" +1077,$\mathsf{Pr}r(\{\omega \})= 1/100$ +1078,$q(F(x))$ +1079,$X_i(a')$ +1080,$\mathsf{Pr}r(X<0)=0$ +1081,$\partial P_/\partial x_i = 1-\partial D_i /\partial x_i$ +1082,$\mathsf{Pr}r(B=1)=p$ +1083,$\bar\delta$ +1084,$\rho(Y)=\rho_m(Y)$ +1085,$d \ge 1$ +1086,$L^p$ +1087,$\kappa_T$ +1088,"$[x,x+dx]$" +1089,$d={d:.3f}$ +1090,$\{\omega\}\in\mathscr{G}\ \forall\omega\in\Omega$ +1091,$1-F(q(p));\alpha)$ +1092,$\tilde p=\tilde p(p)$ +1093,$\mathsf{TVaR}_\pi(X) := X_{N-1}$ +1094,$dp=$ +1095,$\exp(n(e^\zeta-1))$ +1096,$\lim_{s\to 0} g(s)=r>0$ +1097,$10^6A_{75}=508676.91$ +1098,$gc$ +1099,"$[0,\infty)$" +1100,$\tau(\theta)=\tan(\theta)$ +1101,$\text{E}(G)=a\theta$ +1102,$Y=h(Z)$ +1103,$\mathsf{E}[X\cdot Z\circ T] < \mathsf{E}[X\cdot Z]$ +1104,$\bar Y=n^{-1}\sum Y_i$ +1105,$X_{-4}=x$ +1106,$F(q^-(p))=p$ +1107,$k=2$ +1108,$t_2=s^\star$ +1109,"$X_4, X_5$" +1110,$A = 8.14864$ +1111,"$\bar P_{0,1}$" +1112,$\{X\le a\}$ +1113,$S(x_{(j)})(x_{(j+1)}-x_{(j)})$ +1114,$\sum M_i\Delta X$ +1115,$x_1 < \cdots < x_n$ +1116,$8.617 \times 10^{8}$ +1117,$t_*s$ +1122,"$\mathsf{TVaR}_{p=1}=\mathrm{ess\,sup}$" +1123,$\lambda>0$ +1124,$\mathsf{Pr}r(B=1)=\mathsf{Pr}r(X>x)=p$ +1125,$\sim$ +1126,"$[a,b]$" +1127,$A_1 \supseteq A_2 \supseteq \cdots$ +1128,"$\rho(X) = \sup_{\zeta\in A} \langle \zeta, X \rangle$" +1129,$P(a) = \nu S(a) + \delta = \nu (S(a) + \rho)$ +1130,$V(\mu)=\mathsf{Var}(\mathsf{CP}_2)=\lambda(\mu/\lambda)^2x_2=\mu^2(x_2/\lambda)$ +1131,$b\!\urcorner$ +1132,$\mathsf{E}[v(X)] \le \mathsf{E}[v(Y)]$ +1133,$\rho(W_1\wedge a_0)-\bar P_0$ +1134,$\rho(X)=r \ge T_{m_2}(X)-v(m_2)$ +1135,$\kappa(\theta)=\log \int e^{\theta y}f(y)dy$ +1136,$1/(\alpha-1)=1-p$ +1137,$\nu=1-\delta$ +1138,$\mu_c$ +1139,$E[Xi | X=x]$ +1140,$S(X_0)$ +1141,"$Y = u_{1,2}(Y')$" +1142,$g(s)=(s/1-p)^\alpha\wedge 1$ +1143,$\mathbf{B}(t)$ +1144,$RY$ +1145,$\mathbf {t-1}$ +1146,$\mathsf{Var}(X) = \sigma^2 \mu^p$ +1147,$g(s)=s^{0.659}$ +1148,"$\bar Q_{0,0}$" +1149,$26 \rightarrow 2\times 4^2 + 2\times 4 + 1=41 \rightarrow 60 \rightarrow 83 \rightarrow 109\rightarrow\dots$ +1150,$k=n$ +1151,"$t=0.06405%. The prior has a material influence on the posterior mean. This makes the posterior mean a ""conservative"" estimate of $" +1152,$\mathbb{R}^n$ +1153,$\bar P = \bar S + \bar M$ +1154,$\phi(t) = g'(1-t)$ +1155,"$[0,x]$" +1156,"$P_X(a,b]=\mathsf P(X\in (a,b])=F(b)-F(a)$" +1157,$1_{U>s}$ +1158,$\mathsf{E}[X\tilde Z]$ +1159,$P/S-1$ +1160,$M:=\mathsf E[XZ]-\mathsf E[X]$ +1161,$\inf \Delta$ +1162,$H+C = H+ (1-H)\equiv 1$ +1163,"$a_i=a(X_i, p^*)$" +1164,$g'(t)<1$ +1165,$\rho^E(X)$ +1166,$a_i=x_i(\partial a/\partial x_i)$ +1167,$\mu+\mu\sqrt{\mu+2-2\sqrt{1+\mu}}$ +1168,$\square$ +1169,$\alpha\equiv 0$ +1170,$\mathsf{E}[1_A]$ +1171,$\kappa_i(x)=mt/(m+n)$ +1172,$L^i = \mathsf E[X^i]$ +1173,$p>0.5$ +1174,$\Delta Q_{ro}(a) = a-a_{ro}$ +1175,$q(p)=e^{\mu+z_p\sigma}$ +1176,"$4,617,916 | -21.4% | $" +1177,"$k=0,\dots, n$" +1178,$D(t)$ +1179,$k\ge n$ +1180,$V_1$ +1181,$\theta=\theta'$ +1182,$y=a$ +1183,$B(1_{U>0.95})=B(1_{U\le 0.05})=h(0.05)=1-g(1-0.95)=0.0203$ +1184,$\Delta_0 s_0$ +1185,$g_1(s)=d+vs$ +1186,$n+2$ +1187,"$(s_m,g_m)=(1,1)$" +1188,$\tilde R_i=R_i-\delta_i A_i$ +1189,$\bar\nu(a)=1/(1+\bar\iota(a))$ +1190,$\frac{1}{\sqrt{2\pi y^3}}e^{-1/2x}$ +1191,"$f:[0,1]\to\Omega$" +1192,$y_6$ +1193,"$\{L,\dots,m\}$" +1194,$\mathsf{Pr}(D ∪ C^c) = 1$ +1195,$\{X>a\}$ +1196,$m(1-\frac{m}{N})$ +1197,$\alpha(\cdot)$ +1198,$\dfrac{m}{p}\left(1+\dfrac{m}{p}\right)$ +1199,$g(0+)$ +1200,$\mathbf \Omega$ +1201,$\mathsf{Var}(s)=1/\sigma^2$ +1202,$t=0=1$ +1203,$\rho(Y)=g(pq)$ +1204,$P=\rho_{PH}(X)$ +1205,$\mathbf{x}'$ +1206,$\rho^a$ +1207,$(1-t)/t$ +1208,$1-gS$ +1209,$c=1.124$ +1210,$\mathsf{E}[X_i]/x$ +1211,"$\mu f=\int f\,d\mu=\int f(x)\mu(dx)$" +1212,$c(S)$ +1213,"$\mathsf{cov}(h^i, Y(\mathbf{X})) = \mathsf{E}_P[h^iY(X)]$" +1214,$\phi\in \mathcal E$ +1215,$\rho(X)=\mathsf{E}[X\theta]$ +1216,"$X',Y'$" +1217,$g'(S_{X\wedge a}(X\wedge a))$ +1218,$A_n \uparrow A$ +1219,$Y_m>x$ +1220,"$d=1,2,\dots$" +1221,$X(t)=(1-t)X_0 +tX_1$ +1222,$P=L+r(P-a)=vL + da$ +1223,$S=g(S)=1$ +1224,"$i=0,\dots,n-1$" +1225,$\mathsf{E}_{\mathsf{Q}}[X_i \mid X]$ +1226,$\kappa_i(t)=E[X_i \mid X=t]$ +1227,"$p\not\in\{0, 1, 2\}$" +1228,$a:=\lim_{m\downarrow 0} V(m)/m$ +1229,"$(Y,\mathcal{B})$" +1230,$\mathsf{VaR}_p(X)=q^-(p)$ +1231,$s(1)=s_3=1$ +1232,$s_j$ +1233,$\rho(X+\epsilon Y)-\rho(X)$ +1234,$\forall t\in E$ +1235,$\kappa_{2}$ +1236,"$(Alice)+(0,-2)$" +1237,"$X_-:=\max(-X,0)$" +1238,$B=X-A$ +1239,$\iff P +\rho_i(F_i) < \rho_i(X_i) \iff P < \rho_i(X_i) - \rho_i(F_i)$ +1240,$p_Y>0.5$ +1241,"$\mathsf{CP}(\lambda, X) = X_1+\cdots +X_N$" +1242,$1-\tilde p$ +1243,$. Insurance interpretation: $ +1244,$1- \nu F(x)$ +1245,$\mathsf{E}[Y]=1$ +1246,"$\langle \mu,Y \rangle - \langle \mu,X \rangle = \langle \mu, Y-X \rangle \ge 0$" +1247,"$Y_{t,d+1}$" +1248,$\mathsf{E}(X_ig'(S))$ +1249,$ is not differentiable at $ +1250,$P(x)/Q(x)$ +1251,$X_i(a) = X_i(X\wedge a) /X$ +1252,$w_i\ge 0$ +1253,$X_{-1}=C_1 + \cdots + C_N$ +1254,$\lim_{s\downarrow 0} g(s)=d$ +1255,$\omega\in Z$ +1256,$80=9.56 + 70.44$ +1257,$i\in I$ +1258,$\sum_i \kappa_i(x)=x$ +1259,$\mathsf{CoTVaR}(X_i)$ +1260,$cv=0.557$ +1261,$\rho(X)=\lim_n \rho(X_n)$ +1262,$0.01$ +1263,"$(0,0)$" +1264,$a=\mathsf{TVaR}(p^*)$ +1265,$(4-|t|)^q$ +1266,$500 = (\$ +1267,$\pi=1$ +1268,$\mathsf{E}[X]=\mathsf{TVaR}_0(X)$ +1269,$g'(1-s)=\phi(s)$ +1270,$\rho_{TVaR}$ +1271,$S(x)=\exp(-\int_x^\infty h(t)dt)$ +1272,"$\mathcal{M}\subset\mathscr{P}[0,1]$" +1273,$34.05$ +1274,$q_k$ +1275,$1/\sqrt{\alpha}$ +1276,$X > X_k$ +1277,$m(A)=0$ +1278,$m(x) = \nu S(x) + \delta = \nu (S(a) + \rho)$ +1279,$\epsilon>0$ +1280,$a_i'$ +1281,$p_B$ +1282,$c_k$ +1283,$D\rho_X(\cdot)$ +1284,$PICK ONE$ +1285,$\partial a/\partial v_1$ +1286,$q_X(p)=\mu+\sigma z_p$ +1287,$\rho_1(X)$ +1288,"$\xi(\phi^{-1}(t)\mid t) = 1,\ \forall t\in M$" +1289,$X_t=\mathsf{E}[Y\mid\mathscr{F}_t]$ +1290,"$\rho(G(\bar x))=\langle \zeta_{\bar x}, G(\bar x) \rangle$" +1291,$\liminf\rho(X_n)$ +1292,$q_B \le q_C$ +1293,$\mathsf{Pr}(B\mid\mathcal{A})(\cdot)$ +1294,$g(p)\ge p$ +1295,$a(t)=a(X(t))$ +1296,"$\mathsf{cov}(X_1, N | G = const_j) f_G(const_j)$" +1297,$\alpha<\omega_c$ +1298,$t_f$ +1299,$\nu\ll P$ +1300,"$L(e,t)$" +1301,"$\mathsf{CP}(\lambda,\text{gamma}(\alpha,\beta))$" +1302,$\prec_n^*$ +1303,$\forall X\ \exists U\ [\forall Y\ \forall x\ (x\in Y \wedge Y \in X)\rightarrow x\in U]$ +1304,$\mathsf{VaR}$ +1305,$a=q_p(\mathbf{x})$ +1306,$\mathsf{E}[\mathsf{E}[Z\mid X]]=\mathsf{E}[Z]$ +1307,$E_1\cap E_2 = \mathsf{var}nothing$ +1308,$\alpha(2)=0$ +1309,$\mathsf{E}[XZ]=\mathsf{E}[X\mathsf{E}[Z\mid X]]=0$ +1310,"$12,966,000 | 19.0% | $" +1311,$r-\mu$ +1312,"$(n,\lambda)$" +1313,$z_i \ge \zeta$ +1314,$x=q_{\mathbf{v}}(s)$ +1315,$\nabla^i(\phi(X)) = \mathsf E_q[X^i]$ +1316,$\theta\in\tilde\Theta$ +1317,$K=B^a=g^{ak}$ +1318,$w(Z)/\mathsf{E}[w(Z)]$ +1319,$p=0.271$ +1320,$q(p)=\mathsf{VaR}_{p}(X)$ +1321,$L=\mathsf{E}[X]$ +1322,$\mathsf{E}[X_i \mid X=q(1-g^{-1}(1-\tilde p))]$ +1323,$\bar F$ +1324,$X_t = X_{t-1} + \epsilon_t$ +1325,"$=\displaystyle\int_B^{\phantom{X}} \mathsf{var}phi \,d\mathsf{Pr}_T\quad$" +1326,$x_{i2}$ +1327,$\iota = (P-L) / (a-P)$ +1328,$+l$ +1329,$\forall A\in\mathsf{E}E$ +1330,$m(s) := (1-s)\wedge m(s)$ +1331,$p^*={p_star:.3f}$ +1332,$0280]}$ +1337,$P_j+Q_j<\Delta X_j$ +1338,$Z' - \bar\zeta_tZ$ +1339,$X_c$ +1340,$\Delta X$ +1341,"$\mathsf{cov}(X_i,X)$" +1342,$\mathsf{var}phi(X + Y) \le \mathsf{var}phi(X) + \mathsf{var}phi(Y)$ +1343,$2^{20}\approx 1$ +1344,$(\beta g(S))'(x)=-\mathsf{E}[X_i\mid X=x]g'(S(x))f(x)/x$ +1345,$SD(G')=\nu$ +1346,$w(z)$ +1347,$\mathsf{E}[XZ \mid \mathcal{G}]$ +1348,$\mathscr F_1=\sigma(I)$ +1349,$\check g((1-t)^2)=(1-k)+k(1-2t+t^2)=1-2kt+kt^2$ +1350,$(2.1) \cdot (-0.5) = -1.05$ +1351,$uv$ +1352,$\forall x[\exists y(y\in x)\rightarrow \exists y(y\in x \wedge \neg\exists z(z\in x \wedge z\in y))]$ +1353,$g(0.01)=0.1$ +1354,$E_\mathsf{Q}(X_i \mid X)$ +1355,$d(1-d)=v(1-v)=dv$ +1356,$\mathrm{EL}$ +1357,$\mathbf M$ +1358,$S\subset T$ +1359,$e_y$ +1360,$R^2=92\%$ +1361,$\gamma=0.633$ +1362,$\theta'$ +1363,$1.65 - 3.1$ +1364,$\mathsf{Pr}r(X < x) \le 0.4 \le \mathsf{Pr}r(X\le x)$ +1365,$A\in\mathcal{G}$ +1366,$g'(S(x)) = (1-p)^{-1}1_{x >\mathsf{VaR}_p(X)}$ +1367,$\sigma(X)>\sigma(Y)=0$ +1368,$\tau=0.5$ +1369,$\lambda$ +1370,$\mathcal{M}_\rho$ +1371,$p(a) = 1 - \nu F(a)$ +1372,$\beta((a-X)^+)$ +1373,$0\le f<1$ +1374,$g(0^+)>0$ +1375,$d=1/(1+r)$ +1376,$Z=\tilde X_2$ +1377,$10^{-11} - 10^{-15}$ +1378,"$(1-S(x), x)$" +1379,$(\mu_X-r_f) / \sigma_X \ge (\mu_Y-r_f) /\sigma_Y$ +1380,$\mathcal R^h$ +1381,"$\int_0^1 x^2 j(x)\,dx$" +1382,"$697.6 billion in 2016, $" +1383,$P_P$ +1384,$\beta<\alpha$ +1385,$\sigma_0=\sigma_1$ +1386,$\mathsf{TVaR}_0(X) \le c \le \mathsf{TVaR}_1(X)$ +1387,$Z=g'S(X)$ +1388,$1_{U_X\ge p}$ +1389,$\rho(X)=\sum_i \mathsf{E}_\mathsf{Q}[X_i]$ +1390,$O(n)$ +1391,$l(y;\mu)=\log(c(y))+y\tau^{-1}(\mu)-\kappa(\tau^{-1}(\mu))$ +1392,"$p\in [1, \infty]$" +1393,$iota^*$ +1394,$x_7$ +1395,$1/6$ +1396,"$Y\sim N(\mu, \sigma^2)$" +1397,$A=\mathsf E[X]N + A_0\succeq \mathsf E[X]N$ +1398,$X_{2}(a)$ +1399,$\mathsf{E}[Y_{d}]=\sum_{s>d} \mu_s$ +1400,$\Delta_s=g'(s-)-g'(s+)$ +1401,$(g_j-s_j)/(1-g_j)$ +1402,$ for all $ +1403,$250k and \$ +1404,$a^{\star}(X)$ +1405,$P_i = x_i\mathsf{E}_Q[X_i)] - D_i$ +1406,$2\nu$ +1407,"$(1-p, 1]$" +1408,$\mathbf {X_3}$ +1409,$\mathsf{Pr}r(X(\mathbf{x})>a) = S(\mathbf{x}; a)=S(a)$ +1410,$\mathscr{G}=\sigma(\mathcal{A})$ +1411,$\theta = C/(C+T)= C/N$ +1412,$Q(a) = (L-a)V(a) = (L-a)^+$ +1413,$\omega > \omega_I$ +1414,$x>a'$ +1415,$g(\omega_I)$ +1416,$\nabla p$ +1417,$Z_1$ +1418,$X(\omega_1)1$ +1428,$0.5a\}=0$ +1449,$e = P/C$ +1450,$\mathit{MV}(a)$ +1451,$t=-2$ +1452,$\mathsf x\mathsf{VaR}_p(X):=\mathsf{VaR}_p(X)-\mathsf{E}[X]$ +1453,"$t,t'$" +1454,$\phi_i(a)\mathsf{E}(Y\wedge a) = \mathsf{E}(X_i(a))$ +1455,$f(s) \ge s$ +1456,$a_c$ +1457,$s^*$ +1458,$\frac{1}{\sqrt{2\pi}\sigma}\exp(-y^2/2\sigma^2)$ +1459,$S>0$ +1460,$X-P$ +1461,$g_\mu$ +1462,$g'$ +1463,"$100,000, and a maximum premium of \$" +1464,$M_1dX$ +1465,$g(p)/p-1$ +1466,$\rho(X-a)=\rho(X)-a$ +1467,"$750,000,000). The deposit shall be made subject to the approval of the commissioner under those rules and regulations that he or she shall promulgate. The deposit shall be maintained at a deposit value specified by the commissioner, but in any event no less than one hundred thousand dollars ($" +1468,$k/n$ +1469,$t=-\log(1-p)$ +1470,$f(s) = \alpha(1-\alpha)(1-s)^{\alpha-1}$ +1471,"$\iota: x\mapsto (x, Tx)$" +1472,$0 < \alpha \le 1$ +1473,"$\nu \in\mathscr{P}[0,1]$" +1474,$a\ll \sum_i a_i$ +1475,$M = P - \mu_U= 0.505$ +1476,$a(W)=\mathsf{E}[W] + 4\sigma(W)$ +1477,$\iota^\ast$ +1478,$m_X(s)\to\infty$ +1479,$Q(a) = (X-a)V(a)$ +1480,$1/(1-\alpha)$ +1481,$\rho_e$ +1482,$0 \ge \rho(-X+a)=\rho(-X) + a \ge -\rho(X) +a$ +1483,$Z_a$ +1484,$\alpha=1.2$ +1485,$900 and one claim of \$ +1486,$\mathsf{E}[X_i ; X \le a]$ +1487,$p_n=\mathsf{Pr}r(N=n)$ +1488,$E[s]=0.1160$ +1489,$B(1/2)$ +1490,$D>L$ +1491,$\rho(-X)$ +1492,$X=X_c + X_n$ +1493,$m\ge 1$ +1494,"$[a,a+da]$" +1495,$\bar Q(a) =a-\bar P_g(a)$ +1496,$\rho(X)\ge -\rho(-X)\ge a$ +1497,$X^{(d)}_i(a):=(X_i-d)^+$ +1498,$n=9$ +1499,$F_0(x)(\omega) = F(x)$ +1500,$Q-0.2$ +1501,"$ makes the left tail thinner, the right tail thicker, and increases the mean. The effect on the right tail is manageable because it is thinner than a normal, @Zolotarev1986, @Carr2003a. As $" +1502,$a_Y=b_Y=r$ +1503,"$X_1,X_2$" +1504,"$x\in\Omega,L, t\in M$" +1505,$f\in L^1(\mathbb R)$ +1506,$a'(x)=a(1)$ +1507,$g(s)=0.1995$ +1508,$\sum_i X_i(a) = X\wedge a$ +1509,$\mathsf{E}_Q[X]>\mathsf{E}[X]$ +1510,$\iota^i$ +1511,"$F(p)=\mu([0,p])$" +1512,$\mathsf{E}_{\mathsf{Q}}[Y\mid X]\mathsf{E}[Z\mid X] = \mathsf{E}[YZ \mid X]$ +1513,$a\mapsto n=g^a\pmod{p}$ +1514,$F_2\prec_2 F_1$ +1515,$X_i(v_i)$ +1516,"$g'>0, g''<0$" +1517,$V(m) = m^3V^*(1/m)$ +1518,$0 \le \rho(0) = \rho(X-X) \le \rho(X) + \rho(-X)$ +1519,$a<0$ +1520,$a_p\approx o_p \approx q(1-(1-p)/\lambda)$ +1521,$3.2 \times 10^{18}$ +1522,$\mathcal{B}bb P$ +1523,$q(0)=0$ +1524,$a_i=\rho_i(\tilde X_i)$ +1525,$\rho(X-\rho(X))=\rho(X)-\rho(X)=0$ +1526,$t\mapsto v^t$ +1527,$n>0$ +1528,$r+v(m)$ +1529,$g^{ak}=(g^k)^a$ +1530,$\rho(xX)=x\rho(X)$ +1531,"$(fun5.north east)+(\medspc,\medspc)$" +1532,$\theta'=0.5$ +1533,$p = 1-g^{-1}(1-\bar p)$ +1534,$\alpha (1-s)^\alpha/(1-s)$ +1535,$\mathcal Q\subset\mathcal M(\mathsf P)$ +1536,$v_r = (1+r)^{-1}$ +1537,$Z\circ T_i$ +1538,$\mathit{ROE}(s) = r_f + Ck(s)$ +1539,$x<\mathsf{VaR}_p(X)$ +1540,"$P:\mathscr{F}\times M\to [0,1]$" +1541,"$(3,2)$" +1542,$q(0.75)$ +1543,$\Theta_s$ +1544,$\int c(y)dy = 1$ +1545,$Z(\omega)> 0$ +1546,"$\mathsf{cov}(m_X(S), m_Y(S))\ge 0$" +1547,$\lambda=$ +1548,$Z(1000)=(1-0)/(0.1-0)=10$ +1549,$g'_\tau(s) = g'(s)/(1+\tau)\ge 0$ +1550,$53.565-52.2=1.365$ +1551,$0.125 \cdot 8 = 1$ +1552,$l_p=\nu_p-\nu_{1/2}\sqrt{\bar p}$ +1553,$\rho(X_0)=\mathsf{E}[X_0Z]$ +1554,"$426,541,469 | | $" +1555,$b(\theta)=e^{-\kappa(\theta)}$ +1556,$r_h-\mu_L=r-r_L$ +1557,$\cap$ +1558,$a_2'$ +1559,$\mathit{AEL} = 0.01$ +1560,$q=q(p)$ +1561,$y^{\ast}-x^{\ast} < \epsilon$ +1562,$P=L + \delta (a - L) = L + \iota Q$ +1563,$AB$ +1564,$\mathcal Q_i(X)$ +1565,$p_1=0$ +1566,$\mathsf{Pr}r(\|U -\mu_U \| \ge k\sigma_U) \le k^{-2}$ +1567,$X(\omega)= 1-\sqrt{1-\omega^2}$ +1568,$\Theta^X$ +1569,"$u=s_1, v=s_0$" +1570,"$Y_{t,d}$" +1571,$\iota K$ +1572,$j(x)=1/x^{2}$ +1573,$\rho(0) = 0$ +1574,$\mathsf{Pr}r(X>x) = k x^{-\alpha}$ +1575,$q(p)=F^{-1}(p)=\mathsf{VaR}_p(X)$ +1576,$X \preceq_{sl} Y$ +1577,$Z'=ZT$ +1578,$A=X_1 + \cdots X_N$ +1579,"$\mathbf {g(S)\,\Delta X'}$" +1580,$\mathbf s$ +1581,$dF(x)$ +1582,$Z_\mathit{lin}$ +1583,$f$ +1584,$A\in\S$ +1585,"$\mathscr{G}amma = \{(\omega,\omega)\}$" +1586,$\eta_i >0$ +1587,$X(\mathbf{v}) = \sum_i X_i(v_i)$ +1588,$c=\sup_{0\le\alpha<1} \dfrac{\int_\alpha^1 \sigma_2}{\int_\alpha^1 \sigma_1}$ +1589,$\mathsf{E} X + \inf_x \{\alpha_1\mathsf{E}[(x-X)^+] + \alpha_2\mathsf{E}[(X-x)^+] \}$ +1590,$0=\mathsf{Pr}r(X<1)<\mathsf{Pr}r(X\le 1)=1/6$ +1591,"$ ""the standard way to obtain the $" +1592,$R(a)$ +1593,$> \mathsf{VaR}$ +1594,$g_0 \le 1-\alpha$ +1595,$D\rho_X(X_2)$ +1596,$\rho_{m'}(Y) < 89$ +1597,$L(X)=(1-p)^{-1}1_{X\ge x_p}(X)$ +1598,"$[s_1,1]$" +1599,$p^*$ +1600,"$(p, \mathsf{E}[X_i\mid X=q(1-g^{-1}(1-p))])$" +1601,$\mathsf{E}[X] +\lambda\mathsf{E}[(X-\mathsf{E} X)^+]$ +1602,$1/r$ +1603,$\mu^g$ +1604,$\rho(X) = \max_{\mathsf Q\in \mathcal Q} \ \mathsf{E}_\mathsf{Q}[X]$ +1605,$\lambda=\dfrac{1}{1+\rho}$ +1606,$\Omega_i$ +1607,$\mathbf {g_2(s)=s^{0.5}}$ +1608,"$T(y)=(y, y^2)$" +1609,$\kappa_T(y)=-\sqrt{-2y}$ +1610,$X = (x_{ij})$ +1611,$X\succeq Z$ +1612,$p_00$ +1632,$w=0.06405$ +1633,$\mathsf Q_k(B_k)=\mathsf{P}(B_k)/\mathsf{P}(B_k)=1$ +1634,$\mathcal{Q}=\mathcal{M}$ +1635,$\rho_t(X)$ +1636,$-1.350$ +1637,$X(\omega)=x$ +1638,$1-l-(\nu-l)=\delta$ +1639,"$^{\,3,5}$" +1640,$\sum (y_i-\bar y)^2$ +1641,$ +1642,$=\mathsf{E}[X]/(1-p^*)$ +1643,$\mu=r-\sigma^2/2$ +1644,$r_X=\mathsf{TVaR}_p(X)$ +1645,$x\neq 0$ +1646,$. If the insurer has a single insured there is no notion of default: the insured has purchased a policy covering losses up to a limit $ +1647,$g=f+\epsilon 1_B>f$ +1648,$\mathsf{E}[X_i(a)] = \mathsf{E}[X_i \mid X \le a]F(a) + a\mathsf{E}[X_i/X \mid X > a]S(a)$ +1649,$a-X$ +1650,$\rho_1$ +1651,$\rho(-H)=\rho(C)-1=-0.05$ +1652,"$\mathsf{CP}(\lambda_i, x_i)$" +1653,$\theta=0.5$ +1654,$P_i=\mathsf{E}_\mathsf{Q}[X_i]$ +1655,$\mathscr{G}(\omega)=\{\omega\}$ +1656,$\mathrm{NEF}(c)$ +1657,$q_2(t)=t^2$ +1658,"$(X_1,\dots, X_n)'$" +1659,$\eta_i-1$ +1660,"$485,000 which corresponds to a return period of 1.29 years. Line 2 has positive margins across all layers. Line 1 has a peak margin of 9.23% at a portfolio loss of $" +1661,$\mathcal{B}B(S)$ +1662,$A\in\mathcal F$ +1663,$p={strict_p}$ +1664,$g\circ S$ +1665,$\frac{m^2}{\lambda}(1+\frac{m}{\lambda})$ +1666,$\{y\mid c(y)\neq 0\}$ +1667,$\zeta=(1-p)^{-1}1_A$ +1668,$0\le x < 1/6$ +1669,$X\mid X>$ +1670,$2\square^2 + 2\square + 2$ +1671,"$A_k=X_{k,1} + \cdots + X_{k, N}$" +1672,$ro$ +1673,$s=0.047$ +1674,$\tilde p=1-(1-p)^{1/b}>p$ +1675,$Q(a) = 1 - P(a) = 1 - g(S(a))$ +1676,$I_1$ +1677,$\mathsf{E}(X_i \mid G=q)=:\mathsf{E}_q(X_i)$ +1678,$g(s) = \mathsf{Pr}hi(\mathsf{Pr}hi^{-1}(s) +\lambda)$ +1679,$\lambda=\sum_i\lambda_i$ +1680,$\phi(x)=-\int_x^1 (s-x)^{n-1}d\tau(s)$ +1681,$m(x)=S(x)+d_iF(x)+(v-\nu^*)\sqrt{F(x)S(x)}$ +1682,$\partial a/\partial v_i$ +1683,"$\mathbf {X\,\Delta g(S)}$" +1684,"$t\in[t^*,1]$" +1685,$p^2/4$ +1686,$e^{\mu_A}$ +1687,$a=(X\wedge a) + (a-X)^+$ +1688,$x_n(\mathrm{Po}(\lambda_n) - \lambda_n)$ +1689,$\mathsf{TVaR}_{p^\star}(X)=c$ +1690,$\mathsf{E}_{\mathsf Q}[Y]$ +1691,"$(rep.south) + (0.5, -1.85)$" +1692,$(X\wedge l)(\omega)=X(\omega)\wedge l$ +1693,$g(S(s))$ +1694,$\psi=1_\mathscr{G}amma$ +1695,"$\{g_j = g(s_j): j=1,...,m\}$" +1696,$X_2' = X_2+\cdots +X_n$ +1697,$\mathsf{TVaR}_p(X)$ +1698,$P\ge (\mathsf{E}[X] + \iota a)/(1 + \iota)$ +1699,$B=B(p)$ +1700,$f_X(x)$ +1701,$X(\omega)>a$ +1702,$\mathsf{ES}(X)=q(p)$ +1703,$A = X_1 + \cdots + X_N$ +1704,$\rho(A_k)\ge \mathsf{E}[A_k] = k\mathsf{E}[N]$ +1705,$\bar\iota=\iota$ +1706,$g_2(s) = 2s/3 + 1/3$ +1707,$X_1+X_2=X$ +1708,$\bar\theta_s$ +1709,"$Ca(Mg,Fe)Si_2O_6$" +1710,$x>\mathsf{VaR}_p(X)$ +1711,$0<\omega_I<1$ +1712,"$x_{1,2}$" +1713,"$(-1,-1/2)$" +1714,"$|, inf$" +1715,$\nu+\delta=1$ +1716,$X_2=0.3 + 0.7X_2'$ +1717,$p/\mathsf{E}[p]=p(1+r_f)$ +1718,$a=a(x)$ +1719,$\iota_k=M_k/Q_k$ +1720,$1\wedge s/(1-p)$ +1721,$Q=\nu (a-L)$ +1722,$\mathsf{E}_{\mathcal{B}bb{Q}}[Y\mid \mathcal{G}] \mathsf{E}[Z \mid\mathcal{G}] = \mathsf{E}[YZ\mid \mathcal{G}]$ +1723,$\mathbf {X_{2}/X}$ +1724,$a_i = a(X_i; X)$ +1725,$y_c$ +1726,$R_1(t) \$" +1751,$0.354 \cdot 8 = 2.83$ +1752,$R^2=0.86$ +1753,$\mathbf v$ +1754,$K_Q=19.473$ +1755,$M=g(S)-S$ +1756,$0.87$ +1757,"$\mathbf {\omega_1},\dots,\mathbf {\omega_n}$" +1758,$83.3=100/1.2$ +1759,$X=X_i + \hat X_i$ +1760,"$\rho_1, \rho_2$" +1761,$10 million occurrence limit for ABC's fleet of 800 power units. You manage the captive's net exposure through a combination of per occurrence reinsurance and an aggregate stop loss. Your goal is buy reinsurance so the 99th percentile of your net losses is less than $ +1762,"$s, s_i$" +1763,$f\to\infty$ +1764,$M(t):=\mathsf{E}[e^{tX}]$ +1765,$\rho(\tilde X)$ +1766,$1-F_i(x) = x^{-\alpha} L_i(x)$ +1767,$\mathsf{Pr}r(B=0)=1-p$ +1768,$\phi(1-p)=g'(p)$ +1769,$g''(s)<0$ +1770,$Y\succeq Z$ +1771,${{}_tp_x} \mu_{x+t}$ +1772,$X^{}$ +1773,$\rho=\rho_g$ +1774,$X_n(\omega)=n$ +1775,"$\mathsf{LI, COH}$" +1776,$1 towards claims if $ +1777,$\int_0^1 dp$ +1778,$q_{\tilde X}$ +1779,$g_{\min{}}$ +1780,$\delta=0$ +1781,$0x\}$ +1784,$m(1+\frac{m}{p})^2$ +1785,$V=m(L(1+e)P+rS) + (eL+\rho S)$ +1786,"$2,500,000 | \$" +1787,$0 x\}$ +1797,$\lambda=0.5$ +1798,$dF=-d(g\circ S)=$ +1799,$\| X_n \|_\infty \le 1$ +1800,$\beta_i(x) / \alpha_i(x) > 1 > S(x) / g(S(x))$ +1801,$X=X_1+X_2$ +1802,$\mathsf{E} X= a_1s_1 + a_2 s_2$ +1803,"$\zeta=(1,2,3,4,5)$" +1804,$\hat F$ +1805,$\rho(Z)=\int_0^1\eta(\tau)\mathsf{VaR}_\tau(Z)d\tau$ +1806,"$ xx billion, of which California workers compensation deposits account for $" +1807,$R^2_a$ +1808,$p\mapsto e^l/(1+e^l)$ +1809,$\epsilon_+$ +1810,$0.7 \ge p < 0.8$ +1811,$\int xdF(x)=\int xf(x)dx$ +1812,"$c(1,2) - c(2)$" +1813,$n=20$ +1814,"$[x, y]$" +1815,$g(s)+g'(s)(1-s)\ge 1$ +1816,"$\mathcal E'_{X,r}$" +1817,$Y_0$ +1818,$|exag?_[Xt].*(?0$ +1867,$Q\in\mathcal{Q}$ +1868,$\sum_i P_i(a)=P(a)$ +1869,$X\le l$ +1870,$\mathsf{E}[Y]=\mu=np$ +1871,$(\alpha_i S)'(x)=-\mathsf{E}[X_i\mid X=x]f(x)/x=-\kappa_i(x)f(x) / x$ +1872,$P'_1(1)<0$ +1873,$d\theta/d\mu=1/V(\mu)$ +1874,$q_Z$ +1875,$\bigtimes_i X_i$ +1876,$\lambda = \dfrac{E( r_{M} ) - r_{f}}{\sigma_{rM}}$ +1877,$\mathsf{E}[Z \mid X]\preceq_2 Z$ +1878,$\iota = M / Q = \delta / \nu$ +1879,$\delta(F(x))=\delta$ +1880,"$\{\dots,s_k,s_{k+1},\dots\}$" +1881,$f = J/2^N$ +1882,$X(r)=1/r$ +1883,$L>d$ +1884,$\alpha < 1$ +1885,"$\theta,\dots$" +1886,$>0$ +1887,$\tilde \rho(X)=\mathsf{E}(X) + \inf_t \rho(X-t)$ +1888,$\test$ +1889,$-br-v=0.258$ +1890,"$(3,3)$" +1891,$\Xi$ +1892,$-X_2$ +1893,"$(\mathsf{Pr}_X, \sigma(T))$" +1894,$10^{16}$ +1895,$K_i=\dfrac{p_i+R_i(r_i+a_i)}{1-a}$ +1896,$X^{n}$ +1897,$il$ +1898,$P_1$ +1899,$e^t$ +1900,$-\frac{1}{2}$ +1901,$($ +1902,$10^2$ +1903,$\mathsf{E}[u(R - X)]=0$ +1904,$k>\max(N)\max(|X|)$ +1905,$Z\in\mathcal Q$ +1906,$u_1>0$ +1907,$0\le R^2\le 1$ +1908,$X-\sum f_i(X)$ +1909,"$\displaystyle\int_\Omega g(X(\omega), \omega)\mathsf{Pr}r(d\omega)$" +1910,$q \leq p$ +1911,$a_1+a_2$ +1912,$\tau=0$ +1913,$\displaystyle\int_0^1 \mathcal{A}VaR(p)\mu(dp) = \displaystyle\int_0^1 \dfrac{1}{1-p}\displaystyle\int_{p}^1 q(s)ds \mu(dp) =\displaystyle\int_0^1\displaystyle\int_0^s \dfrac{\mu(dp)}{1-p}q(s)ds=\displaystyle\int_0^1\displaystyle\int_{1-s}^1 \dfrac{\mu(dp)}{p}q(s)ds=\displaystyle\int_0^1\phi(s)q(s)ds$ +1914,$})=1-\mathsf{Pr}r(\text{No events $ +1915,$e^{-k(x/\alpha)^{\alpha/(\alpha-1)}}a$ +1933,$X(t)$ +1934,$1-\mathit{EL}$ +1935,$f(\alpha):=\mathsf{E}[X^\alpha-Y^\alpha]$ +1936,$\delta A_i$ +1937,$\partial\rho(X)=\{\zeta\}$ +1938,$\rho(X_{-1}\wedge a)$ +1939,$Q(a)=\nu N(a)$ +1940,$\lambda\ge 0$ +1941,$\mathsf{E}\_\mathsf{Q}[X]$ +1942,$\phi(\tilde r)^2$ +1943,$X_t=a_t$ +1944,$x - y$ +1945,$m=L+2$ +1946,$1.0$ +1947,$s>p^*$ +1948,"$s=8.75, 50, 83.3$" +1949,$\alpha=\infty$ +1950,$F(a)=p$ +1951,$\lim_{s \to 1}{\mathsf{E}[ r_{s} ] = - 1}$ +1952,$m^*$ +1953,$(-1)^nf^{(n)}(x)<0$ +1954,"$X_{0,t}$" +1955,$Z_1=q_Z(U)$ +1956,$J(0)=\infty$ +1957,$\mathsf{E}[X\mid \mathcal F_t](\omega)=\sum_{i \le t} \omega_i/2^i+2^{-(t+1)}$ +1958,$\rho(X) = s\mathsf E[X] + d\max(X)$ +1959,"$[0, 2\pi$" +1960,$\mathsf{TVaR}_{p^*}(X)$ +1961,$\hat\rho(A_k) =\rho(\rho((X+k)^{\oplus N})) = \rho(\rho(X^{\oplus N})+kN)= \hat\rho(A_0) + k\rho(N)$ +1962,$n-k-1$ +1963,$0 = s_0 < s_1 < s_2 < s_3 = 1$ +1964,$X=1_B$ +1965,$s\approx 0.15$ +1966,$f_G$ +1967,$\mathsf{E}[X_{d}]$ +1968,"$(-\infty,0)$" +1969,$\mathsf{E}[X_T]=\mathsf{E}[X_0]$ +1970,$\displaystyle\int_0^\infty xdF(x)$ +1971,$\text{AEP}(L)=1/y$ +1972,"$e',s', r', Q$" +1973,$S=\bigcup_j D^n_j$ +1974,$d \bar S/da$ +1975,$f(p)=(1-p)\phi'(p)=-(1-p)g''(1-p)$ +1976,$\mathsf{E}\_\mathsf{Q}[(X\wedge a)(a)]$ +1977,$\nabla \zeta=0$ +1978,$\rho(U)$ +1979,$\mathsf{VaR}_p(X) = \mathsf{E}[X] + \pi(X)\mathsf{SD}(X)$ +1980,$f(y;\theta)=(-\theta)e^{\theta y}$ +1981,$\mathsf{E}_{\mathsf Q}[X_i\mid X\le a](1-g(S(a))) + a\mathsf{E}_{\mathsf Q}[X_i/X\mid X >a]g(S(a))$ +1982,$F_g(x)$ +1983,$g(s)=1-(1-s)^{{{p}}}$ +1984,$U(1)=1$ +1985,$\bar P_x:=\bar A_x / \bar a_x$ +1986,$\alpha_2 S$ +1987,$2^8-1=255$ +1988,$f(z)=E[e^{izX}]$ +1989,$\mathsf{TVaR}_p(X)=r$ +1990,$P(A | \mathcal{G})(\omega)$ +1991,$-\phi(d^*)<0$ +1992,$l'>l$ +1993,$10^{-12}$ +1994,"$\mathsf{TVaR}_1(X)=\mathrm{ess\,sup}[X]$" +1995,$\int_0^\infty x^{-1}e^{-x/\mu}dx=\infty$ +1996,$F(a+)=\lim_{x\downarrow a} F(x)$ +1997,$P^T(F\mid t)$ +1998,$h(s)=1-g(1-s)$ +1999,$g(s) = vs + d$ +2000,$\mathsf E_g$ +2001,$N=kg m/s^2$ +2002,$f''(\omega)=2s/\omega^3 >0$ +2003,$r=0.025$ +2004,$\rho_\phi=\mathsf{E}$ +2005,$\rho(X)=\int_\Omega X(\omega)\theta(\omega)dP(\omega)$ +2006,$ = \mathsf{E}_{\mathsf{Q}}[X_i\mid X= x]$ +2007,$C(t)$ +2008,$\mathbf {gS}$ +2009,$p_{i^*} < p^*\le p_{i^*+1}$ +2010,$P = 1.5$ +2011,$X = \sum_t D_t$ +2012,$\gamma$ +2013,"$p\in (0, 1)$" +2014,$X_t - X_s$ +2015,$\mathsf{Pr}hi(z)$ +2016,$0.999999999$ +2017,$X_t>a_t$ +2018,$a_x=4$ +2019,$\mathsf{Pr}(A\cap B)=\mathsf{Pr}(A)$ +2020,$U + (U+x)$ +2021,$B(b)>0$ +2022,$D\subset\mathbf C$ +2023,$x\mapsto x^k$ +2024,$P^T(A\mid t)$ +2025,"$\mathsf{MON,NORM}$" +2026,$P \le \dfrac{\mathsf{E}[U]}{\lambda} \approx \dfrac{\mathsf{E}[X]}{\lambda}$ +2027,$-\rho(-X)\le \mathsf{E}[X]$ +2028,$\bar P_t$ +2029,$V'$ +2030,$\log_{10}$ +2031,"$k=0,\dots,n-1$" +2032,"$f:[0,1]\to[0,1]$" +2033,$1-s_j$ +2034,$1=ps_g + (1-p)s_b$ +2035,$a(X_i;X)\le \rho(X_i)$ +2036,$M(a)=g(S(a))-S(a)$ +2037,"$397,308,200** payable in installments over the three-year life of the contract. The minimum payments remaining under this contract as of December 31, 1999 are $" +2038,$t<1<0.5z_p)=p$ +2118,$\int_{\mathsf{E}[X]}^\infty (x-\mathsf{E}[X])^2 f(x)dx$ +2119,$\theta_1$ +2120,$g'(S(x))>1$ +2121,"$p:\Omega\times \mathscr{F} \to [0,1]$" +2122,$U(\omega)=\omega=0.\omega_1\omega_2\dots$ +2123,$0.0476/(1-0.0476)=0.05$ +2124,$\rho:\mathcal{X}\to \mathbb{R}$ +2125,$\mathsf{E}[L]/\mathsf{E}_Q[L]$ +2126,$\Delta X_k = X_{k+1}-X_k$ +2127,$\omega = 1-\sqrt{1-s}$ +2128,$p_1+p_2=1$ +2129,$\bar P_1$ +2130,$V(c)=0$ +2131,"$k=(0.04, 0.4)$" +2132,$\rho(X)>\max(X) g(0+)=\infty$ +2133,$q_{X}(p)=\sqrt{2}\mathsf{Pr}hi^{-1}(p)$ +2134,$\mathsf{E}(X)=\sum_i x_i$ +2135,"$(p,t)$" +2136,$m=n$ +2137,"$x+y\wedge aX =\min(x+y,aX)$" +2138,$\mathsf{E}[X_i \mid X=a]$ +2139,$(\delta^*-d)\sqrt{FS}$ +2140,$A=X_1+\cdots X_N$ +2141,"$j=5,6$" +2142,$\mathsf{ABOVE}$ +2143,$Q=1-g(S)$ +2144,"$\rho_1,\rho_2$" +2145,$2.0-3.0$ +2146,$-8$ +2147,$\text{E}(G)=M_G'(0)=1$ +2148,$\iota(0.5)=\iota^{\star}$ +2149,$0\le q\le 1$ +2150,$\rho(A) + \rho(B)$ +2151,$P(A)=1-\alpha$ +2152,$\frac{\sum_{y=1}^N p^y \phi(X^y)}{\sum_{y=1}^N p^y}$ +2153,"$\rho(X) = \sup_{\mu\in \mathcal{A}} \langle \mu, X \rangle$" +2154,$\omega\in J$ +2155,$\int_A Xd\mathsf{Pr}$ +2156,$a(X_i; X)\le \sup(X_i)$ +2157,$\hat\rho_N$ +2158,$U(x)$ +2159,$1/\sqrt{\lambda_i}$ +2160,$X \mapsto kX$ +2161,$\bar P_i(a)$ +2162,$(dW_t)^2=dt$ +2163,$f(x)$ +2164,$r(X)$ +2165,$\mathsf{Pr}r(X=2)=0.5$ +2166,${}^{[>81]}$ +2167,$p_0 \le p^\ast \le p_1$ +2168,$1000(1+t)$ +2169,$\rho(X)=\mathsf{E}[Xg'(S(X))]=\mathsf{E}[\sum_i X_i g'(S(X)))]=\sum_i \mathsf{E}[X_ig'(S(X))]$ +2170,"$g(0)=0,\ g(1)=1$" +2171,$\Leftrightarrow$ +2172,$q(1-p)$ +2173,$\phi_{m}^o$ +2174,"$\theta\in[0,1]$" +2175,$g(s)=s^{0.3}$ +2176,$\mathsf{E}_{\mathsf Q}[X\wedge a]$ +2177,$\omega'=\omega$ +2178,$\bar X\ge 0$ +2179,$1-g(s)$ +2180,"$X_{1,0}=\cdots=X_{m,0}=X_0=0$" +2181,$(g)$ +2182,$\alpha<0$ +2183,$\rho(\lambda X) \le\lambda\rho(X)$ +2184,$C^1$ +2185,"$T:(\Omega,\mathscr{F})\to (E,\mathsf{E}E)$" +2186,$\Delta\mathit{MV}$ +2187,$\ge P(1)$ +2188,$d>0$ +2189,$a=$ +2190,$\rho(W_1\wedge a_1 \wedge a_1')$ +2191,$g^a=g^{\log_g(n)}=n$ +2192,"$(2,-\x*0.75)$" +2193,$l(y;\theta)=\log(c(y)) +y\theta-\kappa(\theta)$ +2194,"$\int_0^a F(t)\,dt$" +2195,$\mathsf{E}[X_2Z]$ +2196,$m^2W(m)$ +2197,$\lim_{s\downarrow 0}g(s)>0$ +2198,$Y=y$ +2199,$\mathbf{A}$ +2200,$\mathsf{E}_{\mathsf Q}[X_i]$ +2201,$\phi_W(a)=\mathsf{E}[W/Y \mid Y>a]$ +2202,$x=160$ +2203,$A=g^a \pmod p$ +2204,$P_{act} = P + F_0 > P$ +2205,"$\iota, \iota(p)$" +2206,$q \in G$ +2207,$.} with $ +2208,$p_i(a)=\phi_i(a)p(a)$ +2209,$\bar P(a)\le a$ +2210,$\DeltaR$ +2211,$Y$ +2212,$\mathsf{E}[Z_1\mid X]=\tilde Z$ +2213,"$(s,g(s))=({s:.3g},{gs:.3g})$" +2214,"$\mathbf{x}=(1-t, t)$" +2215,$0.8 \le p < 0.9$ +2216,$P=g(S)$ +2217,$\tau a$ +2218,"$\beta_i(t\mathbf{x}, x)$" +2219,$X_2 = \mathsf E[X\mid \mathscr F_2]=X$ +2220,$v_3 a)$ +2267,$q_L(\tau_\sigma^{-1}(U)$ +2268,$B_2$ +2269,$q(Y)$ +2270,$\omega \le \omega_I$ +2271,"$\mathsf{E}(\min(X_i,a))=\mathsf{E}(X_i\wedge a)$" +2272,$\mathsf{E}_{\mathsf{Q}}[Y]=\mathsf{E}[Yg'(S(X))]$ +2273,$(a_1'-a_1)^+$ +2274,$\mathsf{TVaR}_p(X)=51.156$ +2275,$\kappa_i(s)$ +2276,$\mathsf{E}[YZ]$ +2277,$\mathsf E[e^{r_1Z_1}]$ +2278,$\mathsf{Pr} H\xi$ +2279,$r_P$ +2280,$\tau_\sigma(p)=\int_0^p\sigma(u)du$ +2281,$s=0.02$ +2282,$dx/x$ +2283,$=\mathrm{MV}(T(X))$ +2284,$\mathsf Q(\omega)\ge 0$ +2285,$X_t\not\to X_0$ +2286,$4/6$ +2287,$L^1(\mathbb R)\to L^1(\mathbb R)$ +2288,$\rho(X_0)$ +2289,$\mathbf {\beta_{2}g(S)\Delta X}$ +2290,$MgAl_2O_4$ +2291,$s^{\star} \le s_{R+1} \le s_m=1$ +2292,$1\times 51$ +2293,"$g(s)= \displaystyle\int_0^s \phi(1-p)dp = \min(s/(1-\alpha), 1)$" +2294,$\mathscr{G}amma(\alpha):=\int_0^\infty x^{\alpha-1}e^{-x}dx$ +2295,"$t\in[0,t_*]$" +2296,$\mathsf{E}(X-c_l)_+$ +2297,$L_a^{a+y}(X)$ +2298,$\mathbf {g(S)}$ +2299,${}_b\bar V$ +2300,$h(s) = (100 s)\wedge 1$ +2301,$Y=-\rho_{t+1}(X)$ +2302,$Q(a) = 1-P(a)= \nu F(a)$ +2303,$\to$ +2304,$x\mapsto x^{3/2}$ +2305,$\kappa(\theta)=-\sqrt{-2\theta}$ +2306,"$I_{k, n}=(k/n, (k+1)/n]$" +2307,"$(ccc.south |- mcc.south)+(0,-0.5)$" +2308,$\theta=-\alpha/\mu$ +2309,$\int_0^\epsilon (e^{sx}-1)j_n(x)dx \approx \int_0^\epsilon sxj_n(x)dx$ +2310,$s^\alpha$ +2311,"$(\mu,\alpha)$" +2312,$< 1$ +2313,$p\mapsto \mathsf{TVaR}_p(X)$ +2314,$0$ +2315,$V(\mu)=1/(\tau^{-1})'(\mu)=\mu^2$ +2316,$r=\rho(X)$ +2317,$X_{-2}=C_1 + \cdots + C_n$ +2318,$\iota_U > \iota^*\dfrac{r_i+a_i}{1-a}$ +2319,$g = T_{\nu}(\mathsf{var}phi^{-1}(s)+\lambda)$ +2320,$\iota_U > 3$ +2321,$b<1$ +2322,$D/L$ +2323,$σ$ +2324,$n=g^a\pmod{p} \mapsto a=\log_g(a)$ +2325,"$t\not=0,1$" +2326,$\gamma = 2/\sqrt(a) = 2\nu$ +2327,$LR_{Wang}$ +2328,"$\rho[E[X\mid I], \mu)$" +2329,$0\le \tau\le 1$ +2330,"$ap\prod_{j=1}^{k-1} \left(a(p+k)+j\right)\, \frac{\delta_k}{k!}$" +2331,"$(fun2.north west)+(-\smlspc,\smlspc)$" +2332,$(g(s)-s)/(1-s)$ +2333,$\mathsf{Pr}r(X=1)=s$ +2334,"$80K, \$" +2335,$g$ +2336,$\sigma_U = 1$ +2337,$\mathsf{Pr}(X_n>\epsilon)\to 0$ +2338,$\mathscr{F}=\mathscr{G}$ +2339,$n^{-1}\mathsf{M}'\mathsf{M}=\mathsf{id}$ +2340,$\mathsf{Pr} H\xi=\mathsf{Pr} HX$ +2341,$\mathsf{E}[Y]$ +2342,"$s,t,s^\star \ge s_R > 0$" +2343,$\mathcal F_t$ +2344,$\mu g=0$ +2345,$\mathsf{Var}(Y)=np/(1-p)^2$ +2346,$\nu=\nu(F(a))=\nu(p)$ +2347,$\tau_\sigma(p)=\int_0^p \sigma$ +2348,"$9,100+$" +2349,$\bar\iota(x)$ +2350,$Y=1$ +2351,$\mathsf{E}(X_i\mid X=x)f_X(x)/x$ +2352,$r\to\infty$ +2353,$p_{j-}$ +2354,$1-p_s>0.5$ +2355,$\mathcal{A}_\rho$ +2356,"$[0.628, 0,647]$" +2357,$1.38 \times 10^{-23}$ +2358,$\mathsf{E}[Y\tilde W] = n^{-1}\sum_T \mathsf{E}[Y \cdot W\circ T] = n^{-1}\sum \mathsf{E}[Y\circ T^{-1} \cdot W] = \mathsf{E}[YW]$ +2359,$\{A_i\}$ +2360,"$(0,3)$" +2361,"$(Bob) + (0,-1)$" +2362,$\mathsf{E}[X]+\lambda\sigma(X)$ +2363,$m(1+\frac{m}{a})$ +2364,$\rho(X\wedge a(X))$ +2365,$X_{t \wedge \tau_n}$ +2366,$\{X < X_n\}$ +2367,$B_1$ +2368,$b\le a$ +2369,"$\nu(B)=\int_B X\,d\mathsf{Pr}=\mathsf{E}[X1_B]$" +2370,$1-\alpha = q_- + p_i$ +2371,$X'\Delta S$ +2372,$1 premium with a cat EL of $ +2373,"$(0,t_1+t_2]$" +2374,$x_i=n_i + i\xi$ +2375,$L_d^l(x)$ +2376,$|x|^2$ +2377,"$1/2,1/4,1/4$" +2378,"$1-F(q(p); \theta,\dots)$" +2379,$p = (1-s)$ +2380,$\mathsf{E}[X_i\mid X\le a]F(a) + a\mathsf{E}[X_i/X\mid X >a]S(a)$ +2381,$a(X)=3.769$ +2382,$S(x)\to 0$ +2383,$\alpha(v) \le K$ +2384,"$\beta\in[-1,1]$" +2385,$\max X_i$ +2386,$\rho(L) = F^{-1}(p)g'(1-p)dp$ +2387,$K_\delta(x) = K(x/\delta) / \delta$ +2388,$\mathsf{E}(L_\sigma)= \int_0^1 q_L(s)\sigma(s)ds =:\pi_\sigma(L)$ +2389,$\sigma=2.581$ +2390,$\alpha_i(x) =\mathsf{E}[X_i/X\mid X>x]$ +2391,$\rho(A)\le\rho(A_0) +\mathsf E[X]\rho(N)$ +2392,$\ge\mathsf{VaR}_p$ +2393,$B(X)$ +2394,$\rho(x)=x$ +2395,$s>s^*$ +2396,$\mathbf{m}=(m_j)$ +2397,$\gamma_a(x) = \mathsf{E}[ (a \wedge X) / X | X > x]$ +2398,$L=43.1$ +2399,$\alpha(\mathsf{Q})=\infty$ +2400,$\alpha=\alpha^{\min}$ +2401,$\mathsf{TR}$ +2402,$p_m-p_{m-}$ +2403,$-dS=f(x)dx$ +2404,"$\mathcal{M} = \{ f \mid \|f\|_q\le c, f\ge 0 \}$" +2405,$T=3$ +2406,$\tilde r$ +2407,$r = r_1 + r_2$ +2408,$\mathsf{TVaR}_\alpha(X)=\frac{1}{1-\alpha}\int_\alpha^1 F_X^{-1}(t)dt$ +2409,$\kappa_i(x)$ +2410,$e =$ +2411,$p=F(a)$ +2412,$\mathsf{Pr}(N=n)=e^{-\lambda}\lambda^n/n!$ +2413,$j>L$ +2414,$\sigma^2=1/\lambda$ +2415,"$196,600.80 = \$" +2416,$2\le x\le 8$ +2417,$\mu/b$ +2418,$c(y)$ +2419,"$X \in (X_k,X_{k+1}]$" +2420,$\sum_j g(S_j) \Delta X_j = \sum_j q_j X_j$ +2421,$S(x)/(1-p)$ +2422,$n\times n$ +2423,$D\rho_{X_g}(X_c)$ +2424,$g=s$ +2425,$\rho_g(\cdot)$ +2426,$\Longrightarrow$ +2427,$M(a)=d\bar M(a)/da$ +2428,$g(1-p)$ +2429,$\mathsf{E}[X_i \mid X=\hat x]=\mathsf{E}[X_i \mid X=F^{-1}(\tilde p)]$ +2430,"$EL_a =\mathsf{Pr}r(Y>a) = \mathsf{Pr}r(\max(X_1, \dots, X_N)>a)=\mathsf{Pr}r(\text{one or more events $" +2431,$\iota_k$ +2432,$1-g(S(a))$ +2433,"$(\mathsf{E}[X_i]-\mathsf{E}[X_{i,2}(a)]/\mathsf{E}[X_i]$" +2434,$\bar Q(x)$ +2435,$\lambda \rho(X)$ +2436,$=0$ +2437,$L > a$ +2438,$m_i(s)\to\mathsf E[X_i]$ +2439,$X_0 = \sum_{i = 1}^{N}X_i$ +2440,$x=1.38$ +2441,$p(a) = S(a) + \rho k(a)$ +2442,$10^{9}$ +2443,"$\int_K^\infty \mathbb{P}(|X_i| > t) \, dt$" +2444,$f(n;\mu)=\dfrac{e^{-\mu}\mu^n}{n!}$ +2445,$n\ge 1$ +2446,"$\rho(X) = \sup_{\zeta\in\mathcal{A}} \langle \zeta,X \rangle$" +2447,$-4$ +2448,$b=-1$ +2449,$\mathsf{E}[XZ(X)]$ +2450,$x_1 < x_2 < \dots < x_n$ +2451,"$(s^\star,1)$" +2452,"$\rho(X)=\mathsf{TVaR}_1=\mathrm{ess\,sup}$" +2453,$\mathbf {a_{1}'}$ +2454,$\mathbb{R}\to\mathbb{R}$ +2455,$L(X)=e^{kX}/\mathsf{E}[e^{kX}]$ +2456,$x_h>x=\mathsf{VaR}$ +2457,$l(y;\mu)=y\log(\mu)-\mu$ +2458,$1/p$ +2459,$\rho(m)=\rho(0)-m$ +2460,"$F_X(x, \lambda)=\mathsf{Pr}r(X < x)+\lambda \mathsf{Pr}r(X=x)$" +2461,$\exp(h(y))$ +2462,$a \le b$ +2463,$W_0=0$ +2464,$\mathsf{var}(A)=\mathsf{E}(N)\mathsf{E}(X^2)$ +2465,$\theta=\mu$ +2466,$Y\in L^\infty$ +2467,$V(\cdot)$ +2468,$256=2^8$ +2469,$T_0$ +2470,$dQ/dP$ +2471,$\mu(\{p\})=1$ +2472,$\alpha(Q)=\infty$ +2473,$\sum_i x_iX_i$ +2474,$f_X$ +2475,$1 \times 10^{16}$ +2476,$\mathsf{E}[Y'\mid X']=X'$ +2477,$p(a)$ +2478,$s \to 0$ +2479,$\mathsf{E}[X_T] = \mathsf{E}[X_0]$ +2480,$m_i:=m(\{p_i\})\ge 0$ +2481,$\mathsf{Pr} H\xi = \mathsf{Pr} HX$ +2482,$g(s)\ge 0g(0) + sg(1)=s$ +2483,$\rho(A_k) \le \rho(A_0) + k\rho(N)$ +2484,$\mathcal G\subset\mathcal F$ +2485,$\delta(s)=g(s)g(k/s)-g(k)$ +2486,$Y>a$ +2487,$R(x)$ +2488,$p=(2+\bar\alpha)/(1+\bar\alpha)$ +2489,$\Vert \cdot\Vert$ +2490,$=L/(1+r)$ +2491,$0 \le c \le 1$ +2492,$D^n\rho_{X\wedge a}(\cdot)$ +2493,$E_\mathsf{Q}[X_i]$ +2494,$\rho(X)=T_{m_1}(X)-v(m_1)$ +2495,"$(ckey\x.north west)+(-\boundpad,\boundpad)$" +2496,"$[x,x+dx)$" +2497,$S^*(a)$ +2498,$\mathsf{Pr}r(X\le q_l(p))\ge p$ +2499,$i$ +2500,$\lambda S(a)$ +2501,$a<1$ +2502,$X=X_0+X_{-1}+X_{-2}+X_{-3}$ +2503,$a < 1$ +2504,$g(t)=h(t)\{ l(t) < 1\}$ +2505,$(1+\gamma)F_0$ +2506,"$\mathsf{TVaR}_1=\mathrm{ess\,sup}$" +2507,$μ = t ν$ +2508,"$\eta_{p,\alpha_1}(X) < \eta_{p,\alpha_2}(X)$" +2509,$\sup_i f_i$ +2510,$X(\omega_1)<\cdots < X(\omega_n)$ +2511,"$j=1,\dots,r$" +2512,"$\omega\in [0,0.1)\cup [0.25, 0.35) \cup [0.5, 0.6) \cup [0.75, 0.85)$" +2513,$R^1$ +2514,$h_2$ +2515,$\rho(n^{-1}\sum X\circ T) = n^{-1}\sum \rho(X\circ T)$ +2516,$\tilde X_1 \le X_1$ +2517,"$\mathsf{Y}=(Y_1,\dots,Y_r)$" +2518,$\mu(A) = \nu(A)$ +2519,$S_i(x)$ +2520,$s^U \ge s^{\star}$ +2521,$da$ +2522,$g(s)=vs +d > s$ +2523,$\theta > 1$ +2524,"$[0,1]\to[0,1]$" +2525,$\mu\sqrt{\mu^2+4\mu}$ +2526,$k<1$ +2527,"$397,090,870 | | $" +2528,$T(X):=y\wedge (X-r)^+$ +2529,$10^1$ +2530,"$A = \{ \zeta \mid \|\zeta\|_q\le c, \zeta\ge 0 \}$" +2531,$P_1(t)\le P(1)$ +2532,$\mathsf P(X=\mathsf{VaR}_p(X))>0$ +2533,"$p_0\in(0,1)$" +2534,${}^{[<43]}$ +2535,$r_h$ +2536,$\beta_i(a)$ +2537,$10^{{-6}}$ +2538,$g(S(a))\ge S(a)$ +2539,$a_1' = a_0-X_1$ +2540,$1 layer covering losses at or above the $ +2541,$\sigma^2\mathsf{Po}(\mu/\sigma^2)$ +2542,$\mathsf{E}[X_1] \le R_1(t) \le \rho(X_1)$ +2543,$y_{1}=2$ +2544,$\mathsf{Pr}r[X > A] \le \epsilon$ +2545,$\mathsf E[T_s T_t] \ge \mathsf E[T_s] \mathsf E[T_t]=g(s)g(t)$ +2546,$g(st)= \displaystyle\frac{st}{1-p} \le \displaystyle\frac{s}{1-p}\displaystyle\frac{t}{1-p}=g(s)g(t)$ +2547,"$(3,9,11,\dots, 3)$" +2548,$1100 \le x \le 1250$ +2549,$nG$ +2550,$d=iv$ +2551,$\rho(\tilde X_1)\le \rho(X_1)$ +2552,"$\phi(s) = (1-p)^{-1}1_{[p, 1]}(s)$" +2553,$\rho(X) = \mathsf{E}[gX]$ +2554,$X_t = \mathsf{E}[X \mid \mathscr{F}_t]$ +2555,$a/(1000+X_1)$ +2556,$g(s)$ +2557,$\lambda\downarrow 0$ +2558,$\omega'\in\Delta_\omega$ +2559,"$j=R+1,\dots,m-1$" +2560,$a=P+Q$ +2561,$0 = \Delta_m s_m = \Delta_m = \phi_m$ +2562,$n=2$ +2563,$\mathsf{E}[X_\nu]=\nu\mathsf{E}[X_1]$ +2564,$X_n(\omega)= 1$ +2565,$\mathsf{E}_Q[L]$ +2566,$W$ +2567,$Y = NX$ +2568,$d^*=(\log(A/L) + (r_h-\mu_L + \sigma^2/2))/\sigma\sqrt{T}$ +2569,$\int_0^1 j(x)dx=\infty$ +2570,"$\mathsf P((a,b])=b-a$" +2571,$P(0)=R_0(0)=x_0$ +2572,$\max_{s\le t} X_s - X_t$ +2573,$a-L_0^a(X)$ +2574,$\mathcal F$ +2575,$X_{t_n}\to X_t$ +2576,"$\mathsf{corr}(X_i, X)$" +2577,$X \preceq_n Y$ +2578,$1_{U>0.95}$ +2579,$s\mapsto g(s)$ +2580,$\int_{-f_{\max{}}}^{f_{\max{}}}$ +2581,"$8,954,000 | 93.9% | $" +2582,$0\le d\le 1$ +2583,$x_h(1-p)$ +2584,$\bar S_i(a)=\mathsf{E}[X_i(a)]$ +2585,"$A=(-\infty, x]$" +2586,$S_0=1000$ +2587,$\mathcal B$ +2588,$\mathsf{Pr}r(X < x)=1/6=\mathsf{Pr}r(X\le x)$ +2589,$\mathsf{E}(G^3)=g$ +2590,$\rho_g(X)=352$ +2591,$s_l$ +2592,"$S:(\Omega,\mathscr{F})\to(L,\mathcal{A})$" +2593,$A\in\mathcal F'$ +2594,$>100$ +2595,$\delta F(x)$ +2596,$\mathbf M' = \mathbf M(\mathbf E^t)^{-1}$ +2597,$\Delta p\times T$ +2598,$E[T]=s$ +2599,"$\alpha\in [0,1]$" +2600,$\binom{p+k-1}{k}\delta_k$ +2601,$0\le (-X_n) \le 1$ +2602,"$\rho(X+tY)-\rho(X) = \langle \zeta_t, X+tY \rangle -\rho(X) \le \langle \zeta_t, X+tY \rangle - \langle \zeta_t, X \rangle = \langle \zeta_t, tY \rangle$" +2603,${scale} X$ +2604,$t_1$ +2605,$\rho(1_A)=1$ +2606,$\mathbf {d=2}$ +2607,$E[u_j(W_j - X_j + Y_j - H[Y_j])]$ +2608,$x_l < x < x_u$ +2609,"$U_{i,j}$" +2610,$\mathsf P(X \le q_X(p)) > p$ +2611,$\Delta_1 s_1 = T_L$ +2612,$\Delta=\mathsf{Pr}hi(d*)$ +2613,$s\le 1-p < t$ +2614,$1/\lambda^2$ +2615,$E(X_{-1}(a))$ +2616,$F(x)=1-e^{-x/\mu}$ +2617,$\mathsf P(X=\max(X))>0$ +2618,$x_1=0$ +2619,$\to 0$ +2620,$0.999999$ +2621,$N=4$ +2622,$0 < \nu = 1-\delta < 1$ +2623,$M(t)=\mathsf E[\exp(tx)]=\phi(-it)$ +2624,"$Y=\mathrm{ED}(\mu, \sigma^2)$" +2625,$\hat{s}=0.09297$ +2626,$N=5$ +2627,$U_1 = \min(U)$ +2628,$t = 0$ +2629,${{}_tp_x}=\exp(-\int_0^t \mu_{x+s}ds)$ +2630,$K_T(\theta)=-\log(1-\theta/\lambda)$ +2631,$X_0=1000$ +2632,"$\{\mathbf{X}^{(j)}: j=1,\dots,M\}$" +2633,$i:=r_a=r_l\ge r_f$ +2634,$Q_1dX$ +2635,$q(p)=-\log(1-p)\mu$ +2636,$\mathcal{B}bb Q\in{\cal M}_1$ +2637,"$4,526.00 \$" +2638,$\nu=1/(1+\iota)<1$ +2639,$xS(x)\to 0$ +2640,$P = \mathsf{E}[X] + \pi \mathsf{E}[((X-\mathsf{E}[X])^+)^p]^{1/p}$ +2641,$\rho(W_1\wedge a_1 \wedge (a_0-X_1))=\rho(W_1\wedge a_1)$ +2642,"$\{(s_j,g_j): j=0,\dots,L-1; s_j < s_L\}$" +2643,$s=\sqrt{s_1}$ +2644,$-g''(t)=α(α-1)t^{α-2}$ +2645,$a^i$ +2646,$\beta > \alpha$ +2647,$\sigma(X_1)$ +2648,$X_n\uparrow X$ +2649,$(k+1)\times n$ +2650,$-5.91$ +2651,$v'(x)>0$ +2652,$U + R\iota_U + K\iota_K$ +2653,$\mathsf{Pr}(C) = 1$ +2654,$B_i$ +2655,$du$ +2656,$\mathbf {Z_1}$ +2657,$g^{ak}$ +2658,$t\ge 0$ +2659,$\mathsf{E}[(X_i/X)g'(S(x)) \mid X > x]$ +2660,$\nu<1$ +2661,$g(s)= w + (1-w)s/s_0$ +2662,$x = F^{-1}(1-g^{-1}(1-\tilde p))$ +2663,$\mathcal P$ +2664,"$(s_{i}, g(s_{i}))$" +2665,$q=ps_g$ +2666,$0.004=1-0.996$ +2667,"$[a,a+1)$" +2668,$X_{-3}=C'_1 + \cdots + C'_n$ +2669,$-6$ +2670,$g(0+)\ge 0$ +2671,$i=0.02$ +2672,$^{2}$ +2673,$q(\epsilon)/(1+\epsilon)\approx (q+\epsilon\mathsf{E}_q(X_1) )(1-\epsilon)=q-\epsilon(q-\mathsf{E}_q(X_1))=q-\epsilon E_q(X_2)$ +2674,$\mathsf{EE}$ +2675,"$\forall A\in \mathcal{A},\ f(A) \in \mathcal{B}$" +2676,"$\{1,2,\dots, N\}$" +2677,"$n_{\text{water, initial}}$" +2678,$\sup X$ +2679,$H(x)\not=H(y)$ +2680,"$Y_{t,d>0}$" +2681,$B=2.7\times 10^{-6}$ +2682,"$1,500 in financial assets earns \$" +2683,$X_i = \kappa_i(X)$ +2684,$p\delta(p)/p\nu(p)=\iota(p)$ +2685,$\rho_2(X)$ +2686,$f\ge 0$ +2687,$g_i=g^{-1}(u_i)$ +2688,${}^{[>113]}$ +2689,$10^{-43}$ +2690,$phi$ +2691,$\Delta_m=0$ +2692,"$(s_m=1,1)$" +2693,$\mu_1$ +2694,$X_n= X_g-X_c$ +2695,$tR_1(t)0$" +2721,$\mathbf {X_{1}/X}$ +2722,$\bar\delta(x)$ +2723,$s<0.1$ +2724,"$\rho(X)=\rho(X,\mathsf P)$" +2725,$\mathsf E[U\mid \mathcal F_t]$ +2726,$A=F\triangle Q$ +2727,$x+y^c\wedge a+bz$ +2728,$n=\square^\square$ +2729,"$\mathrm{rcd}(\mathsf{Pr}_S,\sigma(T))$" +2730,$\rho(A+B)=\rho(A)+\rho(B)$ +2731,$X_n(\omega)=1$ +2732,"$f(x,y)=q_\alpha(x) - G(x,y)$" +2733,$\mathbf R_{>0}$ +2734,$\pi(X)=\int_a^{\alpha(X)} g(S(t))dt$ +2735,$\mathsf{TVaR}_{0.75}=4\left( \frac{90}{8}+\frac{98}{16}+\frac{100}{16}\right)=94.5$ +2736,$\mathsf{EE}=n^{-1}\mathsf{M}'\mathsf{M}$ +2737,$\mathsf{VaR}_p(X)=\mu + \sigma \mathsf{Pr}hi^{-1}(p)$ +2738,$\tilde X_1=X_1 + \mathsf{E}[X_2\mid X_1]$ +2739,$\theta=\nu\lambda$ +2740,"$1,353.02 \$" +2741,"$\gamma([0,p])=C(p)$" +2742,$3.55 per $ +2743,$v_i = a_i/a$ +2744,$\mathsf{VaR}_{1-s}$ +2745,$\px=\mathsf{Pr}(\bigcup_i (B\cap A_i)$ +2746,$(\bar P_{x+b} - \bar P_x)\bar a_{x+b}=\bar A_{x+b}-\bar P_x \bar a_{x+b}=: {}_b\bar V$ +2747,$\mathsf{E}[g'(S(X))]=\int_0^\infty g'(S(x))f(x)dx=\int_0^\infty -\frac{d}{dx}g(S(x))dx=g(S(0))-g(S(\infty))=g(1)-g(0)=1$ +2748,$(X \wedge a) / X$ +2749,$s \approx 0$ +2750,$\alpha=1$ +2751,$\phi_{\bar x}$ +2752,$P_X\{X=M\}=0$ +2753,"$p(B, y)$" +2754,$r=\mathsf{TVaR}_p(X)$ +2755,"$u=(u_1, u_2)$" +2756,$-0.209795$ +2757,$\mathsf{E}[D_t \mid \mathscr{F}_{t-1}] = \mathsf{E}[X_t - X_{t-1} \mid \mathscr{F}_{t-1}] = 0$ +2758,$Y_\epsilon$ +2759,$Y+W$ +2760,$\alpha=0.99$ +2761,$\mathsf{TVaR}_{p=0}$ +2762,"$Q\in{\mathcal{M}}_{1,f}$" +2763,$g(0)=r_0$ +2764,$p(1-p)/(\nu-l)^2=0.5(1-0.5)=0.25$ +2765,$V=\sum_s V_s$ +2766,$\mathsf{E}[Z]=\mathsf{E}[\mathsf{E}[Z\mid X]] = 0$ +2767,$\displaystyle\int_0^\infty xf(x)dx = \displaystyle\int_0^\infty S(x)dx$ +2768,$\mathsf{TVaR}_{0.975}$ +2769,$\rho=\sigma=\tau$ +2770,$s<0.20$ +2771,$\mu=21.315$ +2772,$\mathsf{Pr}r(\mathsf{CP}=n)=\sum_{k\ge n}\mathsf{Pr}r(\mathsf{CP}=n\mid N=k)\mathsf{Pr}r(N=k)$ +2773,$r = 0.6565$ +2774,$h_1$ +2775,$P_1 + P_2=\rho(X)$ +2776,$p=\mathsf{Pr}hi^{-1}(4)=3.17\times 10^{-5}$ +2777,$\hat q(p)=q(1-g^{-1}(1-p))$ +2778,$p=0.9982$ +2779,$g(s)=s^{0.72}$ +2780,$\rho(X\_{1}\subseteq X^c\wedge a)$ +2781,$g(s)=d+vs$ +2782,$\mathbf{x}=\mathbf{1}$ +2783,$=\displaystyle\int_0^\infty x dF(x)$ +2784,$P=D$ +2785,$p=0.995$ +2786,$M(x)/(1-S(x))$ +2787,$n \le pN < (n+1)$ +2788,$\mathsf{E}[X\mid \mathcal F_{t+1}]$ +2789,$m=0$ +2790,$v_1$ +2791,"$s\in (0,1]$" +2792,$\mathsf{E}(X)=\int_0^\infty xf(x)dx = \int_0^\infty S(x)dx$ +2793,$P = a-Q$ +2794,$Z_1=Z\circ T$ +2795,$P_1=\mathsf{E}[X_1g'(S_X(X))]$ +2796,$\{X = a\}$ +2797,$\alpha = (2 - \xi)/(\xi - 1)$ +2798,$X=\max(X)$ +2799,$aw$ +2800,$\mi(\mu):=\mathsf{Var}(s)$ +2801,$\mathbf{T}$ +2802,$\partial\bar P/ \partial a$ +2803,"$w(s,t)$" +2804,$X = X_1 + (X - X_1)$ +2805,$\kappa_0(x)$ +2806,$R_i t^*$ +2816,$F(x_0)\ge p$ +2817,"$(fun5.north west)+(-\smlspc,\smlspc)$" +2818,$s_0 \le s_0(s_1)$ +2819,$T_B$ +2820,"$(-\infty,0)\subset\Theta$" +2821,$\mu = \sum \mu_i$ +2822,$F_X(x)$ +2823,$\Omega\times \Omega$ +2824,$\alpha(\mathsf Q) < \infty$ +2825,$\mathscr{G}amma=\mathsf{Pr}hi^{-1}(D)$ +2826,$\rho(-X)=-\rho(X)$ +2827,$\bar R'(x)=R(x)$ +2828,$AR\succ BY$ +2829,$S(X(\omega))$ +2830,$\sup f=1$ +2831,$T_0=T-1$ +2832,"$U_j := (0,\dots,0,1,0,\dots,0)$" +2833,$ on $ +2834,$i\in A$ +2835,"$823,500 = \$" +2836,$L_k+\delta(\Delta X_k-L_k)$ +2837,$(1-s*) - \lambda \Xi(s) = 0$ +2838,$P=vL + da = vs + d$ +2839,$g(v)$ +2840,$U < s$ +2841,$a_i > 0$ +2842,$R_1(t_1^*-\epsilon) C_2$ +2847,$S_X(t)=S_{X\wedge a}(t)$ +2848,$\alpha = 0.642.$ +2849,"$(g^k, Km)$" +2850,"$\mathsf{E}[Y]/a_{X,r}(Y)$" +2851,$\mathbf R_{<0}$ +2852,$d(y;\mu) = 2\left(y\log\left(\dfrac{y}{\mu}\right) - (y-\mu)\right)$ +2853,"$|\mathcal{W}(g,W)|$" +2854,$\theta>0$ +2855,$s^\ast = 1/2$ +2856,"$\lim_{\mu\to 0} V(\mu)/\mu=\delta:=\inf\,\{S\setminus \{0 \}\}$" +2857,$u_j=\Delta_j s_j$ +2858,$v(A)=g(\mathsf{P}(A))$ +2859,$\Delta_iX_s$ +2860,$P(a) = \nu S(a) + \delta = (S(a) + \iota) / (1+\iota)$ +2861,$X=Y/\lambda$ +2862,$a=65$ +2863,"$[0,1]\to \mathbb{R}$" +2864,$\mathsf{TVaR}_p = q(p)$ +2865,$1.25$ +2866,$u_k=\Delta_k s_k=1$ +2867,"$f:(0,1)\to (0,1)$" +2868,$99\zeta$ +2879,"$\{x_1,\dots,x_N\}$" +2880,$R_f$ +2881,$\mathbf {j}$ +2882,$B_{\cdot}$ +2883,$\epsilon^B$ +2884,$\alpha(1-\alpha)(1-s)^{\alpha-1} + \alpha\delta_0$ +2885,$\rho_t(X) \le \mathsf E[\rho_{t+1}(X)\mid \mathscr F_t]$ +2886,"$\int_0^1 x^2j(x)\,dx<\infty$" +2887,$\sigma_s=0$ +2888,$\sup_{\theta\in\Theta} y\theta-\kappa(\theta)$ +2889,$N(\bar x)=N(F(\bar x))$ +2890,$(d\mathsf{Q}/dP)(x) = (1-p)^{-1}1_{x >\mathsf{VaR}_p(X)}$ +2891,$\bar\theta_s = \hat\theta_s + t$ +2892,$\mathsf{Var}(Y)=\mu^2/\alpha$ +2893,$\lambda S(x)$ +2894,$\mathsf{Pr}(T_x<\infty)=1$ +2895,$nt$ +2896,$\mathsf P_i(\mathsf P_j - \mathsf P_{j-1})=\mathsf P_i - \mathsf P_i = 0$ +2897,$\forall x\forall y[\forall z(z \in x \leftrightarrow z \in y)\rightarrow x=y]$ +2898,$\mathcal{F}_t = \sigma(X_s : s \leq t)$ +2899,$\mathsf{Pr}(A)=0$ +2900,$a\ge 1$ +2901,$p_R$ +2902,$\alpha_i(x)=\mathsf{E}[X_i/X\mid X>x]$ +2903,$F_{\bar X}(x)=p_1F_1(x) + p_2F_2(x)$ +2904,$1 \times 10^{10}$ +2905,$T_{(1)}=W$ +2906,$\alpha=(\xi-2)/(\xi-1)$ +2907,$\tilde Z_1:=\mathsf{E}[Z_1\mid X] = \tilde Z$ +2908,$f_t$ +2909,$r\times m$ +2910,$3/4 \pm 1/4$ +2911,$q(p)\phi(p)\times dp$ +2912,$\bar\theta_s<0.5$ +2913,$Z\circ T$ +2914,$\tilde{\mathsf{Q}}$ +2915,$J(0)$ +2916,"$\alpha\in [0,1)$" +2917,$X=X(U_2)$ +2918,$\mathsf{C}=(c_{ij})$ +2919,$A = g^{a} \pmod{p}$ +2920,"$500,000) and 100% (from \$" +2921,$\mathsf{E}[Z]\le 1$ +2922,$f_{Y\mid X>a}$ +2923,$X_t = ct+\sigma B_t$ +2924,$\max v\pi$ +2925,"$c_1,c_2\ge 0$" +2926,"$t\in[t_1^{**},1)$" +2927,$\bar q(s)=(k/q)^{1/\alpha}$ +2928,"$\bar Q_{0,1}$" +2929,$\mathbf {x}$ +2930,"$(fun1.north west)+(-\smlspc,\smlspc)$" +2931,$\mathsf{Pr}r(X=\mathsf{E}[X])=0$ +2932,$\partial a / \partial x_i$ +2933,$M=\delta (a-L)$ +2934,$\log$ +2935,$1-g^{-1}(1-p')$ +2936,"$2,500 deductible. (The \$" +2937,$\sum_i X_i$ +2938,$a_i$ +2939,$a_p$ +2940,$\mid$ +2941,$\mathcal F'\subset \mathcal F$ +2942,$d\tilde p=g'(1-p)dp=\phi(p)dp$ +2943,"$1,000,000,000) multiplied by the percentage representing that insurer's residential earthquake insurance market share as of January 1, 1994, as determined by the board. A minimum of seven hundred million dollars ($" +2944,$\mathsf{E}_\mathsf{Q}$ +2945,$\mathsf{Pr}r(Y>y)^r$ +2946,"$\sigma_1=1, \sigma_2=0$" +2947,$1-p \le st$ +2948,$1\times r$ +2949,$>100\%$ +2950,"$[t_1^*,1)$" +2951,$1$ +2952,$\mathsf{E}(N)=\lambda$ +2953,$(1)(0.25)+(90)(0.25)=22.75$ +2954,$d\mu = \kappa''(\theta)d\theta$ +2955,$\beta_i(x)/\alpha_i(x)> 1 > g(S(x)) / S(x)$ +2956,"$(\omega_1,\omega_2)\in [0,1]^2$" +2957,$9$ +2958,$\mathcal V$ +2959,$\lambda=g(\lambda_{obj})$ +2960,$\mathsf{P}(A) = \mathsf{E}[1_A]$ +2961,"$12,025 | \$" +2962,$X:\Omega\to\mathbb R$ +2963,$E=U+T=-T$ +2964,$s=S(x)=1-p$ +2965,"$\rho(X,a)=\rho(X\wedge a)$" +2966,$P_k$ +2967,$X_1={abs(x.X1):.0f}$ +2968,$i^*\times(n-i^*)$ +2969,"$A_2=[\epsilon, \epsilon]$" +2970,$\alpha<1$ +2971,$(.*?)\$ +2972,$g(P)$ +2973,$\mathbb R^\times$ +2974,$\mathsf{E}_Q(X_i)$ +2975,$sP$ +2976,$\rho(0)=\rho(0+0)=\rho(0)+\rho(0)$ +2977,"$\px=\int_B \mathsf{Pr}(A\mid\mathscr{G})\,d\mathsf{Pr}$" +2978,"$1,150 | \$" +2979,"$U_i,U_j$" +2980,$\rho(Y) := E[Y_j\mid X_j=c]$ +2981,$a-L=\mathsf{TVaR}(X)-\mathsf E[X]$ +2982,$\alpha S$ +2983,$g^o$ +2984,"$623,780,445 | 18.8% | $" +2985,$\{X_t\}_{t \geq 0}$ +2986,"$\tau_{ij}=\tau(X_i,X_j)$" +2987,$r\in\mathbb{R}$ +2988,$P_i \le \rho_i(X_i)$ +2989,$\mathsf{P}(X=X_j)=\Delta S_j:=S(X_{j-1})-S(X_j)$ +2990,$\mu/V(\mu)$ +2991,$r^*$ +2992,$p\leftrightarrow p_1 = 3-p$ +2993,$\iota^*=0.15$ +2994,${}^{[<113]}$ +2995,$\mathsf{TVaR}_{0.8}$ +2996,$\rho=(1-\alpha)\sigma+\alpha\tau$ +2997,$P(X\wedge a)$ +2998,"$(\s,4-\s)$" +2999,$X\le c$ +3000,"$\bar P(\mathbf{x}, a)$" +3001,$P_g$ +3002,$1/m>0$ +3003,$s=s^\star$ +3004,$\{\text{capital structure}\} \times \{ \text{risk} \} \to \text{price}$ +3005,$P(t)=P_0(t)+P_1t)$ +3006,$m'$ +3007,$\rho(Y) := \mathsf{E}[Y]$ +3008,$j)$ +3009,$f(x-)$ +3010,$(X \wedge a_1)\leftrightarrow$ +3011,"$\mathsf{E}[Y]/b_{X,r}(Y)$" +3012,$\mathsf F=(\omega^{ij})_{ij}$ +3013,"$\bar P_i(x_1, x_2, a) / x_i$" +3014,$\bar S_i(a) = \mathsf{E}[x_iX_i\mid X\le a]F(a) + a\mathsf{E}[x_iX_i/X\mid X> a]S(a)$ +3015,"$X, F, S, M, Q, P, a$" +3016,$X(0)=0$ +3017,$\phi(tX)=t\phi(X)$ +3018,$\mathsf{E}[X_iZ]$ +3019,$T\lambda \ll \mu$ +3020,$1 = 1_\Omega$ +3021,$\mathbb{P}(B)=0$ +3022,$T_A\in\mathscr{S}(X)$ +3023,$\mathit{NPV}_{\infty}=2\times 2.5=5$ +3024,$\mathsf P(N=n)=p_n$ +3025,$\hat X_i$ +3026,$\rho=\mathsf{E}$ +3027,$\mathsf E[u(X-\pi +R)]$ +3028,$x\in \mathbb{R}$ +3029,$0\le p<1$ +3030,$P_{g}(A)=0$ +3031,$F^{-1}(s)$ +3032,"$Y_l:=\mathsf{CP}(J(1), X_l)$" +3033,$Q\ll P$ +3034,$(x+t)\mathscr{G}amma(x+t)=\mathscr{G}amma(x+t+1)$ +3035,$2.44 > 2 \times 1.2$ +3036,$X_2={abs(x.X2):.0f}$ +3037,"$(x_i, y_{k(i)})$" +3038,$(I/a + U/R)$ +3039,$500 | > \$ +3040,$X_1 =$ +3041,$P \le \dfrac{S}{\lambda} \approx \dfrac{\mathsf{E}[X]}{\lambda}$ +3042,"$(\x*1.2, 2)$" +3043,$\nu(S(x) + \iota)$ +3044,$\log_g(n)=a$ +3045,"$d=d(X_1,\dots,X_n)$" +3046,"$X_2=0,0,0,0,1,1,1,4,24, 500$" +3047,$I(p)$ +3048,"$r\in [\inf_i \tau_i(X), \sup_i \tau_i(X)]$" +3049,$n^{-1}X'X$ +3050,$a_1 = 5.991$ +3051,$x_{\min{}} = m_0b$ +3052,$(\tau^{-1})'(\mu)=1/V(\mu)=\mu^{-p}$ +3053,$\rho(X) = \sup \{ \rho_\phi(X) \mid \phi\in A \}$ +3054,$E[X^i]$ +3055,"$\rho(X)=\int g(S_X(t))\,dt$" +3056,$a_r$ +3057,$S_{\mathbf{v}}(t)=\text{Pr}(X({\mathbf{v}})>t)$ +3058,$\lambda/r<1$ +3059,$f'_+$ +3060,"$(s,g(s))=(0.2, 0.36)$" +3061,$\mathsf{VaR}_{p^*}$ +3062,$R_L=(L-P)/P$ +3063,$\mathsf{E}[X]=28$ +3064,$\rho(X\mid I)$ +3065,$X\ge a$ +3066,$\mathcal{M} = \sigma(\mathcal{A})$ +3067,$x=y$ +3068,"$logX, d$" +3069,"$\{q_k:k=0,\dots,a\}$" +3070,$(1-r_0)\delta_0 + r_0\delta_1$ +3071,$p$ +3072,$(M-N)\times d$ +3073,$E[kX]$ +3074,$0.3 < s <0.4$ +3075,$j \le L$ +3076,$ if $ +3077,$\mathbb{R}^3$ +3078,"$(E, \mathsf{E}E)$" +3079,$k!$ +3080,$\mathsf{E}[X_i \mid X \le a]$ +3081,$n\mathsf{Var}(X)$ +3082,$Y\mid Y > y_c$ +3083,$F^{(2)}(\mu_X)$ +3084,"$\alpha, \beta, \kappa$" +3085,$R^2=0.90$ +3086,"$\bar S_i = \sum_{j} X_{i,j}p_j$" +3087,$p = \mathsf{Pr}r(U=A) = \mathsf{Pr}r(X\ge A) = 1-F(A)$ +3088,"$k,a$" +3089,$P/S$ +3090,$-0.20979$ +3091,$\{X>x\}$ +3092,$g(s) \ge s$ +3093,$X_1=\mathsf{E}[X\mid C]=X$ +3094,"$827,675,737 | 5.6% | $" +3095,$\mathscr{G}amma=\mu\otimes\Lambda$ +3096,$k_1 u$ +3097,${}^{[>144]}$ +3098,$\mathsf{E}_Q[X-X]=\mathsf{E}_Q[X]+\mathsf{E}_Q[-X]$ +3099,$h_f$ +3100,$\nu(p)=(1+\iota(p))^{-1}$ +3101,$\Lambda = (\mathsf{E}[ r_{U} ] - r_{f})/(\sigma_{U} \sigma_{r_{U}})$ +3102,$\beta=\delta^*-d$ +3103,$(p-\nu)/(\nu-l)$ +3104,$\Delta H$ +3105,$\mathsf{Q}(A)\le g(\mathsf{P})(A))$ +3106,"$p(\omega, \cdot)$" +3107,$\hat X$ +3108,$l(y;\mu)=-\dfrac{y}{\mu}-\log(\mu)$ +3109,$j=6$ +3110,$\mathcal Q_1$ +3111,$0\mapsto 0$ +3112,"$(Alice)+(0,-3.25)$" +3113,$b=0$ +3114,$t_n\to t$ +3115,$U(X)\ge U(Y)$ +3116,$c:\mathbb{R}^n\to\mathbb{R}$ +3117,$\delta=0.13$ +3118,$\int_0^\infty \gamma_a(x) S(x) dx = \mathsf{E}[a \wedge X]$ +3119,$(\delta^*-d)\sqrt{F(x)S(x)}$ +3120,$a/4$ +3121,$X_1> x_1$ +3122,$U(t)$ +3123,$p_s=\mathsf{E}(R(\hat\Theta_s))$ +3124,$\mathrm{sign}(z)$ +3125,$\mathsf{E}(X_i \mid X=\hat x)=\mathsf{E}(X_i \mid X=F^{-1}(\tilde p))$ +3126,$n\ge m$ +3127,$a(X_i)=2.665$ +3128,$\rho(X_j)$ +3129,$\rho(X_n) \uparrow \rho(X)$ +3130,$t = T$ +3131,$gS$ +3132,$d^* = D/L^*$ +3133,$U\le p$ +3134,"$391,874,530 | 4.5% | $" +3135,$\rho_{(g)}(X)$ +3136,"$t=2,3,...$" +3137,$W'_t$ +3138,$10^9 - 10^{12}$ +3139,$f \in C(X)$ +3140,$L_X(v)\le \rho(v)$ +3141,$\mathsf{E}[X_i g'(S(X))]$ +3142,$S(x)=1-\mathsf{Pr}hi((x-\mu)/\sigma)=\mathsf{Pr}hi(-(x-\mu)/\sigma)$ +3143,$\theta=\log\left(\frac{p}{1-p}\right)$ +3144,$\mathsf{E}(Y\sigma(U))$ +3145,$1 = \nu + \delta$ +3146,$G = H - TS$ +3147,$\mu_2$ +3148,$X_n=0$ +3149,$\alpha_i(x) = \mathsf{E}[X_i /X \mid X> t]\not=\mathsf{E}[X_i\mid X> t]/\mathsf{E}[X\mid X>t]$ +3150,$\{\omega\in\Omega\mid H(\omega)\le h \}$ +3151,$P(x)=A(1_{X>x})=g(S(x))$ +3152,$\rho(1_A)$ +3153,$25M limit above \$ +3154,"$\displaystyle\int_{T^{-1}(B)} X \,d\mathsf{Pr}$" +3155,$[(k-1/2)b)-F((k+1/2)b))$ +3156,$1/x^2$ +3157,$\bar S_i(x)$ +3158,"$r_f\ge 0, r>0$" +3159,$172.4\times \exp(2.7^2/2) = 6600$ +3160,$P(a)=1-Q(a)=1-h(F(a))$ +3161,"$[0,t]$" +3162,$M_2\Delta X$ +3163,$\mathsf{E}[g]=1$ +3164,$\tilde S$ +3165,$\mathit{ROL} = 1 - \dfrac{( 1 - \mathit{AEL} )}{1 + \alpha_{0} + \alpha_{1} \mathit{AEL}}$ +3166,$\rho_X(X_j)$ +3167,$M_i(t)\not=C_i(t)$ +3168,$q_k = g(S_{k-1}) - g(S_k)$ +3169,$A \subseteq \mathbb{R}$ +3170,$\tilde\rho(X)=\mathsf{E}_Q(Y)$ +3171,$j_n(x)=j(x)$ +3172,$a_{d}=a(Y_{d})$ +3173,$(\hat f(l/P))_l$ +3174,$q+\epsilon\mathsf{E}_q(X_1)$ +3175,$rpq$ +3176,"$\langle \zeta, Z-\mathsf{E} Z\rangle$" +3177,$_{p^*}$ +3178,$g(S(x))=g(0)=0$ +3179,$QR_Q = aR_A + PR_L$ +3180,$r_A$ +3181,$c=2.5$ +3182,$\tilde X_2 = X_2 -\mathsf{E}[X_2\mid X_1]$ +3183,$> 30$ +3184,$\mathrm{PV}_{r}(C)>0$ +3185,$\mathit{ROL} = \alpha_{0} + \alpha_{1} \mathit{AEL}$ +3186,$\mathrm{Re}(a)>0$ +3187,$\mathcal{X}$ +3188,$ab$ +3189,$X_i/X$ +3190,"$10,897,000 | 21.7% | $" +3191,"$(0, \omega_I]$" +3192,$r_1 = 44594.593127378765$ +3193,$\tau = \inf\{t \geq 1 : M_t = t \}$ +3194,$s(x;\mu) = \partial l/\partial \mu$ +3195,$\bar P_2$ +3196,$X=Y(\mathbf{X}) + Z(\mathbf{X}) = Y+Z$ +3197,$T_1 := \sum_{j=n+1}^{N-1} X_{j} p_{j}$ +3198,$\mathit{MV}_{ro}(a_{ro})$ +3199,"$\gamma_{a,i}(x) = \mathsf{E}[ \{\mathsf{E}[X_i | X] / X\} \ {(X \wedge a) / X} 1_{X>x} ] / \mathsf{E}[ \{\mathsf{E}[X_i | X] / X\} 1_{X>x} ]$" +3200,$x_{max}$ +3201,$Q(u)$ +3202,$g(s)=\mathsf{TVaR}_{.99}$ +3203,$96\le x \le 103$ +3204,$E_g[X_i/X | X > x]$ +3205,"$p=0.98, 0.99$" +3206,$X\ge x$ +3207,"$X_{t,d}$" +3208,$\mathsf{E}(\theta)=\int\theta dP=(1+r_f)^{-1}$ +3209,"$(s_0,g(s_0))$" +3210,$\eta=\mathbf x\beta$ +3211,$Fu})=g(1-u)$ +3219,$w(z)=zg(w(z))$ +3220,"$1,2,\dots, 11$" +3221,$f(y;\theta)=c(y)e^{\theta y - \kappa(\theta)}$ +3222,$\mathbf {n}$ +3223,$n\times m$ +3224,$0.5 \times v^T$ +3225,$k_1^2 uv$ +3226,$\mathcal{F}_t$ +3227,"$P(·, \omega)$" +3228,$\tilde X_1 = E[X_1 \dfrac{X\wedge a}{X}\mid X]$ +3229,$\hat f$ +3230,"$\succ,\succeq$" +3231,$x_0=a/2\pi$ +3232,$0 0$ +3284,"$397,090,870 | -45.1% | $" +3285,$d(d-1)\approx d^2$ +3286,$0.99999$ +3287,$\mathbf {a_2'}$ +3288,$\Omega_0:=\{\omega\in \Omega\mid X(\omega)=\max(X)\}$ +3289,$R_2(1)$ +3290,"$(\Omega, P)$" +3291,$t_00$ +3332,$h_xx$ +3333,$\sigma_1=1/\sqrt{2}$ +3334,$X\not= Y$ +3335,"$j=1,\dots, k$" +3336,$a\ge 10$ +3337,$\mathsf{E}[e^{X_t}]=e^{\mu t + \sigma^2t /2}$ +3338,$\alpha_Y + 2 > \alpha_X > \alpha_Y+1$ +3339,$f^{-1}(\mathscr{F})$ +3340,$\mathsf{TVaR}_\alpha(X)=\dfrac{1}{1-\alpha}\displaystyle\int_{\alpha}^1 q(p)dp$ +3341,$\rho(X) \le 0$ +3342,$\int_0^\infty g(S(x))dx = \int_0^1 q(t)\phi(t)dt$ +3343,$1.25/0.7$ +3344,"$(lee.west |- lee.north)+(0,-2.5)$" +3345,$t_l = l f_{\max{}} / n$ +3346,$\mathcal M'\subset \mathcal M$ +3347,$H_r$ +3348,"$[t_1,1]$" +3349,"$\Delta\,g(S)$" +3350,$\mathsf{TVaR}_\pi(X)$ +3351,$\bar P(a)>\mathsf{E}[X\wedge a]$ +3352,$y^*$ +3353,$\mathcal{G}\subset\mathcal{F}$ +3354,$-\sqrt{x}$ +3355,$\Delta_n$ +3356,$a=a(t)$ +3357,"$D = \{ω : P(N, ω) = 1\}$" +3358,$0\le \hat \theta\le 1$ +3359,$\rho(X)=\mathsf{E}[\zeta_X X]$ +3360,$250K + \$ +3361,$g(s)/s$ +3362,$\rho(X)=\displaystyle\int_0^\infty x g'(S(x))f(x)dx$ +3363,"$(de.east |- lee.north)+(0.375,0.25)$" +3364,"$700,000 | \$" +3365,$\hat\theta_s<0.5$ +3366,$(X)$ +3367,"$\mathsf{E}[X_i(a)] = \displaystyle\int_0^\infty \gamma_{a,i}(x) \alpha_i(x) S(x) dx$" +3368,$\Theta^Y$ +3369,$A_\cdot$ +3370,$(1-p)/(p\nu_p^2)$ +3371,$\sigma=0.175$ +3372,$\{X\ge q(p)\}=\{X \ge 12\}$ +3373,"$[0, 1-p)$" +3374,$s>0.2$ +3375,$F_X=X\mathsf P$ +3376,$f(\mathsf{VaR}_p(X))$ +3377,$A=1$ +3378,$p\to 2$ +3379,"$\mathsf{Pr}(A\mid \mathscr{G}):\Omega\to [0,1]$" +3380,$\rho^{ho}_c$ +3381,$\beta_i(x)g(S(x))$ +3382,$a=a_0+(1+c)\mu$ +3383,$a_{gc}$ +3384,"$g(s) = \min(1,\exp(a+b\log(s)))$" +3385,$\lambda_0$ +3386,$S_t=\exp(\mu t + \sigma W_t)$ +3387,$(P-L)/A$ +3388,$A = \{ω\}$ +3389,"$(brR15 |- lee.south)+(-0.125,-0.25)$" +3390,$g'<1$ +3391,$\sigma=\tau=\rho$ +3392,$g(s)=s^{1/3}$ +3393,$=Q=\mathsf{MV}(a-X)^+$ +3394,$-1/2$ +3395,$\sup(X)<\infty$ +3396,$M_i\Delta X$ +3397,$t_1^* < t_1^{**}$ +3398,$s>s_0$ +3399,"$\forall B\in\mathcal{B}B(\mathbb{R}),\ \int_B \mathsf{E}[X\mid T=t]\mathsf{Pr}_T(dt)=\int_{T^{-1}}(B) X\,d\mathsf{Pr}$" +3400,$D ∪ C^c ⊃ N$ +3401,"$(p, 1-g^{-1}(1-p))=(p,\hat p)$" +3402,$\rho(\tilde X)=\mathsf{E}_{\mathsf{Q}}[\tilde X]$ +3403,$L=\mathsf E[X]$ +3404,$c=(1-\alpha)^{-1}$ +3405,$X(x)=1/x$ +3406,$\lambda \ge 0$ +3407,"$\bar P_{0,0}:=\rho(Y_{0,0})$" +3408,$=g(s)-s$ +3409,$\alpha f/(1-g)$ +3410,$\mathcal{S}$ +3411,$X\le \rho(X)$ +3412,$E[X|X>qp]$ +3413,$X\wedge a / X$ +3414,$\mathsf{Pr}_S$ +3415,$\kappa_i'(x)=1$ +3416,$v(B)$ +3417,$\mathrm{COC}$ +3418,$\kappa = 3.3\mathit{AEL}^{0.82}$ +3419,$q^-=q^+$ +3420,$X_k$ +3421,$E[X_\tau] \neq E[X_0] = 0$ +3422,$g(S(x)) / S(x)$ +3423,$c=\text{Var}(G)=\nu^2$ +3424,$\kappa_{1}$ +3425,$X^{\oplus n-1}$ +3426,$P/L$ +3427,$X=\sum_t B_t/2^i$ +3428,$x\mapsto (x-d)_+^{n}$ +3429,$\bar a_{75}=9.81$ +3430,$1_{X>a}$ +3431,$\mathsf{id}$ +3432,$P\in\mathcal{M}$ +3433,$a_1'=a_0-X_1$ +3434,$g''(s)=-s^{3/2}/4$ +3435,$\alpha + 1$ +3436,$a_t\approx (1-t)a_0+ta_t$ +3437,$p_i+R_i(r_i+a_i)$ +3438,$X_1 = aX + b$ +3439,$P/E[U]$ +3440,$\mathcal E=\mathsf{var}nothing$ +3441,$10^{-15} - 10^{-18}$ +3442,$m(\frac{m}{\lambda}-1)^2$ +3443,"$\mathcal G=\{g(s) \mid s\in[0,1],\ g\leftrightarrow m\in\mathcal E_{X,r} \}$" +3444,$r_i$ +3445,"$(0,R)$" +3446,$a1$" +3449,"$391,874,530 | | $" +3450,$R_0(1-t)>P(0)$ +3451,$\pi=1.2613$ +3452,$X=U$ +3453,$\prec_2^*$ +3454,$s=1-\alpha$ +3455,$p_R<0.5$ +3456,$P\approx \mathsf{E}[A(1)] + k\mathsf{Var}(A(1))/2$ +3457,$T_x\ge n$ +3458,$N=\sum_i N_i$ +3459,$=M_y=$ +3460,$\hat Z\tilde Z_{xn}$ +3461,$\mathsf E[X_i \mid X]$ +3462,$\{ x \mid F(x) \ge p \}$ +3463,$\lambda=0.25$ +3464,$kT$ +3465,$\mathsf{TVaR}_{0.5}$ +3466,$X_i\dfrac{X\wedge a}{X}$ +3467,$R_L$ +3468,$W'$ +3469,"$p_{k,n}=F((k+1)/n)-F(k/n)=\mathsf{Pr}r(X\in I_{k,n})$" +3470,$Z\mid X$ +3471,$g(0.25) < 1$ +3472,"$\mathbf{r}=(1,r_1,\dots,r_k)$" +3473,$\ln(10)=2.302585$ +3474,$\alpha_1SdX$ +3475,$f_{\mathbf{x}}$ +3476,"$4,500 | \$" +3477,$A/L-1$ +3478,$g(s)=s^{0.9}$ +3479,$L_X(X)=\rho(X)$ +3480,$\mathsf E[s_i]=nr_i\mathsf E[Z_i/\tau_n]=r_i$ +3481,$a_h=2-a_l$ +3482,$1-f$ +3483,$\mathbf {\Delta g(S)}$ +3484,$L_t$ +3485,"$\mathsf{TVaR}_p(X(x_1,x_2))=(x_1 + x_2)\mathsf{TVaR}_p(Y)$" +3486,$F\subset A$ +3487,$z_1=\mathsf{Pr}hi^{-1}(u_1)$ +3488,$X_s = X_{s_1} + X_{s_2}$ +3489,$\int_0^s \mu(dt)/(1-t)$ +3490,$K$ +3491,$P/(A-P)=P/Q$ +3492,$\int_0^1 f(s)ds = 1 - \alpha < 1$ +3493,$\mu=\tau(\theta)=\dfrac{ne^\theta}{1-e^\theta}$ +3494,$D_t = Y_t - Y_{t-1}$ +3495,$g_4(s)=s^{0.9}$ +3496,$eL + \rho S$ +3497,$\mathbf {\Delta X}$ +3498,$K=\mathsf{xTVaR}_p(X) = \mathsf{TVaR}_p(X) - \mathsf{E}(X)$ +3499,$s+\delta p$ +3500,"$(\sqrt k, \sqrt k)$" +3501,$(1+\rho)\mathsf{E}[C]$ +3502,$\mathsf{E}[X] + \pi \mathsf{E}[((X-\mathsf{E}[X])^+)^p]^{1/p}$ +3503,$(P-L) / (A-P)=$ +3504,"$g(s) = \max(g_{\text{min}}, g^0(s))$" +3505,$\mathsf{TVaR}_{0.5}(X_2)=45.5$ +3506,"$(x_i,y_i)$" +3507,$\bar{X^a} = \mathsf{E}[X^a|X]$ +3508,$\mu^{!g}$ +3509,$\mathsf P(X=X(\omega_0))>0$ +3510,$0.25 + U/4$ +3511,$\sqrt{0.1}=0.316$ +3512,$\mathbf {D^n\rho_{X\wedge 30}(X_1)}$ +3513,$N(a)=a-\mathsf{E}(X\wedge a)$ +3514,$\hat p:=1-g^{-1}(1-p)$ +3515,$g^a \pmod{p}$ +3516,$\sigma^2=1$ +3517,$\delta(p) F(x)=d_iF(x) + (v-\nu^*)\sqrt{FS}$ +3518,$\theta^2/2$ +3519,$\Theta=\{\theta \mid \kappa(\theta)<\infty \}$ +3520,$c(X) = u^{-1} ◦ E [u(X)]$ +3521,$X_1(v_1)$ +3522,$VaR_{0.98}(uX_1 + vX_2)$ +3523,$\mathsf{E}[X_i]\le D\rho_X(X_i)\le \rho(X_i)$ +3524,$g(s) = \mathsf{Pr}hi(\lambda + \mathsf{Pr}hi^{-1}(s))$ +3525,$\mathscr{F}_1$ +3526,"$p,0\le p\le 1$" +3527,$F=$ +3528,"$\pi(X,a)=\int_0^a S(x) + \delta(F(x))F(x)dx$" +3529,$\phi(X) = \mathsf{TVaR}_{0.99}(X)$ +3530,$\beta\to (\alpha+1)/\mu$ +3531,$\mathsf{TVaR}_{0.65}$ +3532,$D_t = (X_t^2 - t) - (X_{t-1}^2 - (t-1)) = X_t^2 - X_{t-1}^2 - 1$ +3533,$t_2$ +3534,$u'(0)=1$ +3535,$\mathsf{Pr}r(L'\ge 270)=65.1\%$ +3536,"$3.129=\lambda \sigma(Y_{0,0})$" +3537,$g(S_4)=0.5$ +3538,$N=r_a$ +3539,$\tilde S(x) = g(S(x))\ge S(x)$ +3540,$T\lambda=\mu$ +3541,$r_f = 0.01$ +3542,$G(x)=\mathsf{Q}(\{X\le x\}) = 1-g(1-F(x))$ +3543,$f(\theta)$ +3544,$Z\in L^0$ +3545,$p(x) = \mathsf{Pr}r(\{\omega\mid X(\omega) = x\})=\mathsf{Pr}r(X=x)$ +3546,$G=c_k(x)$ +3547,$\Theta_i^Y := \mathsf{TVaR}_{1-s_i}(Y)$ +3548,$\mathsf{Pr}r(X\le \mathsf{VaR}_p(X)) \not = p$ +3549,$q(w)=1_{w>\mathsf{Pr}hi^{-1}(p)}$ +3550,$\max(X)<\infty$ +3551,$\mathsf{E}_\mathbb{Q}(X_i) = \mathsf{E}_\mathbb{Q}( \mathsf{E}_\mathbb{Q}(X_i \mid X))$ +3552,$G\in\mathscr{G}$ +3553,$\mu_0$ +3554,$P = \rho(X)$ +3555,"$58,435,829 | 16.7% | -$" +3556,$g(s)=s^\rho$ +3557,$g'(0)=\infty$ +3558,$\theta_1=\theta_2$ +3559,$\omega'\not=\omega$ +3560,$\rho((X-a)^+)=0.273$ +3561,$B\subset\R$ +3562,$\mathsf{Pr}=\mu P_t$ +3563,$f_{\max{}} = 1 / b$ +3564,$q^-(p) := \sup\ \{x \mid F(x) < p \} = \inf\ \{ x \mid F(x) \ge p \}$ +3565,$r\lambda\mathsf{E}[X]$ +3566,$\mathsf{E}[h(T)X]=\mathsf{E}[h(t)\mathsf{E}[X\mid T]]$ +3567,$X_{t\wedge T}$ +3568,$\mathbf {M_{1}}$ +3569,$M=P-s$ +3570,"$U_1, U_2$" +3571,$g^2(s_{R+1})$ +3572,$f(0.x_1x_2x_3...) = 0.x_1x_3\dots$ +3573,$a_{0}=a(Y_{0})$ +3574,$m(m-1)(m\frac{a+1}{a}-\frac{1}{a})$ +3575,$x_2:=\mathsf{E}[X^2]$ +3576,$T_1 := X_{n+1} + \cdots + X_{N-1}$ +3577,$\int_0^1 \frac{dx}{x}=\infty$ +3578,$t_0^*\mathsf{E}[X_1]$ +3588,$C_p - C_v = R$ +3589,$λ^*(N)=1$ +3590,$\rho(X)$ +3591,$\nu>1$ +3592,$\omega>0$ +3593,$\text{E}(G^r)=\theta^r\mathscr{G}amma(a+r)/\mathscr{G}amma(a)$ +3594,"$6 billion, or nearly three times the industry's early estimates. Besides the damage from aftershocks, they said claims increased when inspectors saw damage to home interiors and foundations that were not visible in early, cursory inspections. Among the largest companies, State Farm Insurance now projects losses of nearly $" +3595,"$\langle \nabla X,\zeta \rangle$" +3596,$\mathsf P(I_t=1)=s_t=1-p_t$ +3597,$\alpha=\mathsf{E}[X_i\mid X]$ +3598,$t_1<0.5$ +3599,$\beta_2 gS$ +3600,$d=r/(1+r)$ +3601,$f_Y(y)=f_X(\log(y))/y$ +3602,"$q_X(U), q_Y(U))$" +3603,"$100,000-\$" +3604,$(1-\alpha)\phi_{R+1}^u$ +3605,$\mathsf{Var}(X_\lambda)=\lambda\mathsf{Var}(X_1)$ +3606,$a:=\rho(X)$ +3607,$R = P-L$ +3608,"$\{ f\in L^1\mid \mathsf{E}(f)=1, \forall X\in\mathcal{A}, \mathsf{E}(Xf)\ge 0 \}$" +3609,$\bar S_i(a)$ +3610,$P_k=g(S_k)\Delta X_k=\nu_k S_k+\delta_k$ +3611,$\mathsf{VaR}_{p}( \cdot \mid \mathcal F_t)$ +3612,"$\rho^*(\mu)\ge \sup_t \{ \langle \mu,X_t \rangle - \rho(X_t) \} \ge \sup_t \{ \langle \mu,X \rangle -t \langle \mu,\bar X \rangle - \rho(X) \}=\infty$" +3613,$x_1\mathsf{VaR}_p(X)]$ +3642,"$\rho(X)=\displaystyle\int_0^\infty g(S(x))\,dx$" +3643,$Q = C + lg$ +3644,"$r_f, r>0$" +3645,$\mathsf Q(A)>0$ +3646,$q(p')=q(p)$ +3647,$P_T$ +3648,$\mathsf{P}(\omega_H)=p$ +3649,$m^3/\lambda^2$ +3650,$G_1$ +3651,$R_2(t) = \bar P^a_2(t)/t$ +3652,"$1 minus the corresponding tranche payoff, $" +3653,"$X_1, X_2$" +3654,$\Delta(X\wedge a)$ +3655,$\mathsf{E}[W\mid X]=n^{-1}\sum_{T\in\mathscr{S}(X)} W\circ T=:\tilde W$ +3656,"$384,596,293 | 14.7% | $" +3657,$(X-a_1)^+ \wedge a_2=(X-a_1)^+$ +3658,$V(0)=0$ +3659,$g(1-p)=1-\tilde p$ +3660,$\mathsf{E}[X\wedge a] = \dfrac{k}{\beta-1}F(a)-\dfrac{a}{\beta-1}S(a)$ +3661,"$\px=\int_B \sum_i\mathsf{E}[X_i\mid \mathscr{G}]\,d\mathsf{Pr}$" +3662,"$10,000 | \$" +3663,$\mathbf {\beta_{2}}$ +3664,"$(1-g(S(x)),x)=(p,q(1-g^{-1}(1-p))$" +3665,$T_x\ge x$ +3666,$\mathsf E[X\mid\mathscr F]\preceq_2 X$ +3667,$X'=X\wedge a$ +3668,$\pi_X(t_{2j})\le \pi_Y(t_{2j})$ +3669,$\ge \mathsf{E}[X]$ +3670,$\mathsf{E}[A\wedge L]$ +3671,$\rho(X) = \max\ E[XZ]$ +3672,$g_o$ +3673,"$20,000 plus the difference between \$" +3674,$1-\tilde p=\tilde p(1)-\tilde p(p)=\int_p^1 (d\tilde p/dp)(s)ds = \int_p^1 g'(1-s)ds = \int_0^{1-p} g'(s)ds = g(1-p)-g(0)=g(1-p)$ +3675,$E(X\wedge a)=\int_0^a tf(t)dt + aS(a)$ +3676,$\Theta$ +3677,$(\delta^{\star}-d)\sqrt{S(x)F(x)}$ +3678,$X^\star$ +3679,$0.2$ +3680,"$\{\rho^x_h(s):s\in[s_R,1]\}$" +3681,$\prec_1^*$ +3682,$=\mathsf{E}[X_i \mid X\le a]$ +3683,$M_{2}$ +3684,$g'(1-s)ds$ +3685,$\mathsf{E}[g(X_n)]\to \mathsf{E}[g(x)]$ +3686,$\rho(X\mid \mathcal F_1)$ +3687,$\mathsf P_{s+1}X_{s-i}$ +3688,$T\ge n$ +3689,$g(s^2)$ +3690,$b=0.5$ +3691,$g_2$ +3692,$S \to 1$ +3693,$S_{t+dt}=S_t + \mu dt + \sigma dW_{dt}$ +3694,"$[p_{j-},p_{j+}]$" +3695,"$P(A, ω)$" +3696,$v=4/5$ +3697,$1-g(1-p)$ +3698,$dW_t\approx W_{t+dt}-W_t$ +3699,$\delta_k$ +3700,$\pi(X) = \rho(X\wedge \alpha(X))$ +3701,$\tau_i\in \mathcal R'$ +3702,$F_0=2.5$ +3703,$\mathsf{Pr}(A_0)=0$ +3704,$L_\sigma=L_1$ +3705,$m^3$ +3706,$\sum_{i=1}^{k} x_i$ +3707,$\mathscr{G}amma(z + 1) = z\mathscr{G}amma(z)$ +3708,$\dfrac{\partial S}{\partial v}=\mathsf E[X_i \mid X]$ +3709,$\mathsf{E}[X_t X_s] - \mathsf{E}[X_t X_{s-1}] - \mathsf{E}[X_{t-1} X_s] + \mathsf{E}[X_{t-1} X_{s-1}]$ +3710,${kex:.3f}$ +3711,$\bar P_{0}$ +3712,$\{N=n\}$ +3713,"$\mathscr{G}amma(S,T)\in\mathcal{A}\otimes\mathcal{B}$" +3714,$x^+$ +3715,$0\le x\le 1$ +3716,$s(0)=s_0=0$ +3717,$(1-p)^{-1/2}/4$ +3718,"$[0, t_0]$" +3719,"$ falling in equally sized bands, are generally not equally likely: they reflect the specific distribution of $" +3720,"$(t_3,t_4)$" +3721,$S_{X(t)}(x)$ +3722,$\lambda=\sum_i \lambda_i$ +3723,$1-S$ +3724,"$D^f\rho_{X\wedge a,X}(X_i(a))$" +3725,$A_j$ +3726,$( x_{(j)}-x_{(j-1)} )$ +3727,$T(y)=y$ +3728,$(k+1)/n$ +3729,$z(x)$ +3730,$1_{U<0.2}$ +3731,$P(t) = (1-t)P(0)+tP(1)$ +3732,"$\displaystyle\int_0^a \kappa_i(x)g'(S(x))f(x)\,dx + a\beta_i(a)g(S(a))$" +3733,$t\not=0.5$ +3734,$\log(\sqrt{2\pi})=0.399090$ +3735,$\rho_{m'}(Y) > \rho_m(Y)$ +3736,$\mathsf{E}[X_i (X\wedge a)/X]$ +3737,$x_0=0$ +3738,$a=80$ +3739,$\rho(Y)$ +3740,$-\sigma^2/2$ +3741,$\mathsf{E}[F_m^n] / \mathsf{E}[F_0^n]$ +3742,$\mathsf E[X]=\mathsf E[\mathsf E[X\mid I]]$ +3743,$A_0$ +3744,$s+T$ +3745,$L a) \le \epsilon$ +3848,$P(X_{-1}(a_{gc}))={{mvp_gc}}$ +3849,$x_0$ +3850,"$(L,\mathcal{A})=(\Omega,\mathscr{F})$" +3851,$\sigma^2c^{2-p}\to \infty$ +3852,$\bar B\setminus B$ +3853,$\mathsf{Pr}r(X<2)=1/6<\mathsf{Pr}r(X\le 2)=1/3$ +3854,$\rho(X)=\mathsf{E}[Xg'S(X)]$ +3855,"$\mathsf{TVaR}_p(Y)\in R_{Y:X,r_X}$" +3856,$\rho(X)=\mathsf{E}(XZ)$ +3857,$\Delta X=X_1$ +3858,$db+v_1<0$ +3859,$\mathsf{TCE}_p(X) := \mathsf{E}[X \mid X \ge \mathsf{VaR}_p(X)]$ +3860,$\delta(p)=1-\nu(p)$ +3861,$D^n\rho(\cdot)$ +3862,$\alpha_2SdX$ +3863,$1-\alpha_i(x)S(x)$ +3864,"$I_x=[x, x+dx]$" +3865,$x=1$ +3866,$\mathbf R_{< 0}$ +3867,$t=s_R$ +3868,"$\lambda=0.045,\ 0.0625,\ 0.085,\ 0.125,$" +3869,$f_i$ +3870,$a = \sum_i a_i$ +3871,$\rho>1$ +3872,$1/4$ +3873,$\nabla \zeta$ +3874,$\Delta=a'-a$ +3875,$\mathcal V(X)=\frac{1}{1-p}\mathsf{E}[X^+]$ +3876,$\mathit{NPV}_{\infty}=a_xF_0$ +3877,$M_F$ +3878,$V_1(m)=m^3V(1/m)$ +3879,"$=\displaystyle\int_B^{\phantom{X}} \mathsf{E}[X\mid T=t] \,\mathsf{Pr}_T(dt)\quad$" +3880,$h(p)\le p$ +3881,$p=0.25$ +3882,$t \ne 0$ +3883,$g(0+)=\mu(\{1\})$ +3884,$\mathcal{F}_k$ +3885,$E\in\mathcal{A}_y\ \forall y\in Y\setminus N$ +3886,$P(x)=g(S(x))$ +3887,"$dx,dt,ds$" +3888,"$350,000,000), or if at any time the authoritys available capital is insufficient to pay benefits and continue operations, the authority shall have the power to assess participating insurance companies subject to the maximum limits as set forth in this section and Section 10089.30. The assessment shall be limited to the amount necessary to pay the outstanding or expected claims and claim expenses of the authority and to return the authoritys available capital to three hundred fifty million dollars ($" +3889,$c\ge \mathsf{E}[cZ]$ +3890,$D_g=\infty$ +3891,$1-1/2 + 1/3-1/4+\cdots$ +3892,"$I_{(x,x+y]}(X):=(X-x)^+\wedge y$" +3893,$g(S(x))=1-\tilde p$ +3894,"$\min(X, a)=X\wedge a$" +3895,$\mathsf{E}[X\wedge a]$ +3896,"$\mathsf{E}[X_{t,d}\mid \mathcal F_{\tau}]$" +3897,"$\mathrm{ED}^*(\theta, \lambda)$" +3898,$p_1=3-p$ +3899,$\mathsf Eg(X) \le g(\mathsf EX)$ +3900,$\rho(X+\lambda Y)\ge \rho(X) +\lambda \mathsf{E}[gY]$ +3901,$W_t = B_t/\sqrt{T}$ +3902,$\mathsf{E}_Q(X_iY_i)=\mathsf{E}_Q(X_i\partial Y/\partial X_i)$ +3903,$\mathrm{Sh}(\omega)$ +3904,$\text{Var}(G)=c$ +3905,$t<0.25$ +3906,$\mathscr{P}$ +3907,$(1-\lambda m)^2/\lambda^2$ +3908,$\mathsf{COH}$ +3909,$0\le \lambda\le 1$ +3910,"$\forall E\in\mathcal{A},\ \forall F\in \mathcal{B}$" +3911,$h_\epsilon\to h$ +3912,$q(\omega)=\dfrac{1}{1+r}\dfrac{Q(\omega)}{P(\omega)}$ +3913,$\mathsf{E}[XZ]$ +3914,$p=F$ +3915,"$S\,\Delta X$" +3916,$\lambda' = (\mu^{2 - \xi})/ (2 - \xi)$ +3917,"$\psi:(L\times M, \mathcal{A}\otimes \mathcal{B})\to\mathbb{R}$" +3918,$X\in L_p$ +3919,$\mathsf{Pr}(A\cap B)=0$ +3920,$f_0 * f= f$ +3921,"$\ddot{a}_{x, n\!\urcorner}$" +3922,"$\mathsf b=(b_0,b_1,\dots, b_{n-1})$" +3923,"$f: [0,1]\to [0,1]$" +3924,$1-s$ +3925,$\beta_i(t)$ +3926,$\mathsf{Pr}si(x)=1-\exp(-e^x)$ +3927,$f_i(X)$ +3928,$X\preceq_2 Y$ +3929,$V(\mu)\propto \mu^p$ +3930,$30+10t$ +3931,$d(y;y)=0$ +3932,$\mathsf Q \ll \mathsf P$ +3933,$p_i\not=p_j$ +3934,$\rho_g(X)=\mathsf{E}_\mathsf{Q}[X]$ +3935,$1-g(S_k)$ +3936,$g'(s)\ge 0$ +3937,$S\mathsf{Pr}=\mathsf{Pr}_S$ +3938,$\rho(X) = \mathsf{E}(X) + c\mathsf{E}(X-\mathsf{E}(X))_+$ +3939,$e^m$ +3940,$F'=f$ +3941,$2.764\times 10^3$ +3942,$A=g^a$ +3943,$q(1-g^{-1}(1-p)$ +3944,$50Z$ +3945,$0 \le p \le 1$ +3946,$\mathsf{E}[\theta]=1$ +3947,$\partial L/\partial \mu$ +3948,$\mathcal{G}$ +3949,$X_t=ct + \sigma B_t$ +3950,$\mathsf{j}(90)=6$ +3951,$\mathsf{Pr}(A\mid \mathscr{F}_1)$ +3952,$\beta_i(x)/\alpha_i(x) q(p)] \ge q(p)$ +3968,$Z=AX + (1-A)Y$ +3969,$v\mathrm{EL}+da\ge \mathrm{EL}$ +3970,$c=q(\alpha)=VaR_\alpha(X)$ +3971,"$\mathsf{var}(W)=\sum_{d\ge 0} \mathsf{var}(Y_{-d,d})$" +3972,$L_i$ +3973,$\sup_t \mathbb{E}[|X_t|^p] < \infty$ +3974,"$u_1,u_2$" +3975,$x' - \mathsf{E}_\mathsf{P}[X]$ +3976,$Q_j=1-g(S_j)$ +3977,$F^{-1}$ +3978,$a^i = \nabla_i a$ +3979,$x_{\min{}}\not=0$ +3980,$(0.333...)(0.15)$ +3981,$\rho(\rho(X))=\rho(X)$ +3982,"$100K, and pays the first \$" +3983,$F_i$ +3984,$^{***}$ +3985,$0.5 < t_1 < t_2$ +3986,$r_x$ +3987,"$a_{0,t}' = a_{0,t}$" +3988,$\mathsf Q_{X}$ +3989,$X\wedge a\Delta g$ +3990,$g(s)=(\iota+s)/(\iota+1)$ +3991,"$5,873,000 | | -$" +3992,$\mathsf{P}(\{\omega_i \}\cap B)=0$ +3993,$K = A - P$ +3994,$P = \sum_k P_k$ +3995,"$\mathsf{P}(a,b]=b-a$" +3996,$da1_{X>x}$ +3997,$\mathsf{E}[X_1]\le R_1(t)\le P(1)$ +3998,"$(X_1, \dots, X_d)$" +3999,$F(q^-(p_0))=p_+>p_0$ +4000,$\mathsf{E}[X\mid \mathcal F_t]$ +4001,$a(X(t))=a(t)$ +4002,"$\displaystyle\int_0^a \kappa_i(x) f(x)\,dx + a\alpha_i(a)S(a)$" +4003,$\lim_{\gamma\to\infty} \rho_\gamma$ +4004,${}^{[<98]}$ +4005,$a=(1-f)^2/\nu^2=(1-f)^2/c$ +4006,$\lambda V_X(m/\lambda)$ +4007,"$Y_n=\max(X_1,\dots,X_n)$" +4008,$\iota' := (\iota_U/\iota^*)(1-a)$ +4009,$v={v}$ +4010,$\mathscr{F}FF$ +4011,$z_p^{(2)}$ +4012,"$j = 1, 2, \ldots, M$" +4013,$(0.304-0.2)/(1-0.304) = 15$ +4014,$\bar Q_i(a)=a_i - \bar P_i(a)$ +4015,$\rho(X)=E_\mathsf{Q}(X)$ +4016,$\dfrac{q(\epsilon)}{1+\epsilon}$ +4017,$\| Y \|_{\sigma_2} \le c \| Y \|_{\sigma_1}$ +4018,$\{X_t > a(t)\}$ +4019,"$\int_B \mathsf{E}(X\mid\mathscr{G})(\omega)\,\mathsf{Pr}(d\omega)$" +4020,$s<0.3$ +4021,$s \ge p$ +4022,$0{s}$ +4033,$a=\mathsf{VaR}_{1-\tau}(X)$ +4034,$\mathsf{E}[X_1\mid X < 2^{-m}]$ +4035,$\sup \{ \mathsf{E}[X\mid A] \mid \mathsf{Pr}r(A) > 1-p) \}$ +4036,$\mathbf {g_1(s)=s^{0.4}}$ +4037,$xS(x)\vert_0^\infty =\lim_{x\to\infty} xS(x)=0$ +4038,$t\to\infty$ +4039,$1 < x < 2$ +4040,$\mathsf{Pr}r(N\in I + kP)$ +4041,$\mathsf{E}[X]/P$ +4042,$N=2$ +4043,$\sigma_a$ +4044,$a_1=\int_0^1 (\partial a/\partial x_1)dt=\partial a/\partial x_1$ +4045,$\hat\rho(A_k) = \hat\rho(A_0) + k \rho(N)$ +4046,"$L=\min(\max(X-a, 0), y)$" +4047,"$\rho_{m_Y}\in\mathcal R^c_{X,r}$" +4048,$\zeta_1=0.85$ +4049,$-t^2/2$ +4050,$C_v = \frac{3}{2}R$ +4051,$(1-t)X_1 + tX_2$ +4052,$\iff\rho$ +4053,$\tilde Z_X:=\mathsf{E}[Z\mid X]$ +4054,$\mathsf E[X_i \mid X](\omega) = \mathsf E[X)i \mid X=X(ω)]$ +4055,${lnn}$ +4056,$g(1-F(x))=1-p$ +4057,$X^{<1>}$ +4058,"$k=6,7,8,9$" +4059,$0.417/0.4=1.0425$ +4060,$U_X < p$ +4061,$\delta=0.130$ +4062,$\rho(W_2)$ +4063,$\partial f(x_0)$ +4064,$\mathsf{E}_\mathsf{Q}[\lambda X] = \lambda \mathsf{E}_\mathsf{Q}[X]$ +4065,$P(a) = \nu S(a) + \delta = \nu (S(a) + \iota)$ +4066,$M=\delta a'$ +4067,$e^{\theta y}$ +4068,$C_2(0)\approx \mathsf{E}[X_2]$ +4069,$A - \mathsf E[A] = A_0 + (\mathsf E[X]N - \mathsf E[A])$ +4070,$M_1$ +4071,$7.35 \times 10^{22}$ +4072,$-br-v_i<0$ +4073,$0\le m\le N$ +4074,$S(x)=e^{-x/\mu}$ +4075,"$X_{j}, j=1,\dots, N$" +4076,$\mathbf {M_1\Delta X}$ +4077,$V(m)=m(1+m)$ +4078,"$\phi:(E, \mathsf{E}E)\to(\mathbb{R}, \mathcal{B}B(\mathbb{R}))$" +4079,"$(s,g(s))=({s0:.3g},{gs0:.3g})$" +4080,$\px=\mathsf{Pr}(B\cap A)$ +4081,$t>0$ +4082,$R^2_j$ +4083,$\sum v_iX_i$ +4084,$t_12$ +4097,$X(\psi)=X(\omega)$ +4098,$S \implies \ g$ +4099,$\mathsf{skew}(G)=\mathsf{skew}(G')$ +4100,${10\choose 5} = 252$ +4101,$\lambda_{\text{max}}$ +4102,$2^{10}$ +4103,"$X=X(x_1,\dots,x_n)=x_1X_1 + \cdots + x_nX_n$" +4104,$10^{-10}$ +4105,${}^{1}$ +4106,$A(-X) + B(X)=0$ +4107,$\sigma(\mathcal{A}\otimes \mathcal{B})$ +4108,"$I_{(x, x+dx]}$" +4109,$E[\tau] < \infty$ +4110,$\sum_i m_i=1$ +4111,"$I_{i,n}$" +4112,$\mathbb{P}$ +4113,$1000(a/(1000+X_1))$ +4114,$F_T(x)(\omega) = 0$ +4115,$1 + \mu^2 + \sqrt{1+\mu^2}$ +4116,$0.8\times 1.2 = 24/25$ +4117,$g(s)-s$ +4118,"$s_i, i=R+1,\dots,m$" +4119,$\rho(A_k) \ge k\mathsf{E}[N]$ +4120,$\mu_i$ +4121,$0\times\infty=0$ +4122,$\alpha > -2$ +4123,"$4.7\times 10^{21} / 10^{19} = 470 \text{\,seconds} \approx 8\text{mins}$" +4124,$\exp(\mu-\sigma^2/2)$ +4125,$\ge 0$ +4126,$>1/a_t$ +4127,$\bar M_t$ +4128,$\mathsf{E}[M_t | \mathcal{F}_s] = M_s$ +4129,$\nu(dx)=c(x)dx$ +4130,$5.186592 \times 10^{19}$ +4131,$P(a) = E_g(X\wedge a) =$ +4132,$\mathsf{Q}\in\mathscr{P}$ +4133,$c_{max} := c_r + \Delta_{i^\star} s^{\star}\Theta_{i^\star}^X$ +4134,$g(0+)>0$ +4135,$U(2)=0$ +4136,$R_1(0)=\bar P^a_1(0)$ +4137,$\mathsf{E}_{\mathsf{Q}}[(X-a)^+] \le \rho((X-a)^+)$ +4138,$1-p_0$ +4139,$\log(\mathsf{E}[e^{\pi X}])/\pi$ +4140,$\bar\theta=0.5$ +4141,$\mathsf{Var}(\mathsf{CP}_2)=\lambda(\mu/\lambda)^2x_2=\mu^2(x_2/\lambda)$ +4142,$L_a$ +4143,$^{128}$ +4144,$\nu < p$ +4145,$x:3x:9x$ +4146,$T_1$ +4147,$a_l-1<0$ +4148,$Np=67.45$ +4149,$\mathit{AEL}$ +4150,$\alpha+1=1/(p-1)$ +4151,$(a-X)^+=a-X$ +4152,$S_X(y)$ +4153,$\bar\iota>0$ +4154,"$Y_{t,0}$" +4155,$d=0$ +4156,$\sup_\mathsf{Q} \mathsf{E}_\mathsf{Q}[X]$ +4157,$E_2 = 0$ +4158,"$S\subset \Omega=\{1,\dots,N\}$" +4159,$P^i$ +4160,$A=A(J)$ +4161,$g'(1-p^* )=1$ +4162,$c(y)=$ +4163,"$\Omega=\{\omega_1, \ldots, \omega_6\}$" +4164,$\lim_{x\to\infty} xg(S(x))=0$ +4165,$\mathsf{E}[f(S_t)]=e^{\mu t + \sigma^2t /2}$ +4166,$\delta^*$ +4167,"$(I_1,\dots,I_n)$" +4168,$\mathcal{F}_{t-1}$ +4169,$-0.00002$ +4170,$\mathbb R/A$ +4171,$s_I$ +4172,"$\max(a_0, a_1) < a'< a$" +4173,$\phi'(s)=f(s)/(1-s)\ge 0$ +4174,$q(1)=\infty$ +4175,$\kappa_2$ +4176,$\mathbf {d}$ +4177,$\mathsf{VaR}(X)=1000$ +4178,$(1+\gamma)^{t-x}$ +4179,$\rho(\lambda X)$ +4180,$v(m)$ +4181,$35.40 | \$ +4182,$v=1/2$ +4183,"$r \in [\mathsf{TVaR}_0(X), \mathsf{TVaR}_1(X)]$" +4184,$s^{\alpha}$ +4185,$r=0.05$ +4186,$P(x) = g(S(x))$ +4187,$\iota=$ +4188,$F_1^{-1}(\omega_1)-F_2^{-1}(\omega_1)$ +4189,$S(x)+R(x)$ +4190,$1-L/P = (P-L)/P$ +4191,$\rho(X)\le\lim \rho(X_n)$ +4192,$0 \le d \le 1$ +4193,$S_Z$ +4194,$\zeta\in\mathcal{M}$ +4195,$3.40 per $ +4196,$d^\ast = 2g^\ast-1$ +4197,$a(X(\mathbf{v}))=a(\mathbf{v})$ +4198,$U(X)$ +4199,$0.8\le p<0.9$ +4200,$\sigma=1.333$ +4201,$\mathit{ROL} = Q_{\alpha_{1}}( \alpha_{0} + {Q_{\alpha_{1}}}^{- 1\ }( \mathit{AEL} ) )$ +4202,"$c\in[0,1/2]$" +4203,$q_-=\sum_{z_i > \zeta} p_i$ +4204,$\mathsf{E}[X] + \pi \mathsf{Var}(X)$ +4205,"$C = \{ω:P(\{ω\}, ω) = 1\}$" +4206,$\dfrac{d}{da}$ +4207,$\alpha_i(a)=\mathsf{E}[X_i/X \mid X > a]$ +4208,$V_{G_\nu}$ +4209,$(\theta)$ +4210,$\mathsf{E}(X ; B)$ +4211,$\delta = g(s)g(t)-g(st)$ +4212,$\rho(\lambda X)=\lambda \rho(X)$ +4213,"$\lambda p_{k,n}$" +4214,$s=0.25$ +4215,$k+1$ +4216,$E(G-E(G))^3 = 2a\theta^3$ +4217,$Q_j = 1 - g(S_j)$ +4218,$\omega_2$ +4219,$\sigma=0.075$ +4220,$b\!\!\urcorner$ +4221,$-\pi < \phi \le \pi$ +4222,$J(1)$ +4223,$\rho(X) = \mathsf{E}[X] + \lambda \mathsf{E}[(X-\mathsf{E}[X])^+]$ +4224,$\ge 5000$ +4225,"$F(\omega, x) = \mathsf{Pr}(X\le x\mid\mathscr{G})(\omega)$" +4226,$E_2\not=0$ +4227,$X:\{\text{Explicit Events}\}\to\mathbb{R}$ +4228,$\tilde px)$ +4242,$H_k(X)=H_{g_k}(X)$ +4243,$0.9999\dots=0.9 \times (1 + 0.1 + 0.001 +\cdots) = 0.9 \times (1 - 0.1)^{-1} =1$ +4244,$2***52 \le f 2^N < 2**53$ +4245,$ and derives $ +4246,$1-g(p)$ +4247,$p^\star \le p_i < p_j$ +4248,$T(y)$ +4249,$\rho_t(X) \ge \mathsf E[\rho_{t+1}(X)\mid \mathcal F_t]$ +4250,$g(s) = s^\alpha$ +4251,$\hat\rho(A_{k_0}) \ge \rho(A_{k_0})$ +4252,$1/V(\mu)$ +4253,$K(t):=\log\mathsf{E}[e^{tY}]=\kappa(t+\theta)-\kappa(\theta)$ +4254,"$c_i=\displaystyle\int_0^1\dfrac{\partial c}{\partial x_i}(tx)\,dt$" +4255,$E_{\mathcal{B}bb{Q}}[Y]=E[Yg'(S(X))]$ +4256,"$\nu(B)=\int_B P(A\mid\mathscr{G})\,d\mathsf{Pr}$" +4257,$\mathsf{E}[\cdot]$ +4258,"$1,000 | 0 | 0 | \$" +4259,"$(F(x),x)=(1-S(x), x)$" +4260,$\theta(\mu)=\arctan(\mu)$ +4261,$\mathbf {D^n\rho_{X\wedge 30}(X_2)}$ +4262,"$100 million loss event, without specifying details. Events defined by values of $" +4263,$S(x)=e^{-\beta x}$ +4264,$\mathsf{E}[X_i\mid X=x]$ +4265,$\mathsf{E}_{\mathsf Q}[X_i \mid X=x] = \mathsf{E}[X_iZ \mid X=x]/\mathsf{E}[Z \mid X=x] = \mathsf{E}[X_i \mid X=x]$ +4266,$\sigma=1.0$ +4267,$-3$ +4268,$A = \bigcup_{n=1}^{\infty} A_n$ +4269,$Z_8$ +4270,$\tilde\Theta_1$ +4271,$01$ +4280,$\rho(X)=\mathsf{VaR}_{0.995}(X)-\mathsf{E}[X]$ +4281,$U\le u$ +4282,$\sum_i v_i=0$ +4283,$\bar\delta a$ +4284,$y>0$ +4285,$a_1'$ +4286,$X_i(a)\not= X_i\wedge a_i$ +4287,"$10,000 | | $" +4288,$x_1\leftrightarrow y_2$ +4289,$da\to 0$ +4290,$\alpha=\frac{2-p}{1-p}$ +4291,$\mathsf{var}phi(t) = P^\omega_t X(\omega) = \mathsf{E}[X\mid T=t]$ +4292,$y^{\ast}-x^{\ast} \ge \epsilon$ +4293,$\mathsf{E}[\cdot\mid X]$ +4294,$X=X\mid\theta$ +4295,$X^i(\epsilon)$ +4296,"$e', s', r'$" +4297,$\rho(X_i)<\infty$ +4298,$\tilde p=1-g(1-p)$ +4299,$\lambda_t S(a)$ +4300,$\log(x)$ +4301,$R_i > C_i$ +4302,$\mathbf {X_2(a)}$ +4303,$X+100$ +4304,$g(s) = \dfrac{r_o+s(1+r_K)}{1+r_o+r_Ks}$ +4305,"$w'_{i,j}$" +4306,"$\mathscr{P} =\{1+\lambda(\zeta-\mathsf{E}\zeta) \mid \zeta\ge 0, \|\zeta\|_q\le 1 \}$" +4307,$K_h(t):=k(h+t)-k(t)$ +4308,$\mathsf{S}=(\rho_{ij})$ +4309,$h'$ +4310,$0.61u(52)>0.63 u(50)$ +4311,$q^-(p)=\mathsf{VaR}_p(X)$ +4312,$\xi(A\mid\cdot)$ +4313,$\mathsf{E}[D_t \mid \mathscr{F}_{t-1}] \leq 0$ +4314,$\omega^n=1$ +4315,$L/P$ +4316,$\mathsf{Pr}r(X\wedge a > a)=0$ +4317,"$827, the IRR becomes 10%; if it is \$" +4318,$X=X(\bar x)=G\circ F(\bar x)=GF(\bar x)$ +4319,$\kappa_i(x)=E[X_i \mid X=x]$ +4320,$PV=kNT$ +4321,$Y=c\in \mathbb R$ +4322,$\sigma=0.1980$ +4323,"$u\in[0, 1-p]$" +4324,$e_x$ +4325,$\nu=1$ +4326,$\delta_{p_j}$ +4327,$s^*=1-p^*\le 1$ +4328,$\mathsf{E} X + c\mathsf{E}[\vert X-\mathsf{E} X \vert^p]^{1/p}$ +4329,$g'(1-s)+g(0+)\delta_1$ +4330,$\mathsf{E}=n^{-1}\mathsf{M}'\mathsf{M}$ +4331,$\sigma_1 Z_1 + \cdots \sigma_T Z_T$ +4332,$1 for each $ +4333,"$1 between any of the layers, then $" +4334,$L_a^{a+da}=a-\mathit{RV}$ +4335,$M_{1}\Delta X$ +4336,$g(s) \approx m_0+(1+m'(0))s$ +4337,$\pi(x)$ +4338,$\mu^3+2\mu^2+\mu^2\sqrt{\mu^2+2\mu}$ +4339,"$=(0.231, 0.002, 0.257, \dots, 0.008)$" +4340,$\mathsf{TVaR}_{0.95}(X)=3699$ +4341,$\Delta_{i} X_{s-i} = (\mathsf P_{i+1} - \mathsf P_{i})X_{s-i}$ +4342,$T_*=10^7\mathrm K$ +4343,$\nabla\rho(X)$ +4344,$qq$ +4345,$q_{X+Y}=q_X+q_Y$ +4346,$sZ_i$ +4347,$\mathcal K$ +4348,$\mathsf{E}_{\mathsf Q}[X_j]$ +4349,$\sup_\mathsf{Q} (\mathsf{E}_\mathsf{Q}[X] - l(Q))$ +4350,$\displaystyle\int_0^\infty xg'(S(x))f(x)dx = -xg(S(x))\mathcal{B}ig\vert_0^\infty + \displaystyle\int_0^\infty g(S(x))dx=\displaystyle\int_0^\infty g(S(x))dx$ +4351,$\alpha>0$ +4352,$e^{-\theta x}$ +4353,$\gamma=0.421$ +4354,$λ^*(N) = 1$ +4355,$\int_{1-p}^1 \phi(t)dt =\int_0^p \phi(1-t)dt=g(p)$ +4356,$D=(X-a)^+$ +4357,"$(\Omega,\mathscr{F},\mathsf{Pr})$" +4358,"$\displaystyle\int_B \mathsf{Pr}(A\mid\mathscr{G})\,d\mathsf{Pr}$" +4359,$b^2-c<0$ +4360,$\rho_i(X_i)$ +4361,$c(S)=g(\mathsf{Pr}r(S))$ +4362,$\tilde{\tilde R_i}$ +4363,$U(0)=2$ +4364,$(1/\mathsf{LR_{dist}}-1)/(1/\mathsf{LR_{plan}})-1$ +4365,"$uv\in (s_1, s_0]$" +4366,"$X_1, \dots, X_n$" +4367,$n=\sum_i n_i$ +4368,$0 \le t \le 1$ +4369,$x_1+y_1 \le x_2+y_1\le x_2+y_2$ +4370,$a = 8.1484$ +4371,$\alpha$ +4372,$\mathit{NPV}$ +4373,$v-\nu^*=(\iota^*-i)/v\nu^*$ +4374,$\alpha-A(n)$ +4375,"$t=2,3,\dots$" +4376,$\mu_X\le\mu_X$ +4377,$a_2 = 2.157$ +4378,$\mathsf{Pr}r(\mathsf{var}nothing)=0$ +4379,$\sum_i a_i=a$ +4380,$1/g'(0)$ +4381,${}^{[<104]}$ +4382,$0 \le g(s) \le 1$ +4383,$\mathbf x$ +4384,$\nu=f\mu$ +4385,"$L^\infty(\Omega, \mathsf{P})$" +4386,$0\le Y\le 1$ +4387,$R(a)=\delta N(a)$ +4388,$\int X_n=1$ +4389,"$Y_{0,1}$" +4390,$\mathsf{E}(X) = \int_0^1 q(p)dp$ +4391,$t\mapsto P(t)$ +4392,$t=T_x2$ +4419,$\int_0^a$ +4420,$\rho(X)\ge 0$ +4421,$a=R+Q$ +4422,"$\px=\int_B \sum_i X_i\,d\mathsf{Pr}$" +4423,$\mathcal M$ +4424,$\mathsf{E}(W/X | X\ge x)$ +4425,$g=h$ +4426,"$\rho(X,\mathsf P_I)(\omega)=\rho(X, \mathsf P_{I(\omega)})$" +4427,$\int_0^1 F^{-1}(p)dp$ +4428,$X(\omega)\wedge a$ +4429,$1_\omega$ +4430,$\phi_{R+1}^s$ +4431,$X_m$ +4432,$\rho(X+c)=\rho(X)+c$ +4433,"$2\mu, 2\sigma$" +4434,$D_i^n$ +4435,$\Delta A$ +4436,$\mathsf{E}_\mathsf{Q}[X_1]$ +4437,"$L,P,M,Q,a,LR,PQ,COC$" +4438,$p \in G$ +4439,$g(S(x))=\exp(-\alpha H(x))$ +4440,$0.375/1.5 = 0.25$ +4441,$V(m)\sim c_0m^p$ +4442,$\mu_x = -d\log({{}_tp_x})/dt = \lim_{t\downarrow 0} {}_tq_x/t$ +4443,$\mathsf{E}[|X|]<\infty$ +4444,$A_n$ +4445,$\iota^*$ +4446,$P(\Omega)=1$ +4447,$V(\mu)\propto \mu$ +4448,$ | > \ | \$ +4449,$1/\sqrt{\alpha}\to 0$ +4450,$u'''' \le 0$ +4451,$I_x$ +4452,$\mathsf E[X_i]$ +4453,$h(X)=(X-\mathsf{E} X)$ +4454,"$\rho(Z)=\sup_{\zeta\in\mathcal{A}} \langle \zeta, Z \rangle$" +4455,$\rho(X+m)=\rho(X)-m$ +4456,$X_{1c}$ +4457,$\mathsf{TVaR}_{p_0}(X)$ +4458,$\xi(\cdot\mid t)$ +4459,$K = A^{k}=g^{ak} \pmod{p}$ +4460,$\nu=\nu(a)<1$ +4461,$I=A$ +4462,$c^*(y):=c(y)e^{l(y;y)}$ +4463,$\mathsf{Var}(X)$ +4464,$X_0=X_1=0$ +4465,$\mathscr{O}(\eta)$ +4466,$K_i = \mathsf E[X_i \mid X \ge a] - \mathsf E[X_i]$ +4467,$t0$" +4484,$n^{-1}\mathsf{M}'\mathsf{M}$ +4485,$e^{-r_Dt}$ +4486,"$(s_0, 1]$" +4487,$\mathsf{E}[e^sX]<\infty$ +4488,$r_L$ +4489,"$\rho^E(t_1,t_2)(X)=c$" +4490,$N_t$ +4491,$T_*\propto |v|^2\propto GM/R$ +4492,$\kappa(\theta)=\theta-\theta\log(-\theta)$ +4493,$\sum_i a_i=0$ +4494,$\lim_{\mu\to 0} V(\mu)=0$ +4495,$\rho_{PH}$ +4496,$2.576\times 11$ +4497,$\mathsf{E}[N]=2.0$ +4498,"$X_1,\dots,X_m$" +4499,$\bar h$ +4500,$g'(S(x))dF(x)$ +4501,$\nu (1-s)$ +4502,$Q_P$ +4503,$h_\epsilon$ +4504,$\rho(X) = \mathsf{E}(X) + \| (X-\mathsf{E} X)_+ \|_p$ +4505,"$S=[0,2\pi]$" +4506,$\sum_i E[X_i|anything]\le _{cx} \sum X_i \le_{cx} F_{X_i}^{-1}(U)$ +4507,$s<0.03$ +4508,$X\succeq Y$ +4509,$\rho(kX)$ +4510,$\Omega = \{\rho_g\}$ +4511,$P_Q = \mathsf{P}[(X-a)1_{X>a}]$ +4512,$Y=aX+b$ +4513,$g(s^2)=d+O(s^2) > g(s)^2=d^2 + O(s^2)$ +4514,$X_j^3$ +4515,$\| Z \|^*= \sup\ \{ \mathsf{E}[YZ] \mid \| Y \| \le 1 \}$ +4516,$\sigma^2 = \lambda \alpha(\alpha + 1) / (\beta^2\mu^p)$ +4517,$e^{-rT}$ +4518,$RA = RA_0 + \cdots + RA_{T-1}$ +4519,$\{ P_t\}_t$ +4520,$-\rho(-X_i) \le \mathsf{E}[X_i] \le \rho(X_i)$ +4521,$a(w+h_2)$ +4522,$s < 0.1$ +4523,"$, apply the advanced form of the Lagrange Inversion Formula\footnote{$" +4524,$\nu=1/(1+\iota)=1-\delta$ +4525,$X=X_1+X_2+X_3$ +4526,"$375,088,155 | | $" +4527,$\bar a_x$ +4528,$k=-\log(p)/u$ +4529,$S=R^{-1}$ +4530,$\mathsf{E}[kX]$ +4531,$(10t+10t)/2$ +4532,"$\min(\alpha_X, \alpha_Y) >2$" +4533,$g^ag^k=g^{a+k}$ +4534,$d\tilde p=g'(S(s))f(s)ds$ +4535,$2\lim_{\mu\to 0} V(\mu)/\mu^2=\lim_{\mu\to 0}V''(\mu)$ +4536,$Q=Q_X$ +4537,$\lim_{\theta\uparrow 0} -\theta\log(-\theta) =0$ +4538,$(3\times 6 + 2\times 2)/ 8 = 11/4$ +4539,$\mathsf P(A)=1-p$ +4540,"$p(t, B)=\mathsf{Pr}(S\in B\mid T):E\times \S\to [0,1]$" +4541,$\int_0^1 xj(x)dx< \infty$ +4542,$Y_t\to 0$ +4543,$\rho(A)>\hat\rho(A)$ +4544,$s^\ast=1/2$ +4545,$\mathsf{E} X + c\mathsf{E}[\vert X-\tau \vert^p]^{1/p}$ +4546,"$p\neq 1,2$" +4547,$4\times 4$ +4548,$P_idx_i$ +4549,$R_4$ +4550,$0 \le s \le p$ +4551,"$\mathsf{LOSS}=\mathsf{LR}\,\mathsf{PREM}$" +4552,$d^\ast$ +4553,"$w,N,K$" +4554,$\mu_d$ +4555,$dF(x) = dp$ +4556,$v_{res}\sqrt{(1+v^2)/n}\approx v_{res}v/\sqrt{n}$ +4557,$r$ +4558,$\mathsf{Pr}r(N=n)=e^{-\lambda}\lambda^n/n!$ +4559,$a=\mathsf{VaR}_p$ +4560,"$u_1, u_2$" +4561,$\tau(\tau^{-1}(\mu))=\mu$ +4562,$2.576\sigma_d$ +4563,$m_j / r_j$ +4564,$w_u$ +4565,"$\mathsf p=(p_0,\dots,p_{n-1})$" +4566,"$\mathsf{E}[Y_{0,0}]+\lambda\sigma(Y_{0,0})=58.129$" +4567,$u_L=T_L$ +4568,$\exp(a\arcsin(z))=\sum_n \frac{p_n(a)}{n!}z^n$ +4569,$\pi = \mathsf E[PR]$ +4570,$A \in\mathscr{F}$ +4571,$q(\epsilon)\approx q + \epsilon\mathsf{E}_q(X_i)$ +4572,"$d,v\ge 0$" +4573,$\rho(X)=-\rho(-X)$ +4574,$m \times 1$ +4575,$V(X)$ +4576,$z_p=(p/(L(x)k)^{1/\alpha}$ +4577,$\mathsf{TVaR}_p(X)=(12(0.9-p) + 2.5)/(1-p)$ +4578,$d=1/5$ +4579,"$(\omega^{ij})_{i,j}$" +4580,$p=\mathsf{Pr}hi((a-\mu)/\sigma)$ +4581,$s < s_0$ +4582,$\mathsf{biTVaR}$ +4583,$g'(S(x))f(x)$ +4584,"$(A.north east) + (-0.07mm,0)$" +4585,$\mathsf{E}[\theta]=\int\theta dP=(1+r_f)^{-1}$ +4586,"$\mathcal F'=\{\mathsf{var}nothing, \Omega \}$" +4587,$E(XZ \mid \mathcal{G})=ZE(X \mid \mathcal{G})$ +4588,$X_0= \sum_iX_i$ +4589,$\mathsf{E}[X]=\int_0^\infty S(x)dx$ +4590,$s= \phi (\xi - 1) \mu^{\xi - 1}$ +4591,$Y_t=t-N_t$ +4592,$l(kX)\le\rho(kX)$ +4593,$1-U$ +4594,$\mathsf{E}_\mathsf{Q}[X+c]=\mathsf{E}_\mathsf{Q}[X]+c$ +4595,$\not=$ +4596,"$[a, a+da]$" +4597,"$126,772.29 - \$" +4598,$1_Af_t(X)=1_Af_t(1_AX)$ +4599,$\lambda < 0$ +4600,$N\times d$ +4601,$a>a(f)$ +4602,$S_{\mathbf{x}}$ +4603,$p=p_{j_\pm}$ +4604,$a>0$ +4605,$\mathit{MV}_{gc}(a_{gc})=a_{gc}-P(X\wedge a_{gc})=5583.9$ +4606,$\tilde X+X$ +4607,$\kappa_1(10)$ +4608,$p(a)=\nu S(a) + \delta = S(a) + \delta F(a) = 1-\nu F(a)$ +4609,"$D_n,D_n^*$" +4610,$\int_1^\infty x^{\alpha-1}dx=1/\alpha$ +4611,$\bar P(\infty)$ +4612,$A_1(u_1)$ +4613,"$P=\mathrm{EL} + r\,Q$" +4614,$\nu(B)=\mathsf{Pr} 1_B\xi$ +4615,$\mathbf {g_3(s)=s^{0.7}}$ +4616,$\alpha_X > \alpha_Y + 2$ +4617,$\mathsf Q(\omega)=Z(\omega)\mathsf{Pr}r(\omega)$ +4618,$m_p$ +4619,$X_1\wedge a$ +4620,$f(w|s)$ +4621,$f=dQ / dP\ge 0$ +4622,$\mathsf{E}(\mathsf{Pr}i)$ +4623,$\hat \rho$ +4624,$X_0=\mathsf{E}[X]$ +4625,$B(b)\approx -b\mu_x$ +4626,$\mathbb{R}^2$ +4627,$\check g(1-t)= (1-k)+k(1-t) = 1-kt$ +4628,$a_t/t = \tau a_0 + a_1$ +4629,$n'=7$ +4630,"$I_i\in\{0,1\}$" +4631,$10^0$ +4632,$X(\mathbf{1})$ +4633,$Y\ge X$ +4634,$\mathsf E(X)$ +4635,$s_1Z_1 + s_2 Z_2 \sim Z$ +4636,$(34.05-23.81) / (100-34.05)=15.5$ +4637,$t_0a$ +4653,$f(t)=\overline{f(-t)}$ +4654,$p\delta_p$ +4655,$|Y_n|\le 1$ +4656,$x=8$ +4657,$t+2$ +4658,$m^2/\lambda$ +4659,$0\le\alpha\le K$ +4660,$a(\cdot)$ +4661,$\phi(0)=0$ +4662,"$(0,0,0,0,0,0,0,0,5,5)$" +4663,$\alpha_1$ +4664,$\mathbf {Q_1\Delta X}$ +4665,$t_0=0.296$ +4666,$\mathsf{TVaR}_p(X)=80$ +4667,$-norm equal to 1. (Note that $ +4668,$(\sum_i \nu_i)(\kappa(t+\theta) - \kappa(\theta))$ +4669,$E[Y_N]\le 0$ +4670,"$\int_0^{100} g(S(x))\,dx$" +4671,"$g, g', g''$" +4672,$h(X) = \prod_{i=1}^{n} \frac{1}{x_i!}$ +4673,$\mathsf{TVaR}_{0.6975}$ +4674,$(2k+1)/n$ +4675,$u=ug(1)=ug(1)+(1-u)g(0) \le g(u)$ +4676,$\mathrm{PV}_{r=i}(C)=0$ +4677,$X\wedge a\not\in \mathbf{X}$ +4678,$ which is an $ +4679,$\bar S(a)=\displaystyle\int_0^a S(x)dx$ +4680,"$[0,1]\times[0,1]\times\cdots$" +4681,$\tau=1$ +4682,"$X_1,\dots,X_r$" +4683,$\mathsf{E}(N)=$ +4684,$\square\rho_i$ +4685,"$(X,\mathcal{A})$" +4686,$c(y)=\dfrac{1}{\sqrt{2\pi}}e^{-y^2/2}$ +4687,$F^{\times}_{23}$ +4688,$u_L = \Delta_L s_L$ +4689,$\phi(s)=\displaystyle\int_{1-s}^1\dfrac{\mu(dp)}{p}=\int_0^s\dfrac{\mu(dp)}{1-p}$ +4690,$C=X+W$ +4691,$\rho(1_A) \le \rho(1)=1$ +4692,$13$ +4693,$\mathsf{E}(G^r)=\theta^r\mathscr{G}amma(a+r)/\mathscr{G}amma(a)$ +4694,$\rho(X)<\rho(Y)$ +4695,"$\mathrm{DM}^*(\theta, \sum_i \lambda_i)$" +4696,"$\rho(X),\rho(Y)\le 0$" +4697,$\mathsf{P}_X(A) :=\mathsf{Pr}r(X\in A)$ +4698,$q(p) \times \phi(p)dp$ +4699,$q(p)=\inf\{x \mid F_X(x)\ge p \}$ +4700,"$1_{(0,1)}$" +4701,"$(\Omega,\mathcal F, \mathsf{P})$" +4702,$z_p=q_h(p)$ +4703,$\mathsf{E}[X \mid \mathscr{G}]$ +4704,$\pi$ +4705,$A=(1-p)x$ +4706,$ It is the reciprocal of $ +4707,$μ$ +4708,$\delta_i$ +4709,$\int_0^\infty S(x)dx$ +4710,$\mu = w_1 \delta_{\alpha_1} + w_2 \delta_{\alpha_2}$ +4711,$y^*-x^* \ge \epsilon$ +4712,$\theta_s = \hat\theta_s + T$ +4713,$g(s)=s^{0.5}$ +4714,$x^{a-1}e^{x/\theta}$ +4715,$L^tf^*=m$ +4716,$1-m$ +4717,$\rho(X) = \mathcal{N}_{\tilde X}(X)$ +4718,$A(X)\not= B(X)$ +4719,$X-(1+r)T$ +4720,$(X\wedge a)$ +4721,$\mathsf{E}_{\mathsf Q_t}$ +4722,$A\subset Y$ +4723,$P = \mathsf{E}[X] + \pi\mathsf{Var}^+(X)$ +4724,$R_0(t)$ +4725,$P(1)=\mathsf{E}[X_1]<\infty$ +4726,$\sum_i x_i\mathsf{Pr}r(X=x_i)$ +4727,$c < c_{min}$ +4728,$g'(S(x))\ge 1$ +4729,$\lambda=0.0725$ +4730,$1-\tilde p=g(S(x))=g(1-p)$ +4731,$1 \times 10^{13}$ +4732,"$\mu=19.0, \sigma=2.58$" +4733,$\mathsf{E}[YZ\mid X]=Z\mathsf{E}[Y\mid X]$ +4734,$E_0$ +4735,$X=k+1$ +4736,$\mu \mathsf P^g_I X$ +4737,$s\ge 0.2$ +4738,$\beta \ge 1$ +4739,$\mathscr{F}\subset\mathscr P(\Omega)$ +4740,$\mathsf E[A] = \mathsf E[\mathsf E[X^{\oplus N}]] \le \mathsf E[\rho(X^{\oplus N})]$ +4741,$p=\mathsf{Pr}r(\theta>0.5)$ +4742,$x=q_X(1-s)=\mathsf{VaR}_{1-s}(X)$ +4743,"$\mathsf{TVaR}_0,\mathsf{TVaR}_1$" +4744,$v = 1/(1+r)$ +4745,$Q(A)=0$ +4746,"$\mathsf{VaR}_p(CP(\lambda, X)) = \mathsf{VaR}_{1 - \lambda(1-p)}(X)$" +4747,$s_1x)$ +4754,$\bar S(a)=\mathsf{E}[X\wedge a]$ +4755,$\nu Z_1$ +4756,$\sum_t Q_t$ +4757,$\phi(1-t)$ +4758,$1=v+d$ +4759,$g(s)=\nu s + \delta$ +4760,$u'>0$ +4761,"$\rho(X)=\int g(S(x))\,dx$" +4762,"$Y_m=\max(X_1,\dots,X_m)$" +4763,$\sum (1-S)\Delta (X\wedge a)$ +4764,$X\le \alpha$ +4765,$\alpha(\mathsf P)=0$ +4766,$|t|$ +4767,$\mu=-1-\dfrac{1}{\theta}$ +4768,$q=F^{-1}$ +4769,"$(0.5,1.5)$" +4770,"$\bar P(\mathbf{v}, a)$" +4771,$\mathsf{Var}(Y) \ge \mathsf{Var}(X)$ +4772,"$(D.south east)+(0.2, 0.05)$" +4773,"$(4,3)$" +4774,$\tilde X_j$ +4775,$\mathsf{E}[Y_\theta]$ +4776,"$c = 1.0, 1.5$" +4777,$\mathsf{P}(B)=0.5$ +4778,$X(x+\epsilon)$ +4779,$y$ +4780,$\mathsf{E}[e^{sX_{m/n}}]=\mathsf{E}[e^{sX_{1}}]^{m/n}$ +4781,$\kappa'(\theta)>0$ +4782,$\ge \mathsf{VaR}$ +4783,$c_x\approx\lambda$ +4784,$B^c_k$ +4785,"$\omega\in [0,1]$" +4786,"$X_1,\dots,X_n$" +4787,$10\times 10^6$ +4788,$\rho_m(Y)\le b_Y$ +4789,$b=1/1.2=0.83$ +4790,$v\in V$ +4791,$\sum_x x q_x$ +4792,"$. It falls into the compensated IACP, case 3 group, discussed in Part III, and takes any real value, positive or negative, despite only having negative jumps. It has a thick left tail and thin right tail. Its mean is zero, but the variance does not exist. A tilt with $" +4793,$a\theta=1$ +4794,$\mathsf QV$ +4795,$\mathrm{LR}$ +4796,"$\rho(X)=\max(\rho_1(X), \rho_2(X))$" +4797,"$x_1,1$" +4798,$\sigma^2=1 / \lambda$ +4799,$x\le 1$ +4800,$p_j = 1/M$ +4801,$a_{\min}$ +4802,$x\times f(x)dx$ +4803,$m=(K^{-1}Km)$ +4804,$X_1\le a_1$ +4805,$X_{t+dt}=X_t + \mu dt + \sigma dW_{dt}$ +4806,$\rho(\tilde X+X)=\rho(\tilde X)+\rho(X)$ +4807,$1/a_0$ +4808,"$(\langle X(\epsilon), \zeta_\epsilon \rangle - \langle X, \zeta \rangle)/\epsilon = \langle (X(\epsilon)-X)/\epsilon,\zeta \rangle = \mathsf{E}_Q(\nabla X)$" +4809,$\sigma_1=0.85$ +4810,$\mathsf{E}[ X_i]= M^{-1} \sum_{j}X_{i}^{< j >}$ +4811,$1/6\le x < 2/6$ +4812,$\mathsf{E}[Y]=\mathsf{E}[N]\mathsf{E}[X]$ +4813,$(1-m)(1+\frac{(1-m)^2}{a^2})$ +4814,$\int xdF$ +4815,$B=2\mathbb Z + \xi\mathbb Z$ +4816,$\pi_i:=0.5$ +4817,$\mathsf{E}[X_i/X \times D]$ +4818,$275.654\times 10^3$ +4819,$h\left(\displaystyle\int_\Omega g(X(\omega))\mathsf{Pr}r(d\omega)\right)$ +4820,$\rho(X)=\int P(x)dx$ +4821,"$s_g, s_b$" +4822,"$\mathrm{ess\,sup}(X)g(0-)$" +4823,$f(x)=\pm x$ +4824,${}^{[>40]}$ +4825,$p<1$ +4826,$\beta'_i(x)=0$ +4827,"$c\in[0,1]$" +4828,$\iff \mathcal A_{t+1}\subseteq \mathcal A_t$ +4829,"$\alpha_p = 1- (\| (X-\eta_{p,\alpha})^+\|_{p-1} / \| (X-\eta_{p,\alpha})_- \|_{p})^{p-1}$" +4830,$\nu=\nu_p$ +4831,$U_L\rho_L^E(t_1)$ +4832,$\bullet$ +4833,$b_l \le 1 \le b_h=2-b_l$ +4834,"$(s_1,g(s_1))$" +4835,"$\int_0^\infty z(x)\,dF(x)=1$" +4836,$\alpha_{1}$ +4837,$\mathrm{CV}(Y_1)/\sqrt{t}$ +4838,$\mathsf{Pr}(A\cap B)=\mathsf{Pr}(A)\mathsf{Pr}(B)=\int_B P(A)\mathsf{Pr}_T(dt)$ +4839,$\partial \zeta_{\bar x}/\partial x_i$ +4840,$x=100$ +4841,$\mathbf {p}$ +4842,"$\rho(Y)=\rho_{m_Y}(Y)\le b_{X,r}(Y)$" +4843,$R_1(t)>P(1)$ +4844,$\Delta S_5$ +4845,$F(x) < p \iff q^-(p) > x$ +4846,$RM = g(\tilde X)$ +4847,"$\mathsf{corr}(X_i, Z):=\dfrac{\mathsf{Cov}(X_i,Z)}{\sigma_{X_i}\sigma_Z}$" +4848,"$f(x,t)$" +4849,$10^3 - 10^{-1}$ +4850,"$500,000\*0.2108 = \$" +4851,$a_i=\mathsf{E}[X_i\mid X\ge \mathsf{VaR}_{p^**}(X)]$ +4852,$F_0 = P_{act}-\mathsf{E}_{rn}[U]$ +4853,$\beta_1g(S)dx$ +4854,$v_i$ +4855,$\mathit{ROL} = \dfrac{1 - \exp( - ( \alpha_{0} + \alpha_{1} \mathit{AEL} ) )}{1 - \exp( - ( \alpha_{0} + \alpha_{1} ) )}$ +4856,$g(S(x))\approx S(x)\approx 1$ +4857,$\sum_ x x\mathrm{Po}(j(x)dx)$ +4858,$p=F(a)=1-S(a)$ +4859,"$G(x+th, \omega+d\omega) = c_k(x+th)$" +4860,$g\mapsto\rho_g$ +4861,$R=g^k\pmod p$ +4862,$\mathsf{E}[p]$ +4863,$e[d_t d_s] = e[d_t]e[d_s]$ +4864,$Z=\sum Z_i$ +4865,"$g:[0,1]\to[0,1]$" +4866,$A(\lambda X)=\lambda A(X)$ +4867,$\Delta$ +4868,$\mathsf{TVaR}_0=max(X)$ +4869,"$e(f, y)$" +4870,$\text{LOSS}=\text{LR}\ \text{PREM}$ +4871,$\rho(X)\le \rho(Y)$ +4872,$h(p)=1-g(1-p)$ +4873,$\mathsf{QCX}$ +4874,$-\kappa_T(y)$ +4875,$f(p)=(1-p)\phi'(p)$ +4876,"$\{H,T\}$" +4877,$1_G(\omega)$ +4878,$\rho(\lambda X + (1-\lambda)Y)\le \lambda\rho(X)+(1-\lambda)\rho(Y)$ +4879,$d+v=1$ +4880,$K=g^k$ +4881,$b-a$ +4882,$\sigma={sigma:.5f}$ +4883,"$\rho(X) = \mathrm{ess\,sup}(X) = \mathsf{TVaR}_1(X)$" +4884,$dP/dt$ +4885,$\theta=f(\mu)$ +4886,"$(\Omega,\mathscr{F},\mu)$" +4887,$W_2$ +4888,$\mathsf v = (\hat F(t_l))_l$ +4889,$A_X = 5.976$ +4890,$\alpha(\mathsf Q) = 0$ +4891,$\rho_{1/2}$ +4892,"$s,t \in[0,1]$" +4893,$-k<0$ +4894,$D_g(\mathcal{B}bb Q\mid \mathcal{B}bb P)\ge D_g(\mathcal{B}bb P\mid \mathcal{B}bb P)=g(1)$ +4895,"$X_i(\omega), i=1,...,N$" +4896,$pd_i+(v-\nu^*)\sqrt{pq}$ +4897,$\mathsf{E}[X_i/X]$ +4898,$\int_0^1 Z=1$ +4899,$0.1 / 0.7=$ +4900,$p=3/2$ +4901,$q=0.9215$ +4902,$u = 1-e-l-r^*$ +4903,$r_X= r_f + \beta_X(r_m-r_f)$ +4904,$x_{\min{}}$ +4905,"$\pi : X\mapsto (X, \alpha(X))\mapsto E_g(X\wedge \alpha(X))$" +4906,$\bar S(x)$ +4907,$\rho^a_g(X)$ +4908,$M^3$ +4909,$Q^2=M^2/\iota$ +4910,$af$ +4911,$\mathsf{E}(XZ \mid \mathcal{G})$ +4912,$\chi( s ) = p - \log(s)$ +4913,$\kappa_X(\theta)=\log\mathsf{E}[e^{\theta X_t}]=\log\mathsf{E}[e^{\theta\sigma B_t + ct\theta}]=t(c\theta+ \sigma^2\theta^2/2)$ +4914,$\pi_1$ +4915,$\sigma^2\mu^p$ +4916,$f(y;\theta)=$ +4917,"$C(a)=\int_a^\infty S(x)\,dx + \tau a$" +4918,$l(\mathbf X)=(\sum_i X_i^2)^{0.5}$ +4919,$\mathsf{E}[X_i \mid X\le a]$ +4920,"$M\subset \{1,\dots, n\}\setminus \{i, j\}$" +4921,"$g(s) = \min(1, s / (1-p)$" +4922,$2^\lambda < \kappa$ +4923,$v-l$ +4924,$\Delta^o$ +4925,$p_{k+1} = S_k-S_{k+1}$ +4926,"$(0,1) < 1$" +4927,$\DeltaC$ +4928,$dt$ +4929,$\mathscr{G}_1\subset \mathscr{G}_2$ +4930,$(2\times 90 + 100) / 3$ +4931,$\mathsf E[XY]\not= \mathsf E[X]\times \mathsf E[Y]$ +4932,"$[0, \omega_I)$" +4933,$\delta_{p_i}$ +4934,"$1,353.02 | \ | > \$" +4935,$P_i=v\mathsf{E}[X_i\wedge a] + da$ +4936,$0.5$ +4937,$e^{-\beta s}s^{-1}\approx s^{-1}$ +4938,$a\alpha_i(a)$ +4939,$10 million I **must care more** about a loss of $ +4940,$R^*$ +4941,$c(y)e^{y\theta - \kappa(\theta)}$ +4942,$a_l>b_l$ +4943,$X=C+G$ +4944,$D(E)$ +4945,$\mathbf r$ +4946,$M/Q$ +4947,$S(X)=1-F(x)=\mathsf{Pr}r(X>x)$ +4948,"$s=(0.02, 0.3)$" +4949,$\phi_i = \mathsf{E}(X_i)/\mathsf{E}(Y)$ +4950,"$(\Omega, \mathscr{F}, λ)$" +4951,$K_\delta * f\to f$ +4952,$P(X_{-1}(a_{gc}))$ +4953,"$(\alpha,\beta)$" +4954,$\lambda > 0$ +4955,$E(X-q(X))^+$ +4956,$\mathsf{E}[U]=\mathsf{E}[X]$ +4957,$w(A)\le v(A)$ +4958,$\iota^*\approx 10$ +4959,$LR = EL/(EL+\iota Q)$ +4960,$s\to 0$ +4961,$a=1$ +4962,$\approx (920+961)/2=940.5$ +4963,$p^\star$ +4964,$\omega_c$ +4965,$8$ +4966,$P=\displaystyle\sum_i P_i$ +4967,$t \le g(t) = \displaystyle\frac{t}{1-p}$ +4968,$\max(X)=M$ +4969,$\omega=s$ +4970,$\lambda_{t}$ +4971,$X_n(\omega)$ +4972,$r_y>r_I$ +4973,$^\circledR$ +4974,$X_i < cx/a$ +4975,$\dfrac{e^{\theta x}}{x^{\alpha+1}}$ +4976,$\mathbf{v}'$ +4977,$A\in\tF$ +4978,$\mathsf{E}[X_i]/\mathsf{E}[X] \times \mathsf{E}[D]$ +4979,"$t\mapsto p(t, A)$" +4980,$\frac{1}{2-p}=\frac{\alpha-1}{\alpha}$ +4981,"$\mathsf{corr}(X_i, Z)$" +4982,$x_2(S(x_1)-S(x_2))=x_2f(x_2)$ +4983,"$w(s,t)= \Delta_R s_R/(1-T_{R-1})$" +4984,$\rho^*(\mu)=\infty$ +4985,$4 billion no losses excess $ +4986,$F_I^{n*}$ +4987,$\rho(c)=\rho(0+c)=\rho(0)+c$ +4988,$\dot f(t)=a(x)$ +4989,$\mathcal{A}_3$ +4990,$x_i=1$ +4991,$s>{s0:3g}$ +4992,$8.617 \times 10^{4}$ +4993,$1-\tilde p=g(S(x))$ +4994,$(A+B)/(1-p)$ +4995,$k-\rho_m(X)$ +4996,$V_\kappa$ +4997,$\rho(\cdot\mid\mathcal F_1)$ +4998,$(v-\nu^*)\int_0^a \sqrt{F(x)S(x)}dx$ +4999,$10000 \times (1-\alpha)$ +5000,"$(\omega,\omega')\in\Delta$" +5001,"$p\not\in (0,1)$" +5002,$\mathsf{E}[D_t D_s] = \mathsf{E}[(X_t - X_{t-1})(X_s - X_{s-1})]$ +5003,$\mathsf{Pr}_T(B) = \mathsf{Pr}(T^{-1}(B)) = \mathsf{Pr}(T\in B)$ +5004,$Y(ω)=\mathsf{E}[X\mid\mathscr{G}](ω)$ +5005,"$(X_1, X_2)$" +5006,$\sup X = 1$ +5007,$Q_0^i=0$ +5008,$\le_{\mathrm{cx}}$ +5009,$t=t_0^*$ +5010,$R'_1(1) \ge 0$ +5011,$A(\lambda X)=A(\lambda X)$ +5012,$2^{17}=131072<150000<2^{18}=262144$ +5013,$f(x) \to 0$ +5014,$(\tau^{-1})'(\mu)=1/V(\mu)$ +5015,$\Delta gS$ +5016,$d(y;\mu)=2(l(y;y)-l(y;\mu))$ +5017,$2T+U=0$ +5018,$i(v+\cdots+v^n) = 1-v^n$ +5019,$v-\nu^*$ +5020,$p_1$ +5021,$t\to 1$ +5022,$p(a) = \nu S(a) + \delta = \nu (S(a) + \rho)$ +5023,$10^{20}$ +5024,${}^{[>33]}$ +5025,$i=0.025$ +5026,$\Omega^X_{i_U} \ge c-c_r > \Omega^X_{i_U+1}$ +5027,$E(X_{-1}(a))=\bar S_0(a)$ +5028,$\Omega_a := \{\omega\in \Omega \mid (X\wedge a)=a \}$ +5029,$X=1_{U>0.8}$ +5030,$\mathsf{Pr}r(A\le t)= 1/2 + \mathsf{Pr}r(U\le t) /2 = 1/2 + t/2$ +5031,$\kappa'$ +5032,$\rho_k\to\infty$ +5033,$(X-d)^+$ +5034,$Pr(X > a)$ +5035,$\mu\to 0$ +5036,$P(0)$ +5037,$f(s) = -g''(1-s)(1-s)$ +5038,$t\sigma^2$ +5039,$T:\Omega\to Y$ +5040,$a_i:=\bar Q_i(a)$ +5041,$S=X+Y$ +5042,$\mathsf{E}(Y)$ +5043,"$h=0,1/2, 1$" +5044,$\mathsf{MON}$ +5045,$k=\mathsf E[X]$ +5046,$p(1-p)$ +5047,$a\max(0, \mathit{EGL}_{ro}(a))$" +5112,$\bar Q_{0}$ +5113,$P^i = \mathsf{E}[Z\cdot X^i]$ +5114,$\mathsf{E} X +\lambda {(X-\mathsf{E} X)^+}_1$ +5115,$\mathsf{EPD}_p(X)$ +5116,$F:\Omega\times\mathbb{R}\to\mathbb{R}$ +5117,$\mathsf{TVaR}_p(X) > \mathsf{Var}_p(X)$ +5118,"$(s_R,h_R)$" +5119,$\tilde X_1 + \tilde X_2 = X_1 + X_2$ +5120,$S_Y(a)$ +5121,$\{\omega\in\Omega \mid X(\omega)=x\}$ +5122,"$(s,g(s))=(0.2,0.36)$" +5123,$(v-\nu^*)\sqrt{F(x)S(x)}$ +5124,$\beta_i(a) g(S(a))$ +5125,$g'(p)=\phi(1-p)\ge 0$ +5126,$\tilde p/p$ +5127,"$(A=g^a,a)$" +5128,"$P_1(t)=\int_0^{a_t}\mathsf{E}[tX_1/X_t\mid X_t > x]g(S_t(x))\,dx$" +5129,$-$ +5130,$\Delta X_7$ +5131,"$X_{0,2}$" +5132,$t_1<\cdots 0$ +5147,$-k_i 1_{A_i}$ +5148,"$31.5 million. Nine of Argonaut’s 11 top officers were fired, and Singleton began running the operations from headquarters in Los Angeles. Argonaut, one of the last large companies in the malpractice market, discontinued underwriting individual policies for the 20,000 physicians it covered. It continued to offer coverage to the 25 percent of the nation’s hospitals it covered, but at higher rates and covering fewer risks. In the meantime, the company collected $" +5149,$\hat y_i=b_0+b_1 x_i$ +5150,$F=\mathsf{Pr}hi$ +5151,$\mathsf{Pr}r[X > A]$ +5152,$a=\inf$ +5153,$x_0=\mathsf E[X]$ +5154,$x_1 < x_2$ +5155,$\mathcal F_0$ +5156,$X_1(a/(1000+X_1))$ +5157,$\mathbf {\Delta X'}$ +5158,$S\subset\Omega$ +5159,"$0, r_1, r_2$" +5160,$h=\sin(77 s)$ +5161,$q_p({\mathbf{x}})$ +5162,$\mathbf {S\Delta X'}$ +5163,$10/11$ +5164,"${}^{[<35,23]}$" +5165,$Y_{0}=\sum_{d>0} X_{d}$ +5166,$\rho(X)=\rho(Y)$ +5167,$1/20=0.05$ +5168,$-q(-Y)$ +5169,$n=2^2$ +5170,$q(\omega)> 0$ +5171,$\mathsf P^g$ +5172,"$0, 9(1+\delta), 10(1+\delta)$" +5173,$1-p=\frac{1}{\alpha-1}$ +5174,"$122,500 | 1 | \$" +5175,$\lim J(x)\to\infty$ +5176,$\mathsf{E}[X \mid X \ge q^+(p)]$ +5177,"$(1-s, g(1-s))$" +5178,$a^i = \nabla^i a$ +5179,$M_\oplus \approx 5.972 \times 10^{24} \ \text{kg}$ +5180,$\nabla_i(X) = \mathsf{E}[X^i\mid p>0.99]$ +5181,$1_{U_X\ge p}=1$ +5182,$g'(S)dF(x)$ +5183,$i\not\in S$ +5184,$\omega_i\in B$ +5185,$S\Delta X\wedge a$ +5186,"$\mathbf {X_{i,j}}$" +5187,$\rho(X)=\mathsf{E}[gX]$ +5188,$=Q=\mathrm{MV}(a-X)^+$ +5189,$V=\sum_s v_sB_s$ +5190,$\zeta=0$ +5191,$\phi(0)$ +5192,"$1,839.20 + \$" +5193,$0.5\le p^* \le 0.75$ +5194,"$\mathscr{F},\mathcal{A},\mathcal{B}$" +5195,$\square^\square-1$ +5196,$\{p_j\}$ +5197,$t$ +5198,$c\ge 1$ +5199,$\nu(B)=\mathsf{Pr}(B\cap A)$ +5200,$\max X$ +5201,$\mathsf{TVaR}_{0.99}(X)=119.8=\mathsf{E}(W+Q\mid X\ge 100)=\mathsf{E}(W\mid X\ge 100) + \mathsf{E}(Q\mid X\ge 100)=19.8+100$ +5202,"$\mathsf{MON, NORM, TI}$" +5203,$\{\rho_L^E(t): 0 \le t \le s_L\}$ +5204,$\mathsf{Pr}r(X< x)\le 0.75 \le \mathsf{Pr}r(X\le x)$ +5205,$s > 0.5$ +5206,$\mathsf{E}[Y]=\mu=np/(1-p)$ +5207,$P_c(4450)^+$ +5208,$ws_0^2 + s_0(s_1 - 2\sqrt{s_1}w) - (1-w)s_1 = 0$ +5209,$\mathsf{E}[1_B\mid \mathscr{G}]$ +5210,"$\int_{[a,b]} h(x)dF(x)$" +5211,$\eta=\mathbf{x}\beta$ +5212,$1/\sigma^2$ +5213,$AP=\partial P^a/\partial x_i$ +5214,$\mathsf{CTE}^+$ +5215,$\lambda=0.755$ +5216,"$j=R,\dots,i_L-1,i_U+1,\dots,m$" +5217,"$1,553.08 \$" +5218,$g''(s)=-s^{-3/2}/4$ +5219,$\log(\phi(x)) = -\log(\sqrt{2\pi}) - \displaystyle\frac{x^2}{2\ln(10)}$ +5220,$\bar\iota(a)=\bar\iota$ +5221,$\delta Q_i$ +5222,$m_*$ +5223,$a_x$ +5224,$\{\omega\in\Omega\mid X(\omega)\le x\}$ +5225,$\mathsf{E}[X_1]<\infty$ +5226,"$\mathscr{G}amma(\alpha):=\int_0^\infty x^{\alpha-1} e^{-x}\,dx$" +5227,$g(S_k)-g(S_{k+1})$ +5228,$\tau=\mathsf{TVaR}_t$ +5229,$\rho_1(Y) = \rho_2(Y)$ +5230,$j\not= i$ +5231,$\Delta X_m$ +5232,"$\theta\in(-\pi/2,\pi/2)$" +5233,$z_p^{(2)} \approx z_{p/2}$ +5234,$t=0.25$ +5235,$f(x)=2$ +5236,$F(x)=\int f(x)dx$ +5237,$M$ +5238,$01$ +5274,$-1.240$ +5275,$\mathcal{B}bb Q$ +5276,$00$ +5278,$n=7$ +5279,$\Longleftrightarrow$ +5280,$\rho(X+Y)=\rho(X)+\rho(Y)$ +5281,$Z\circ T=Z\circ T_X$ +5282,$P = \mathsf{E}[Xe^{\pi X}]/\mathsf{E}[e^{\pi X}]$ +5283,$\mathbf {Q}$ +5284,$\mathsf{Pr}(A\cap B)$ +5285,"$(3,\infty)$" +5286,$c(y)=1$ +5287,$\delta = \iota/(1+\iota)$ +5288,$X_2=0$ +5289,"$(\Omega, \mathcal F, \mathsf P)$" +5290,"$(valu\x.south east)+(\boundpad,-\boundpad)$" +5291,$fC$ +5292,"$(X,a_2)$" +5293,$\mathsf{E}_\mathsf{Q}(Y\mid X)\mathsf{E}(Z\mid X) = \mathsf{E}(YZ \mid X)$ +5294,$NPV$ +5295,"$(s_j,g_j)$" +5296,$\bar\mu$ +5297,$q_{X_1}(p)+q_{X_2}(p)=q_{X_1+X_2}(p)$ +5298,$\dfrac{g(s)}{1-g(s)}$ +5299,"$C_{1,\cdot}$" +5300,$\delta_1$ +5301,$g_1(s)=s^{0.4}$ +5302,$P=l + \delta(a-l)$ +5303,$p=0.1$ +5304,$ = a bond with probability $ +5305,$\mathsf{Pr}r(X ≥ x_0) = 1-p$ +5306,$500 paid loss reduces the assets from \$ +5307,$\mathcal R^x$ +5308,$Y\not\succeq X$ +5309,$\mathbf {\sigma}$ +5310,$f(x)\approx 0$ +5311,$k \$ +5315,$\rho(\tilde X_1)=\rho(X_1)+\rho(\mathsf{E}[X_2\mid X_1])$ +5316,$(0.4)$ +5317,$\phi(t)\approx \phi(0)$ +5318,$133 ( 20%·\$ +5319,$B2$ +5324,$(P-L)/L=P/L-1$ +5325,$\sum X_i(a)p$ +5326,$X_2-X_1$ +5327,"$F_n^{-1}(1)=\frac{1}{(n-1)!}\mathsf{E}[\min(X_1,\dots, X_{n-1}]$" +5328,$=\mathsf{E}(X_i/X \mid X \le a)$ +5329,$x=X(1-g^{-1}(1-\tilde p))$ +5330,$\mathsf{E}_\mathsf{Q}[X+tY]$ +5331,$\alpha_X = \alpha_Y + 1$ +5332,$\rho(X\mid \mathcal F_1) =\mathsf E[X g'\mathsf{Pr}r(X>x\mid \mathcal F_1) ]$ +5333,$\mathsf{Pr}r(U\le \omega)=\omega$ +5334,"$\mathsf{TVaR}_{\alpha=1}=\mathrm{ess\,sup}$" +5335,"$100 million in earthquake limits in California, said Johnson & Higgins' Mr. Stuart. Now it's tough to put together up to $" +5336,$-1.365$ +5337,$X(\mathbf{v})$ +5338,$\int gS(x)dx=\int xg'(S(x))P_X(dx)$ +5339,$g(s)g(t)=O(d^2)< g(s)$ +5340,$-0.0012$ +5341,$\mathsf{Var}(\mathsf{CP}_1)=\mu x_2$ +5342,$D_t = X_t - X_{t-1} = \epsilon_t$ +5343,$Y\le a$ +5344,$M = P \mu_U = 0.3$ +5345,$13809$ +5346,$N=n$ +5347,$\beta_i(x) =\mathsf{E}_{\mathsf Q}[X_i/X\mid X>x]$ +5348,$\bar g(s)=1-g(1-s)$ +5349,$x_1$ +5350,$\phi\circ X$ +5351,$\mathsf{P}(X=X_j)$ +5352,$\mathbf {X'}$ +5353,$P^T(\cdot\mid t)$ +5354,$\alpha_i(x)$ +5355,$g''(p)=-\phi'(1-p)\le 0$ +5356,"$10 + 80/3\beta(1.5, 2.5)$" +5357,"$1.434 billion in excess of the CEA' s claims paying ability, representing $" +5358,$g'(S(x))=v$ +5359,$p_s=\mathsf{Pr}r(\Theta_s>0.5)$ +5360,$\frac{1}{1-p}+1=\frac{2-p}{1-p}$ +5361,"$(Alice) + (0,-4)$" +5362,$E[tX_i\mid X=x]f_X(x)$ +5363,$i^\star$ +5364,$\rho(X_n) \downarrow \rho(X)$ +5365,$l(p)= \nu-\sqrt{p(1-p)}$ +5366,$\mathsf{Pr}[\cdot\mid Y]$ +5367,$p-1=22$ +5368,$Q-0.4$ +5369,$s_0>0$ +5370,$\mathsf{Var}(G)=a\theta^2$ +5371,$g(u) = s_1(k_1 - k_0) + k_0 u = w + \displaystyle\frac{1 - w}{s_0} u$ +5372,$n^{-1}\mathsf{M}'\mathsf{M}=\mathsf{I}$ +5373,$j(x)\to\infty$ +5374,$t^*=0.544$ +5375,$r=0.15$ +5376,$X_{k+1}$ +5377,$\mathsf{E}[D_t \mid \mathscr{F}_{t-1}] \geq 0$ +5378,"$A_t = \min(t, T)$" +5379,$g(S_k) + g(1-S_k) \ge 0$ +5380,$\mathsf{E}(X) = \displaystyle\int_0^\infty xf(x)dx = -xS(x)\mathcal{B}ig\vert_0^\infty + \displaystyle\int_0^\infty S(x)dx = \displaystyle\int_0^\infty S(x)dx$ +5381,"$G(X_1,\dots, X_n)'=(Y_1,\dots, Y_r)'$" +5382,$\rho_g(X\wedge a)=\bar P(a)$ +5383,$\prec_3$ +5384,$a>a_{ro}$ +5385,$Y_n\to Y$ +5386,$\px=\mathsf{Pr}(B\cap \bigcup A_i)$ +5387,$X^{\oplus n}$ +5388,${n}-X_2$ +5389,$\sup \{ \mathsf{E}(LZ) \mid Z \preceq \sigma \}$ +5390,$X\wedge 20$ +5391,$k=7$ +5392,$\rho(X_1)$ +5393,$d\mathsf{Q}/d\mathsf{P}$ +5394,$\mathsf{E}_Q(N_i) =$ +5395,"$(3,1)$" +5396,$E(r_{FS}(t))$ +5397,$G=\sum_i N_i(x_i) + C_i(x_i)$ +5398,$203.6 billion in 1999 and $ +5399,$k_2$ +5400,$n < N-1$ +5401,$10^9$ +5402,$ | \$ +5403,$\delta_p=1-\nu_p=\rho_p\nu_p$ +5404,"$(s^\ast, g(s^\ast))$" +5405,$q(p)=c$ +5406,$\bar P_{i}(a)$ +5407,$T=1$ +5408,$\mathsf{Pr}r(X>a)$ +5409,$\mathsf{E}[X\mid\mathscr{G}]=\mathsf{E}[X]$ +5410,"$\mathcal F_t=\sigma(Z_1,\dots, Z_t)$" +5411,$(b-a)U_N$ +5412,$x=0$ +5413,$p\delta_p/p\nu_p=\rho_p$ +5414,"$a(x_1,\dots,x_n):=a(X(x_1,\dots,x_n))$" +5415,$x_p=\mathsf{VaR}_p(X)$ +5416,"$\mathcal{A}=\{A_1,A_2,\dots\}$" +5417,$V(\mu)=\tau'(\tau^{-1}(\mu))$ +5418,$1-t=g^{-1}(1-s)$ +5419,$\hat g(s)=1-g(1-s)$ +5420,$X=q_X(U)$ +5421,$\omega<1/n$ +5422,"$\mathsf{Cov}(X_i, Z)=\mathsf E[X_iZ]-\mathsf E[X_i]\mathsf E[Z]=\mathsf E[X_iZ]-\mathsf E[X_i]$" +5423,$\lambda=1$ +5424,"$\kappa(\theta) = \displaystyle\int_{\mu_0}^\mu \dfrac{m\,dm}{V(m)}$" +5425,$g(st) = \displaystyle\frac{st}{1-p} < \displaystyle\frac{s}{1-p}= g(s)g(t)$ +5426,$\sigma=0.45$ +5427,$x_iX_i$ +5428,$\alpha_i(t)$ +5429,$u''' > 0$ +5430,$q_X$ +5431,$\mathsf{Pr}(0)>0$ +5432,$g(1)=1$ +5433,"$\sigma_1,\dots,\sigma_T$" +5434,$\mu(E\cap U)\ge \alpha\mu(U)$ +5435,$10^{-8}$ +5436,$p^\star = p_i := 1-s_i$ +5437,"$x\in[0,\infty)$" +5438,$\mathsf{TVaR}_{p^*}(Y)$ +5439,$X = A + B$ +5440,$\sup(\lambda X)=\lambda \sup(X)$ +5441,$\mathsf{TVaR}_{0.8}(X)$ +5442,$\sqrt{FS}$ +5443,${6 \choose 2}=15$ +5444,$\kappa_2(X)$ +5445,$r_D=1-D$ +5446,$\theta \neq 0$ +5447,$Y=c$ +5448,${\lambda\alpha(\alpha+1)}/{\beta^2}$ +5449,"$\int_B \mathsf{E}(\sum X_i\mid\mathscr{G})\,d\mathsf{Pr}$" +5450,"$\mu^*(E)=\inf\left\{ \sum_{n\ge 1} \bar\mu(E_n) \mid E_n\in S(R),\ E\subset\bigcup_n E_n \right\}$" +5451,"$(X, a)$" +5452,"$[a, b] \subset \mathbb{R}$" +5453,$k_i(a) = \phi_i(a) k(a)$ +5454,$\lim_{t\downarrow 0}\displaystyle\frac{\rho(X+tX_i)-\rho(X)}{t} = \int x_i g'S(x)f(x)dx$ +5455,"$\langle \zeta, G \rangle=\int q_G q_\zeta$" +5456,$\mathcal S(X)=\mathsf{E}[X]$ +5457,$-0.769$ +5458,"$[\epsilon_1, \epsilon_2] \succeq [0, \epsilon_1+\epsilon_2]$" +5459,$\kappa_{i^*}$ +5460,$\int fdP=1$ +5461,$X\wedge a = \sum X_i(a)$ +5462,$\mathsf{E}[Z\mid X]=0$ +5463,$\bar P_{x+b}$ +5464,$X_{0}$ +5465,$m_X(s) \to \infty$ +5466,$h(x)=\sqrt x$ +5467,$\beta-\alpha$ +5468,$V_t$ +5469,$u^{iv}\le 0$ +5470,$P(gross) = 0.86956\cdot46.6+0.13043\cdot100=53.565$ +5471,$(g^{k})^a = K$ +5472,$\mathsf{E}[g'(S(X))]=\int_0^\infty g'(S(x))dF(x)=\int_0^\infty -\frac{d}{dx}g(S(x))dx=g(S(0))-g(S(\infty))=g(1)-g(0)=1$ +5473,"$p\not\in\{0,1,2\}$" +5474,"$X_1=0,0,0,0,1,1,2,3,20, 400$" +5475,$\mathsf{Pr}r(X_1)$ +5476,$X=Y+Z$ +5477,"$\mathsf{Q}_2(A)=2\mathsf{P}(A\cap (0.5, 1])$" +5478,$\rho(X+tY)\ge \mathsf{E}_{\mathsf Q_X}[X+tY]=\mathsf{E}_{\mathsf Q_X}[X]+\mathsf{E}_{\mathsf Q_X}[tY]=\rho(X)+t\mathsf{E}_{\mathsf Q_X}[Y]$ +5479,$\mu_f$ +5480,$\rho \mapsto \rho(Y)$ +5481,$\lambda < \kappa$ +5482,$\hat\sigma_s$ +5483,$\sum_{i}X_{i}^{( j )} = X_0^{( j )}$ +5484,$(\alpha-1)/\beta$ +5485,$X(\omega)=1$ +5486,$<\mathsf{E}[X]=100$ +5487,"$(-\x*0.75, -2)$" +5488,$X\beta_1(x) g(S(x))$ +5563,$\mathcal{B}B(S)\otimes \mathcal{A}$ +5564,"$\kappa_i(\mathbf{v}, x)$" +5565,$\forall X\in L^p$ +5566,$1/(1+r)$ +5567,$\nu_p$ +5568,$\ge 2$ +5569,$\mathsf{E}_Q[X_i(a)] + \delta Q_i$ +5570,$\tau=\tau_i=0$ +5571,$\bar P=\mathsf{E}[W]+\lambda\sigma(W)$ +5572,$S(x_i)$ +5573,$F_0=2$ +5574,$GM/R$ +5575,$2^{32}=4$ +5576,$\mathsf{Pr}(I_t=1)=s_t=1-p_t$ +5577,${}^nS^{-1}(q)$ +5578,$g(s)=1-(1-s)^m$ +5579,$Y_c=(Y\mid Y > y_c)$ +5580,"$t \in (0,1)$" +5581,$\mu=\tau(\theta)=\tan{\theta}$ +5582,$\mathsf{E}[Y_i\mid X_n]$ +5583,$g-br-v$ +5594,$7 \times 10^{-7} - 4 \times 10^{-7}$ +5595,$\mu^0=1$ +5596,$X \prec_n Y$ +5597,"$400,000 excess of \$" +5598,$\check M\implies\ \check g$ +5599,$F:\mathbb{R}^n \to \mathcal{X}$ +5600,"$(fun1a.south -| fun3a.south east)+(\smlspc,-\smlspc)$" +5601,$1000-950=50$ +5602,$m_1=m_2$ +5603,$Y-X\le 0$ +5604,$L_X(s)/L_Y(s) = k$ +5605,"$393,874,333 | | $" +5606,$\{x\}=x-\lfloor x\rfloor$ +5607,$A(X)$ +5608,$s=q(p)$ +5609,"$\mathcal{R}_{X,r} \to\mathbf{R}$" +5610,$g^{-1}(x)\le s$ +5611,$t=-3$ +5612,"$X\in L^\infty([0,1])$" +5613,$\pi(X)=\log(m_X(\alpha)) / \alpha$ +5614,"$x_l', 8:'$" +5615,$\mathsf{TVaR}_p(X)=\frac{1}{1-p}\int_p^1 F_X^{-1}(t)dt$ +5616,$C'_i$ +5617,$A_1$ +5618,$f(R) = \mathsf{E}[f(X)]$ +5619,"$\mathit{Tw}_p(\mu, \sigma^2)$" +5620,"$\psi(S,t)=1_{\{t\}}$" +5621,"$k, X_k$" +5622,$P(\mu)R(\mu) = Q(\mu)\sqrt{R(\mu)}$ +5623,$a\mathsf{E}[X_i/X \mid X]$ +5624,$q(\psi)$ +5625,$t = 1$ +5626,$\mathsf{Pr}r(X>q(p))=1-p$ +5627,$\mathsf{Pr}r(X_n\in A)=1$ +5628,$id\times\tau$ +5629,"$A_3=[0, \epsilon-k]$" +5630,$\mathsf{E}[X\wedge 0]=0$ +5631,"$X\wedge a:=\min(X, a)$" +5632,$dp=f(s)ds$ +5633,$m(p)=q+p\delta_p$ +5634,$M(a)=\delta F(a)$ +5635,$\rho(\cdot)$ +5636,"$\rho_{m_X}\in\mathcal R^c_{X,r}$" +5637,$wq_Y(p)+(1-w)q_Z(p)$ +5638,"$(s,t)$" +5639,$P=(kN/V)T\propto$ +5640,"$u\in[0,1]$" +5641,"$\mathbf {X\,p}$" +5642,"$200 of losses otherwise payable to any claimant under this subsection. All claims under life insurance policies and annuity contracts, whether for death proceeds, annuity proceeds or investment values, must be treated as loss claims. Claims may not be cumulated by assignment to avoid application of the $" +5643,$X_p =F_X^{-1}(p + (1-p)U_X$ +5644,"$a\wedge b:=\min(a,b)$" +5645,$t_2-\epsilon$ +5646,$1-v^n$ +5647,$1-2c\mathsf{Pr}r(Z>\mathsf{E} Z)$ +5648,$g'(s) = as^{a-1}$ +5649,$Q_{2}\Delta X$ +5650,$\rho(Y)=\rho(X)g(p)=g(q)g(p).$ +5651,$q(1)$ +5652,$\lambda^{1-p}m^p$ +5653,$g^{kS}=R^S$ +5654,$H$ +5655,$\rho \ge \mathsf{E}[X]$ +5656,$\alpha=\alpha(p)$ +5657,$\mathsf{var}phi(X)=\mathsf{var}phi(Y)$ +5658,$\mathsf{VaR}_p(X)=q_X^{-}(p) = \sup \{ x\mid F_X(x) < p \}$ +5659,$\phi(t)$ +5660,$80K; \$ +5661,$A(X+Y)\le A(X)+A(Y)$ +5662,$n\times 2$ +5663,$G\in \sigma(\mathcal{A}\otimes \mathcal{B})$ +5664,$g'(S(x))\ge 0$ +5665,$\check g(s)=1 + k(s-1)=1-k + ks$ +5666,$r_U \ge -1$ +5667,$CV=\nu=\sqrt{a}\theta$ +5668,$\rho_g(V)$ +5669,$\beta_i$ +5670,$\mathsf{Pr}(X\le r\mid\mathscr{G})=E[1_{X\le r}\mid\mathscr{G}]$ +5671,$t=-1$ +5672,$\tau^{-1}(\mu)=-1/(2\mu^2)$ +5673,$\psi$ +5674,"$\mathsf{biTVaR}_{0,1}^w(X)=(1-w)\mathsf{E}[X]+w\sup(X)$" +5675,$\tan\theta$ +5676,$\mathbf {\mathsf{E}[X_i\wedge a_i]}$ +5677,$10^{-2}$ +5678,$\mathsf{E}[W_t]=t$ +5679,$Q_i = \sum_k Q_k^i$ +5680,$\rho(X + \rho(X))=0$ +5681,$a_l$ +5682,$\lambda=(1-\alpha_p)^{-1}$ +5683,$(L-A)^+$ +5684,$-\log(-\theta)$ +5685,$R_1(t) = \bar P^a_1(t)/(1-t)$ +5686,$f(0)=0$ +5687,$F_3$ +5688,$X_t = e^{B_t - t^2/2}$ +5689,"$(X, \Sigma)$" +5690,$\mathsf{Pr}r(X_2)$ +5691,$\rho(X)=g(s)$ +5692,$4$ +5693,$R_1(t)<\rho(X_1)$ +5694,$\rho_m(Y)\le \rho_{m'}(Y)\le b_Y$ +5695,$1-p$ +5696,$C_1$ +5697,$s^\star = (1-T_{R-1})/\phi_R$ +5698,$\mu \mathsf P_I = \mathsf P$ +5699,$\bar P(a+da) -\bar P(a)$ +5700,$t \neq s$ +5701,$\rho(X)=\mathsf{E}[XZ]$ +5702,"$\mathsf P(\rho(X(u_1, U_2)) > x)=\mathsf P(\rho(X(U_1, U_2)) \mid \mathscr F_1)(u_1)$" +5703,$\rho_\min(L_i)=\rho_i(L_i)$ +5704,$g=u^2=0.01$ +5705,$p_0 < p^\star < p_1$ +5706,$\ge 0.98$ +5707,$x>q(p)$ +5708,$\Theta=\{0\}$ +5709,$X_1+X_2\not\in\mathcal A$ +5710,$Z_{\mathit{lift}}$ +5711,$X_1=\mathsf P_1X$ +5712,$\Delta G$ +5713,$\hat F(x)=\sum_{i:x_i\le x} \lambda_i/\lambda$ +5714,$X_1=aX+b$ +5715,$< 30$ +5716,$G(x)= 1-g(1-F(x))$ +5717,$a(X;X)=\rho(X)=\sum_i a_i$ +5718,$\alpha\beta^2$ +5719,$x_t$ +5720,$\mathsf{TVaR}_{p_0}(X)=\mathsf{E}[X \mid A]$ +5721,$\hat q(p)$ +5722,$\Theta_i^X := \mathsf{TVaR}_{1-s_i}(X)$ +5723,$a(X_i; X)$ +5724,$\sigma^2 dt$ +5725,$y_1 \wedge y_2$ +5726,$f^{-1}(A)\in\mathcal B$ +5727,$i=1$ +5728,$g(S_7)=g(0)=0$ +5729,$NPV = F_0$ +5730,$\tau_n$ +5731,$Y=e^X$ +5732,"$200 of losses otherwise payable to any claimant under this subsection other than the federal government. All claims under life insurance and annuity policies, whether for death proceeds, annuity proceeds or investment values, shall be treated as loss claims. Claims may not be cumulated by assignment to avoid application of the $" +5733,$\mathsf{E}[L]$ +5734,$\rho(X)=\max_k \mathsf{E}_{\mathsf Q_k}[X]$ +5735,$xf_i(x)$ +5736,$M=g-S$ +5737,$x\mapsto |x|$ +5738,$\mathsf{var}phi(t X) = t \mathsf{var}phi(X)$ +5739,$\mathsf{E}[X_1]=\mathsf{E}[X_2]$ +5740,"$L_a^{a+y}(x)=\min(y, \max(x-a,0))$" +5741,$\mathbb{R}$ +5742,$1-EL$ +5743,"$L(e,t)=eR_t$" +5744,$\mathsf{VaR}\_p(X)$ +5745,$COC = (P-L) / Q$ +5746,$\mathsf{E}_Q(X \mid \mathcal{G})\mathsf{E}(Z \mid \mathcal{G}) = E(XZ \mid \mathcal{G})$ +5747,$\mathsf{E}[X_i \mid X]$ +5748,$M^i = P^i - L^i$ +5749,$\mathsf{E}[s]=0$ +5750,$\Delta_{1}$ +5751,"$\mathsf{biTVaR}_{0,1-t}$" +5752,$10^{-3} - 10^{-1}$ +5753,$\rho(X)=\mathsf{E}[hX]$ +5754,$\delta = \iota \nu$ +5755,$M_t = t$ +5756,$X_n(0)=1$ +5757,$V(t)$ +5758,$(L^t)^+$ +5759,$P(x) = \sum_i P_i(x)$ +5760,"$P_i(\mathbf{x}, a)=\mathsf{E}_Q[X_i(a)]$" +5761,$\mathsf{TCE}_{0.98}=\mathsf{E}[X \mid X \ge \mathsf{VaR}_{0.98}]$ +5762,$X=X_+-X_-$ +5763,$s*$ +5764,"$t\in[0, t^*]$" +5765,$\tau \leq T$ +5766,"$(0.x_0x_1x_2\dots, 0.y_0y_1y_2\dots, )\mapsto (0.x_0y_0x_1y_1\dots)$" +5767,$\beta_i(x) =\mathsf{E}_{\mathsf{Q}}[X_i/X\mid X>x]=\mathsf{E}[(X_i/X)g'S(X))\mid X>x]$ +5768,"$ power is convolution, giving the hitting probability to the level $" +5769,$a=a(f)$ +5770,$s_2 - s_1$ +5771,$\iota:1$ +5772,"$(\Omega,\mathcal F,\mathsf{P})$" +5773,$R'_1(t)$ +5774,$m(F)=0$ +5775,"$T:(\Omega,\mathscr{F})\to(M,\mathcal{B})$" +5776,$V(\mu)=\mu^2$ +5777,$x^2 A$ +5788,$x\to\infty$ +5789,$A+\gamma$ +5790,$E_0+a_1$ +5791,$X\ge \mathsf{VaR}_p(X)$ +5792,$\tilde X_2 = X_2 - \mathsf{E}[X_2]$ +5793,$\int X_n\to 0$ +5794,$X_i(a)=(X\wedge a)X_i/X$ +5795,"$\phi:(L, \mathcal{A}) \to (M, \mathcal{B})$" +5796,"$\rho^E(t_1,t_2)$" +5797,$\mathsf E[X] = \mathsf{TVaR}_0(X)$ +5798,"$\omega\mapsto Q(\omega, B)$" +5799,"$\omega = [0, 1]$" +5800,$f(\lambda x + (1-\lambda)y)\le \lambda f(x) + (1-\lambda )f(y)$ +5801,"$Z \sim \mathrm{DM}^*(\theta, \nu)$" +5802,"$\mu=12,11,10,10$" +5803,$M = 0.603$ +5804,$k(0)=\log(1)=0$ +5805,$-iv^2$ +5806,$x=\log(y)$ +5807,$X_1=c_1-Y/2$ +5808,$X_t=(1-t)X_0+tX_1$ +5809,$\rho(Total\subseteq X^c\wedge a)$ +5810,$gdX$ +5811,$1=\bar\nu+\bar\delta$ +5812,$\lambda > 1$ +5813,"$i=0,\dots,i^\star$" +5814,$X=q=F^{-1}$ +5815,$\mathsf{E}[X_1] / \mathsf{E}[X]$ +5816,"$t\in[0.7,1.0]$" +5817,$\sum_i$ +5818,$Q(a)$ +5819,$X_\theta$ +5820,"$\mathsf{biTVaR}_{1-s,1-s_R}$" +5821,$v_1<0$ +5822,"$[0,1;\ 0.89, 0.11]$" +5823,$\mathsf{E}[X_ie^{kX}]/\mathsf{E}[e^{kX}]$ +5824,$\max_\mathsf{Q} \mathsf{E}_\mathsf{Q}[X]$ +5825,$\alpha(\cdot))$ +5826,$p_t(A)$ +5827,$d^*(0)=0$ +5828,$\mathsf{Pr}r(X < x) \le 1/6 \le \mathsf{Pr}r(X\le x)$ +5829,$L_p dp$ +5830,$\rho_a(X+c) = \rho((X+c)\wedge a(X+c)) = \rho((X+c)\wedge (a(X)+c)) = \rho((X\wedge a(X))+c) = \rho((X\wedge a(X))) + c=\rho_a(X)+c$ +5831,$S(x)=1-F(x)=\mathsf{Pr}r(X>x)$ +5832,$\Delta_i=\mathsf P_{i+1} - \mathsf P_i$ +5833,$C_i$ +5834,$L^i$ +5835,$g(S(x_{i+1}-))-g(S(x_{i}))$ +5836,$\rho_g = \int g(S)$ +5837,$\rho_h(X)<\rho_h(Y)$ +5838,$S(x)>0$ +5839,$Y\sim$ +5840,$\mathsf{Pr}r(X < x) \le 0.1 \le \mathsf{Pr}r(X\le x)$ +5841,$\mathscr{P}=\{ (1-p)^{-1}1_A \mid P(A)\le 1-p \}$ +5842,$X\ge x_0$ +5843,$n > 2$ +5844,$\mathsf{E}_{\mathsf{Q}}[X] = \rho(X)$ +5845,$\mathsf B(s)$ +5846,$\mathsf{E}[Z]=\mu$ +5847,$t_* < 0.5 < t^*$ +5848,"$X,Y,X+Y$" +5849,$A(1_{X>x_1} + 1_{X>x_2})= A(1_{X>x_1}) + A(1_{X>x_2})$ +5850,$a = 0.5$ +5851,$d\mathsf{Q}/d\mathsf{P} = g'(S(X))$ +5852,$-0.5 \le \epsilon \le 0.5$ +5853,"$(\sqrt{st}, \sqrt{st})$" +5854,$\sum_{n\ge 0} 1_{N>n} X_n$ +5855,$E\subset\Omega$ +5856,$\mathsf{Var}(X+c)=\mathsf{Var}(X)$ +5857,$w_u=1+c(1-\gamma)$ +5858,$001158$ +5859,$\ge 5000 / \text{Probability}$ +5860,$ is not continuous and $ +5861,"$(10,0)$" +5862,$M\times d$ +5863,"$\{0,1,2,N_e-1\}$" +5864,$q\phi$ +5865,"$R_i=\alpha p_i + \beta r_{qp,i} + \gamma\, \text{controls}_i$" +5866,$1.5$ +5867,$K = A -P = 3.798$ +5868,$θ={v}$ +5869,${}^{[<82]}$ +5870,$2 \times 10^{14}$ +5871,${}^{[>66]}$ +5872,"$(L,\mathcal{A})$" +5873,$1M/\$ +5874,$Z(\omega)=0$ +5875,$M(h)=\exp(\mu h + \sigma^2 h^2 / 2)$ +5876,$\mu = \dfrac{m_1 m_2}{m_1 + m_2}$ +5877,$1+2c(1-\mathsf{Pr}r(Z>\mathsf{E} Z)$ +5878,$S_{U}$ +5879,$Q_t=\rho(\mathsf{E}[X\mid t])$ +5880,$L=R$ +5881,$\rho(X\_{2}\subseteq X^c\wedge a)$ +5882,$k=a$ +5883,$\iota^{\star}$ +5884,$\mathsf{Pr}r(X < x)\ge 1/6$ +5885,$1.25 \times 10^{14}$ +5886,$C_i(t) = \partial \bar P^a/\partial x_i$ +5887,$R_1(t)<\mathsf{E}[X_0]$ +5888,$ROE=-m'(1)/(1-m'(1))$ +5889,$g_o:=h_R + (s_{R+1}-s_R)(1-h_R)/(1-s_R) \le g_{R+1} \le 1$ +5890,$R_0(t)>\mathsf{E}[X_0]$ +5891,$s_L < \dots < s_R$ +5892,$\mathsf{SSD}$ +5893,$F(x;\alpha)$ +5894,$\kappa_j(x)/x>\alpha_j(x)$ +5895,"$ is average invested assets, equal to $" +5896,$L_{ij}$ +5897,$(x^{-1}-x^{-3})\phi(x)$ +5898,"$U_n(a, b)$" +5899,"$2,\dots,r$" +5900,$h=m/V(m)$ +5901,$\mu-\nu$ +5902,$f=1_A$ +5903,"$D^n_{\rho_g,X}(Y)$" +5904,$\mathsf{Pr}r(A)=1-p$ +5905,"$\psi(s,t)=1_B(t)\mathsf{E}[X\mid S=s,T=t]$" +5906,$\mathsf{E}[Y_\theta]=y$ +5907,$ is $ +5908,$p=1-g(1-F(x))$ +5909,"$(x_B, g(S(x_B-))$" +5910,$X_1 = \mathsf{E}[X\mid \mathscr{F}_1]$ +5911,"$\rho\in \mathcal R^x_{X,r}$" +5912,$\mathsf Q\in \mathcal Q$ +5913,$\mathsf{E}(L)=\int_0^\infty S(x)dx$ +5914,$l(t)=1$ +5915,$\mathsf{E}[X_1\mid X \ge a]$ +5916,$\tilde X_1 + \tilde X_2 \succeq^2 \tilde X_1$ +5917,$y=\exp(x)$ +5918,$\mathsf{E}_{\mathsf Q}[\kappa_i(X)]$ +5919,$\mathsf{TVaR}_p( X )$ +5920,$\mathsf{E}[1_{X>a}]=\mathsf{P}(1_{X>a}=1)$ +5921,$ and investor equity $ +5922,$\rho_g(X)=c$ +5923,$Y(\omega_1)\le Y(\omega_2)$ +5924,$\{X_k^i\}$ +5925,$0=\rho(0)=\rho(X-X)\le\rho(X) + \rho(-X)$ +5926,$AD$ +5927,$\sec\theta$ +5928,$X_t = \mathbb{E}[X \mid \mathcal{F}_t]$ +5929,"$\langle \zeta_{\bar x}, N_i \rangle$" +5930,$d^*(x)\neq 0$ +5931,$a(X)=a(\sum_i X_i) = \sum_i a_i$ +5932,$E_\mathsf{Q}(X_i\mid X)=E(X_i\mid X)$ +5933,$\beta=0$ +5934,$N=1000$ +5935,$\mathbf {\mathsf{P}(X)}$ +5936,$\Delta^u$ +5937,$\mathsf{TVaR}$ +5938,$\Lambda = \dfrac{M - K r_{f}}{\sigma_{U}^{2}}$ +5939,$\sigma=2$ +5940,$\sigma_1=0.15$ +5941,$X-Y$ +5942,$\mu_{x+t}$ +5943,$\int c(x)dx>1$ +5944,"$\{1,2,3,4,5,6\}$" +5945,$\beta_i(k) = \mathsf E_q[\kappa_i(X)/X\mid X > X_k]$ +5946,$1.231-1=0.231$ +5947,$V_j$ +5948,"$[x_1, x_2]$" +5949,$\rho(X_t)=\mathsf{E}_{\mathsf Q_t}[X_t]$ +5950,$\mathsf{V@R}$ +5951,$\displaystyle\int_0^\infty u(x) g'(S_X(x)) dF_X(x)$ +5952,$f_Y(y)=\lambda f_Z(\lambda y)$ +5953,"$(0,\dots,0,r_0,\dots, r_k)$" +5954,$P_i \ge \mathsf{E}[X_i]$ +5955,$\pi'(k)=...$ +5956,$(1-t)x_0$ +5957,$C_p / C_v$ +5958,"$A\subset[0,\infty)$" +5959,"$(\mathsf{E}_q(X_1)(1-\epsilon\mathsf{E}_q(X_2)/q), \mathsf{E}_q(X_2)(1+\epsilon \mathsf{E}_q(X_1)/q))$" +5960,"$\rho_2(X)=\mathsf{E}[X] + \mathsf{cov}(X,Z)$" +5961,$0.675=(6.258/7.613)^2$ +5962,$U>p$ +5963,$\mathbf {M_{2}\Delta X}$ +5964,$\bar P_g$ +5965,$Q+P>1$ +5966,$g''(s)=0$ +5967,$1/\lambda = \sum_j 1/\lambda_j$ +5968,$e^{\mu+\sigma}$ +5969,$x_l < x=\mathsf{VaR}$ +5970,$\sum_i F_i=F$ +5971,$\delta(s)$ +5972,$\mathsf{Pr}(B\mid \mathscr{G})(ω)$ +5973,$117*:*43 = \$ +5974,$g'(s)<1$ +5975,$\tilde \Theta$ +5976,"$1,250k. However, a large deductible policy with a \$" +5977,$Var[T]=s(1-s)/N$ +5978,"$B(0, p)$" +5979,$X_n\to X$ +5980,$\tilde X-X$ +5981,$g=\mathsf{E}(G^3)=\nu^3 skew(G')+3c+1$ +5982,$\mathsf{E}[X_2\mid X=20]=6$ +5983,$q_{X+c}(p)=c+q_X(p)$ +5984,$B_1 \succ A_1$ +5985,$k+1/2$ +5986,$\mathsf{E}[(X-\mathsf{E} X)^+]={(X-\mathsf{E} X)^+}_1$ +5987,"$i=1,2$" +5988,$X_1 > a_1$ +5989,$\partial\rho(Z)$ +5990,$f(L)=L$ +5991,$X\wedge a=\sum_i X_i(a)$ +5992,"$a=1,2,\dots, 100,101$" +5993,$\theta=(1-f)/a$ +5994,$\mathsf{E}_\mathsf{Q}[X_i \mid X]$ +5995,$<1/a_t$ +5996,"$k\mathsf{E}[(X_i-\mathsf{E} X_i)(X-\mathsf{E} X)]=k\mathsf{cov}(X_i,X)$" +5997,$B_k$ +5998,$\mathsf{Var}(B(p))=p(1-p)$ +5999,$\mathsf{E}[X_i | X > \mathsf{VaR}]$ +6000,$r =$ +6001,$7$ +6002,$=E(X_i \mid X \ge a)$ +6003,"$p_0,\dots, p_{n'}$" +6004,$-\rho(-X) =b-\rho(b-X)$ +6005,$d^*(x)=x^2$ +6006,$\beta_0+\beta_1$ +6007,$l_Q\in{\mathcal{X}}'$ +6008,$\mathscr F$ +6009,$F_i = X_i(1 - (X\wedge a) / X)$ +6010,$(v-\nu^*)\sqrt{FS}$ +6011,"$a_{0,0}$" +6012,$t>0.25$ +6013,$d(y;\mu) = (y-\mu)^2$ +6014,$\lambda\mu_t$ +6015,$\mathsf{Pr}(D) = 1$ +6016,$R_0(t)\ge \mathsf{E}[X_0]$ +6017,"$X_{t-2,2}$" +6018,$z$ +6019,$\mathsf{Pr}i=B-p\nu(p)$ +6020,$\rho=\dfrac{M}{l} = \dfrac{1-\lambda}{\lambda}$ +6021,$Z_s$ +6022,$\partial P_/\partial x_j = -\partial D_i /\partial x_j$ +6023,$\kappa_i(x)= E[X_i\mid X=x]$ +6024,$\hat p$ +6025,$\pm \mathsf{Pr}hi^{-1}(30/31)= \pm1.85$ +6026,$\mathsf{TVaR}_1=\operatorname{ess\ sup}$ +6027,$a-X\le 0$ +6028,$\mathsf{E}[X_1]$ +6029,$V(m)=mW(m)$ +6030,$\rho(X)=-U(X)$ +6031,$p<2$ +6032,$\sum_{j=0}^{n-1} p_j \le \pi < \sum_{j=0}^{n} p_j$ +6033,$\mathsf{P}(A\mid B)=\mathsf{P}(A\cap B)/\mathsf{P}(B)$ +6034,"$g(S)\,\Delta X$" +6035,$\bar P=\mathsf{TVaR}_{p^\ast}(X)$ +6036,$\mu_{x+t}=-\dfrac{d}{dt}\log({}_tp_x)$ +6037,$n\ge 3$ +6038,$a\mapsto g^a \pmod{p}$ +6039,$f(t)=\rho(tX)$ +6040,$\mathsf{TVaR}_{0.95}=(x \times 3 + x_{99} + x_{100})/5$ +6041,$t \le 1-p$ +6042,$697.6 billion underlying Table \ref{tab-equity-what-if} this implies $ +6043,$_k$ +6044,$q_X\le q_Y$ +6045,$S(x_{i-1})-S(x_{i})=S(x_i-(x_i-x_{i-1}))-S(x_i)=-S'(x'_i)(x_i-x_{i-1})=f(x'_i)(x_{i}-x_{i-1})$ +6046,$x_0 = \mathsf E[X]$ +6047,$\rho(X) = \int_\Omega g(S(x))dx$ +6048,$1_Z$ +6049,$\mathbf n$ +6050,"$X,X_i$" +6051,$\mathbf {X_{n}}$ +6052,"$\rho(1000, 3000, 3500)$" +6053,$\{X \ge a\}$ +6054,$st \le 1-p < s$ +6055,"$\omega\mapsto P(\omega, B)$" +6056,$l+t$ +6057,$0< a\le 1$ +6058,$M/\check M$ +6059,${}^{[<81]}$ +6060,$S(x)=(\lambda / (\lambda + x))^\alpha$ +6061,$1/y$ +6062,"$1,418.57 \$" +6063,$0\le v\le 1$ +6064,$(m-1)$ +6065,$\alpha_i(a) S(a)$ +6066,$\mathsf{E}[X^n]$ +6067,$m\in \mathcal{M}$ +6068,$\alpha + Mg/a$ +6069,"$(4-\s, \s)$" +6070,$x^{-\alpha-1}e^{\theta x}$ +6071,$x>0$ +6072,$\check g(s)=1-g(1-s)=1-(d + v(1-s))=vs$ +6073,$-7$ +6074,$\rho(X)=\bar P$ +6075,"$U, V$" +6076,$\tilde M(a)=\bar M(a)-\tau a$ +6077,$\mathcal F'_0\subset\mathcal F_0$ +6078,$(X\wedge a)\Delta g$ +6079,$\alpha_i(X_u)= \mathsf{E}[u_iX_i \mid X_u > F_u^{-1}(p)] = u_i \partial T/\partial u_i$ +6080,$A = P + K$ +6081,$f(S_t)=\log(S_t)$ +6082,$Z(X)$ +6083,$\mathsf{E}_Q(X \mid \mathcal{G}) = E(X \mid \mathcal{G})$ +6084,$t/|t|$ +6085,$g(s)=s^{1/2}$ +6086,$\{s_i\}$ +6087,$0<\alpha\le 1$ +6088,$\phi'(s)ds$ +6089,$\mathsf{Var}(B(p)/p\nu_p)=p(1-p)/(p\nu_p)^2$ +6090,$\mathsf{E}[x_iX_i\mid X(\mathbf{x}) \le a]F_{\mathbf{x}};a) = \mathsf{E}[x_iX_i 1_{X(\mathbf{x}) \le a}]$ +6091,$Y_n=n-S_n$ +6092,$\nu\uparrow\infty$ +6093,$g(u)g(v)$ +6094,$g'(s) \ge 0$ +6095,$e^{ct}$ +6096,$d\mathsf{Pr}i = (r_h-\mu_L)\mathsf{Pr}i dt$ +6097,$\phi_n$ +6098,"$12,000. Project A returns \$" +6099,$\approx 10^{-40}$ +6100,$R_C$ +6101,$s = 1$ +6102,"$\displaystyle\int_0^{1-g(S(a))} \kappa_i(q(1-g^{-1}(1-p)))\,dp + a\beta_i(a)g(S(a))$" +6103,$E[X \mid \mathcal{F}_t] - E[X \mid \mathcal{F}_{t-1}]$ +6104,$\mathsf{E}_\mathsf{Q}(X)$ +6105,$N=X-L_{r_a}^{r_a+r_l}(X)$ +6106,$Y_s=(Y\mid Y\le y_c)$ +6107,"$\mathsf{E}_{P_g}[\mathsf{E}[X_i\mid X]]=\int_{[0,\infty]} \mathsf{E}[X_i\mid X=x]Z(x)P_x(dx)$" +6108,$\rho(X_i)$ +6109,"$R, S$" +6110,$V(\mu)=\mu^2/\alpha$ +6111,$\mathscr{P}=\{\mathsf{Q} \mid d\mathsf{Q}/d\mathsf{P} \le 1/(1-\alpha) \}$ +6112,$s=f'(x_0)$ +6113,"$u_A, u_E$" +6114,$S_k\Delta X_k$ +6115,"$(a,b]$" +6116,$\sqrt{F(x)S(x)}$ +6117,"$X\sim\text{Lognormal}(\mu=19.9, \sigma=2.36)$" +6118,$\alpha_i(x)-\kappa_i(x)/x=0$ +6119,$Q_k$ +6120,$\Omega= \mathrm{int}\ \mathrm{conv}(S)$ +6121,$\mathsf{E}[X_i\mid X]$ +6122,$s+1$ +6123,${}^{[<78]}$ +6124,$105$ +6125,$2x$ +6126,$S(x+a)$ +6127,$6\times 16= 96$ +6128,$X(\omega)=$ +6129,"$\zeta_1,\dots,\zeta_T$" +6130,$\le 0$ +6131,$\bar S$ +6132,"$n,p=0.3$" +6133,$\lambda X \in\mathcal{A}_\rho$ +6134,$0\le x < a$ +6135,$g(x)=x^{0.5}$ +6136,$g(uv)=g(u)g(v)$ +6137,$D_i=\sum_j (\hat y_j - \hat y_{j(i)})^2 / ((k+1)s^2) = (e_i/se(e_i))^2\times h_{ii}/ ((k+1)(1-h_{ii})$ +6138,$E[X_1|X=10]$ +6139,"$r1,1(s)$" +6140,$t=b_i/se(b_i)$ +6141,$\mathbf{B}(0)=\mathbf{P_0}$ +6142,$B(0.5)$ +6143,"$\int_0^1 xj(x)\,dx<\infty$" +6144,$g'(S(x))<1$ +6145,$\hat{\mathsf a}\hat{\mathsf b}$ +6146,$L = 4\pi R^2 \sigma T^4$ +6147,$\phi(p)dp$ +6148,$U(a)=-s$ +6149,$x>1$ +6150,"$f(x;\alpha,\beta)=x^\alpha- e^{-x/\beta} / \beta^\alpha x\mathscr{G}amma(\alpha)$" +6151,"$\mathbf {j, p, S, \kappa_1, \Delta X, \Delta(X\wedge a)}$" +6152,$u = \beta_i(x)S(x)$ +6153,"$X_{i,i}(a)=X_{i,j}\dfrac{X_j\wedge a}{X_j}$" +6154,$x\mapsto \mathsf E[f(X_2)\mid X_1=x]$ +6155,$\zeta_1=1$ +6156,"$r=1,2,3,4$" +6157,$\Delta \mathit{MV}_{ro}(a)$ +6158,$\mathsf E[X^{\oplus n}]\le\rho(X^{\oplus n})$ +6159,"$X_{t-1,1}$" +6160,$\mathsf{E}[X\mid Y]$ +6161,$\mathsf{VaR}_{0.85}$ +6162,$\mathsf{TVaR}_{0.99}$ +6163,$\preceq$ +6164,$g(st) = \displaystyle\frac{st}{1-p} < 1 = g(s)g(t)$ +6165,$\psi(u)=\mathsf{Pr}r(Y > u)$ +6166,"$i=1,\dots, n_r$" +6167,$X=X_1+...+X_n$ +6168,$n_s(1-g(s))$ +6169,$-g''(t) = w_1 \delta_{\alpha_1}/\alpha_1 + w_2 \delta_{\alpha_2}/\alpha_2$ +6170,$j(x)=x^{-5/2}$ +6171,$\rho(B(s_l))$ +6172,"$X_j=\sum_i X_{i,j}$" +6173,$b< \alpha$ +6174,$\rho(Y)\le b_Y$ +6175,$y_{2}=3$ +6176,$\{X > x \}$ +6177,$ and therefore $ +6178,$A_x\setminus A_y$ +6179,$b - a$ +6180,$\pi_g(X)=\int_a^{\alpha(X)} g(S(t))dt$ +6181,"$w_1, w_2$" +6182,$X_n = 1$ +6183,$\zeta\ge 0$ +6184,$\mathbf {\alpha_2}$ +6185,"$650,000 × 0.9 × 1.588 = \$" +6186,$X_i-F_i$ +6187,$Z=q/p$ +6188,"$i=0.02,\ 0.04$" +6189,$\mathbf {d=0}$ +6190,$. If $ +6191,$k'(h)$ +6192,$dF=-dS=$ +6193,$\rho(A)=4.875 > \hat\rho(A)=4.8125$ +6194,$y_3$ +6195,$X\mathsf{Pr}=\mathsf{Pr}(X^{-1}(\cdot)))$ +6196,"$\rho(\lambda P,\lambda R,\lambda a)=\lambda\rho(P,R,a)$" +6197,$\mathcal E:=\{Y \circ T \mid T \text{ PPT} \}$ +6198,$Y'\mid X'$ +6199,$\eta(\theta)=\theta$ +6200,$p \ll 1$ +6201,$Q=(a-EL)/(1+r)$ +6202,$ and $ +6203,$\beta gS$ +6204,$X \wedge a$ +6205,$r=5\%$ +6206,$U_\infty=\lim_N U_N$ +6207,$\lim \mathsf{E}[X_n] = \mathsf{E}[X] < \infty$ +6208,$g(S_n)=0$ +6209,$\mathsf{E}[e^{sX_{m/n}}]=\mathsf{E}[e^{sX_{1/n}}]^m$ +6210,$\mathsf{E}_{\mathsf{Q}}[X] \le \rho(X)$ +6211,$0.1525$ +6212,"$p^*\in[0,1]$" +6213,$U=X$ +6214,$=1.75$ +6215,"$X_1,X$" +6216,$590 million in premium for paying losses $ +6217,$\aleph_1$ +6218,$P^T_S(A\mid\cdot)$ +6219,$\mathsf{NORM}$ +6220,$x=q(p)$ +6221,$\rho(-1_{A^c}) = c < 0$ +6222,$578.37+ 282.76=861.13$ +6223,$\mathsf{E}[Z_1]=1$ +6224,"$X_t=1_{[1,\infty)}$" +6225,$X\preceq Y\implies \rho(X)\le \rho(Y)$ +6226,$a_0>\mathsf{E}[X_0]$ +6227,$c=\mathsf{VaR}$ +6228,$X_d$ +6229,$P(t)>P(1)$ +6230,$a=D+S$ +6231,$\mathbf {\mathsf{E}[X_i(a)]}$ +6232,$\mathsf{Pr}r(q^-(F(X))\not=X)=0$ +6233,$\dfrac{d}{dx}g(S(x))=-g'(S(x))f(x)$ +6234,$\{X_n\}$ +6235,$\mathsf{CONVEX}$ +6236,$\mathbf {x_2}$ +6237,$P_R$ +6238,$dx$ +6239,$S_i$ +6240,$V^{\ast}(1)=p/(1+r-p)$ +6241,$g(p)$ +6242,$\lambda>1$ +6243,$})$ +6244,$\rho_h$ +6245,$\sum_x dx = a$ +6246,$c(n)=\dfrac{1}{n!}$ +6247,$2.439 > 2 \times 1.204=2.408$ +6248,"$iil, d$" +6249,$E(X_i \mid X=a)$ +6250,$c\ge \mathsf{E}[cg]$ +6251,$ϕ(1-t)=g'(t)$ +6252,$\mathbf {t+2}$ +6253,"$1, =$" +6254,$\rho=\rho_\gamma$ +6255,$T^{-1}$ +6256,$\mathcal{B}B(S)\otimes\mathcal{A}$ +6257,"$\mathsf{Var}(X_x)=\mathsf{Var}(\mathsf{CP}(j_n(x)\delta, x))=x^2j_n(x)\delta$" +6258,"$\sigma_1=0, \sigma_2=1$" +6259,$\log(X^2)=2\log(X)$ +6260,$\kappa_X$ +6261,$\mathsf{E}[XZ_j] = (5)(1/10)(8)+(5)(1/10)(9)=8.5=\mathsf{TVaR}_{0.8}(X)$ +6262,$\mathsf{E}[1_A\mid\mathscr{G}]$ +6263,"$\alpha_X\in(\alpha_Y+1/3, \alpha_Y+1/2)$" +6264,$0.5 < t_*0$ +6302,$-t$ +6303,"$\rho(X)=\mathrm{ess\,sup}(X)$" +6304,$Y\le 0$ +6305,$s\downarrow 0$ +6306,$\tilde y$ +6307,$\rho_{t+1}(X)\le\rho_{t+1}(Y)$ +6308,$\mathsf{F}\mathsf{A}=\mathsf{C}$ +6309,$r_X$ +6310,$X=\frac{1}{n}\sum_i X_i$ +6311,$m(m-1)$ +6312,$e^{-kX}/\mathsf{E}[e^{-kX}]$ +6313,$T_s(p) = \mathsf{TVaR}_p(s)$ +6314,$X=X_0+Y$ +6315,"$2^1, 2^3, ...$" +6316,$0\not\in\Omega_p$ +6317,$N'$ +6318,$R>0$ +6319,"$\partial \rho(X)=argmax_{\zeta\in A} \langle \zeta, X \rangle$" +6320,$X_i=X_i(a)$ +6321,$\mathsf{Pr}r(T_x<\infty)=1$ +6322,"$\bar S(a)=\int_0^a S(x)\,dx$" +6323,$\omega_1$ +6324,$g'(S(x))=(1-p)^{-1}$ +6325,$\mathsf{E}(X_i(a))$ +6326,$s/g(s)\le 1$ +6327,$g(s/\omega)g(\omega)$ +6328,$\mathsf{E}[Z\mid X>a]=g(S(a))/S(a)$ +6329,$0 \le f'(z) \le 1$ +6330,$\mathsf{Pr}r(X)$ +6331,$\bar Z = F(\bar x)$ +6332,$\zeta_s = 8$ +6333,$v=1/1.1<1$ +6334,$2.592 \times 10^{16}$ +6335,$x^{-\alpha-1}e^{\theta x}/x$ +6336,$\mu^2/\lambda$ +6337,$\mathsf{E}_\mathsf{Q}[X_i \mid X=x]=\mathsf{E}[X_i \mid X=x]$ +6338,$v^T=21.6\%$ +6339,$m(1+\frac{m}{p})$ +6340,$D \rho(X_0)$ +6341,"$\mathsf{E}_Q[X]=\int x\,Q(dx)$" +6342,$\mathsf{CV}(G) = \mathsf{SD}(G') = \nu$ +6343,$A_n \in \mathcal{C}$ +6344,$ then $ +6345,$\mathcal{R}$ +6346,$h=-\beta$ +6347,$N':=kN$ +6348,$x\mapsto x$ +6349,$\mathsf{E}[|X_t|] < \infty$ +6350,$X\_{1}$ +6351,$a(\mathbf(x))$ +6352,$k-L_0^k$ +6353,$X_3$ +6354,$g'(S(x))$ +6355,"$(Alice) + (0,-3.75)$" +6356,$d=iv=i/(1+i)$ +6357,$\tau_\sigma(\alpha) = \int_\alpha^1 \sigma$ +6358,$10^4$ +6359,"$p\in [0,1]$" +6360,$\rho(\tilde X_1)=\rho(X_1) + \mathsf{E}[X_2]$ +6361,"$A\subset [0, \infty)$" +6362,$X>F_u^{-1}(p)$ +6363,$\mathbf {t}$ +6364,$\rho(X) = \mathsf{E}(X) + c\mathsf{E}( |X-\mathsf{E}(X)|^p)^{1/p}$ +6365,$\sigma(X)$ +6366,$A^c\supset A_1\supset A_2\supset \dots$ +6367,$Y_c$ +6368,$Q_0=\rho(V_0)=\rho(X_1)$ +6369,$m_X$ +6370,$\mathscr{G}\subset\mathscr{F}$ +6371,"$2^0, 2^2, 2^4, ...$" +6372,$u = \alpha_i(x)S(x)$ +6373,$_{gc}$ +6374,"$781,475,899 | -1.2% | $" +6375,$\sigma^2 = \sum \sigma_i^2$ +6376,$U_N$ +6377,$X_1=X_2=Y$ +6378,"$(s_{R+2}, g_{R+2})$" +6379,$v=1/(1+i)$ +6380,$\mathcal M_\rho$ +6381,$\{X_{t \wedge \tau}\}$ +6382,$p_k=F((k+1/2)b)-F(k-1/2)b)$ +6383,$y^{\ast}$ +6384,$l(0.5)=\log(1)=0$ +6385,"$50 of the amount allowed on each claim in the classes under paragraphs II, V, and VI except claims of the guaranty associations as defined in RSA 404-B, 404-H, 404-D, and 408-B shall be deducted from the claim. Claims may not be cumulated by assignment to avoid application of the $" +6386,$\mathbf {\kappa_2}$ +6387,$u=-0.295$ +6388,$(p-\nu)/\nu$ +6389,$k\ge 1$ +6390,$\kappa(\theta)=\dfrac{\theta^2}{2}$ +6391,$\mathsf{Pr}(X_n=1)=1/n$ +6392,$\sum_i a_i=\sum_i a(X_i;X)=\rho(X)$ +6393,$\mathsf{Pr}r(X>a)>1-\alpha$ +6394,$g'\left (S_X(X)\right )$ +6395,$6 \times 10^{}$ +6396,$Xq$ +6397,$a=11$ +6398,$\mathsf{E}[gY]\le 0$ +6399,$\mathsf{TVaR}_0(\cdot)=\mathsf{E}[\cdot]$ +6400,$m > 1$ +6401,$\mathsf{Pr}r(X\le \mathsf{VaR}_p(X)) = p$ +6402,$(x-\mu)^2 - 2\sigma^2x=(x-\mu-\sigma^2) - 2\mu\sigma^2-\sigma^4$ +6403,"$380,000$" +6404,$\lambda=\beta/\alpha$ +6405,$x_120]}$ +6446,$\sigma=1$ +6447,$\beta_D$ +6448,"$\mathit{LOSS}=\mathit{LR}\,\mathit{PREM}$" +6449,$\rho_g$ +6450,$F_\alpha$ +6451,$x=wy + (1-w)z$ +6452,$w=1$ +6453,$k_3$ +6454,$\sqrt{R^2}$ +6455,$Y_{d}=\sum_{s>d} X_{s}$ +6456,"$\mathcal E_{X,r}(Y)$" +6457,$agg$ +6458,$Z(200)=0$ +6459,$\kappa/x$ +6460,$\mathsf{E}[X^k]=\mathsf{E}[Y^k]$ +6461,$2^2$ +6462,$\mathsf{Pr}r(X_n>\epsilon)\to 0$ +6463,$X=q(U_X)$ +6464,$1 million but some $ +6465,$\sigma$ +6466,$R_i = P_i - U_i = 0$ +6467,$K_\theta(t)= e^\theta(e^t -1)$ +6468,$1/V$ +6469,$0s_R\}$" +6472,$100\cdot g(s)$ +6473,$p={reg_p}$ +6474,$\{a_n\}$ +6475,$\omega=0$ +6476,$n=2^m+k$ +6477,"$({a}, {b})$" +6478,"$U:=X\wedge a:=\min(X,a)$" +6479,$10^5$ +6480,$m_j$ +6481,$X\le A$ +6482,$r_h=r+\pi$ +6483,$h'(j)$ +6484,$\mathrm{NE}(\mu)$ +6485,$[F(x)](\cdot)$ +6486,$10^{15}$ +6487,$X_l$ +6488,$\sup_{m\in\mathcal{M}}\rho_m$ +6489,$00$ +6527,$\mathsf{Pr}(B)=0$ +6528,"$g(s)=\displaystyle\int_{1-s}^1 \phi(p)dp = \displaystyle\int_0^s \phi(1-p)dp = \min(s/(1-p), 1)$" +6529,"$f,F,S,q$" +6530,"$135,000, no less than $" +6531,"$(x,y)\not=(0,0)$" +6532,$5 \times 10^{10}$ +6533,$\rho(X)=1$ +6534,$\cdots$ +6535,$250 must be held as surplus. This leaves \$ +6536,$\kappa(\theta)=\log \sum_n e^{\theta n}/n!$ +6537,$1.3 \cdot 0.5 = 0.65$ +6538,$s_0<1$ +6539,$(\alpha-1)(p-1)=-1$ +6540,"$Mg,Fe)SiO_3$" +6541,$\rho_2(X_1)=1$ +6542,$Y_\nu$ +6543,"$W_2=\sum_{t+d=2} Y_{t,d}$" +6544,"$859,470,180 | -8.2% | $" +6545,$(\beta)$ +6546,$\mathcal R^h\subset \mathcal R^x$ +6547,$q_X(p) = X(T(p))$ +6548,$X = g(Z)$ +6549,$\zeta\in\partial \rho(X)$ +6550,$\rho(X)=\rho(X\wedge a) + \rho((X-a)^+)$ +6551,$0=p_0>0$ +6621,$f'_-(x)=\lim_{h\uparrow 0} (f(x+h)-f(x))/h$ +6622,"$\mathsf{TVaR}_1(X)=\mathrm{ess\,sup} X$" +6623,${\mathcal{M}}_1$ +6624,$10^{-18} - 10^{-22}$ +6625,"$, $" +6626,$=\dfrac{g(s)-s}{1-s}$ +6627,$1-S(a)=F(a)$ +6628,$\theta(\mu)=\int_{\mu_0}^\mu dm/V(m)$ +6629,$X_s \ge \mathsf{E}[X_t \mid \mathscr{F}_s]$ +6630,$0\le \alpha\le 1$ +6631,$836.0 - 423.8 = 412.2$ +6632,$Z_k:=q_k / p_k$ +6633,$\epsilon\approx 10^{-2}$ +6634,$\delta=1-\nu=\rho\nu$ +6635,$\mathscr{O}(\zeta)$ +6636,$\mathscr{G}amma\not\in\mathscr{F}\otimes\mathcal{B}$ +6637,$\mathsf{E}[X_1h(X_2)]$ +6638,$\rho_{t+1}(-\rho_{t+1}(X))=\rho_{t+1}(X)$ +6639,$λ>0$ +6640,$g(S_X(x))=1$ +6641,$=\int_0^\infty xf(x)dx = \int_0^\infty S(x)dx = \int_0^1 q(p)dp$ +6642,$\notiff$ +6643,$\lambda=0.045$ +6644,"$i=A,E$" +6645,$C$ +6646,$\mathsf{E}(B)=p$ +6647,$b={b}$ +6648,$\mathsf{E}[N]$ +6649,$\mathcal F^{NS}$ +6650,$P(\alpha(X))$ +6651,$(\partial P_i / \partial x_i)dx_i$ +6652,$U = A = 8.149$ +6653,"$(valu2.south east)+(\boundpad,-\boundpad)$" +6654,$F = G m_1 m_2 / r^2$ +6655,$h(0)$ +6656,$r_P-\mu_L$ +6657,$\mathsf{E}[x]$ +6658,$\pi=\mathsf{Pr}i/p\nu(p)$ +6659,${}^{[<40]}$ +6660,$U(X) a)S(a)$ +6675,$D(x)$ +6676,$V_\nu$ +6677,$c^*$ +6678,"$A, B$" +6679,"$(Bob)+(0,-3.5)$" +6680,$\int_p^1 q_Z(t)dt \le g(1-p)$ +6681,$X_2$ +6682,$(LL^t)^{-1}L^t$ +6683,$p_+$ +6684,$g'S_t(X_t)$ +6685,${Z}_p \le c$ +6686,$y\in Y$ +6687,$\ge c$ +6688,$T_k$ +6689,$l(y;\mu)=-\dfrac{y}{2\mu^2} + \dfrac{1}{\mu}$ +6690,$\log(0.999/0.001)=6.9$ +6691,$\omega'=1$ +6692,$X(\nu)$ +6693,$P_{i}(a)$ +6694,$\hat y_i=b_0+b_1 x_{i1} + \cdots + b_kx_{ik}$ +6695,$\kappa_1'(-\kappa(\theta))\kappa'(\theta)=1$ +6696,$g_\tau$ +6697,$\mathsf{E}_P[dQ/dP]=1$ +6698,$2^{-1}$ +6699,$\approx 0.9999999999$ +6700,$b = B/P = 1/1.2 = 0.83$ +6701,$^{**}$ +6702,$s_n=s_L=s_R$ +6703,$X=\sum_{i=1}^n X_i$ +6704,$J(x)$ +6705,$0<\rho\le 1$ +6706,$X\equiv 1$ +6707,"$\mathbf {g(S)\,\Delta X}$" +6708,$\tilde M_i(a) = \bar M_i(a)-\tau_i a_i$ +6709,$Q(x)=1-g(s)$ +6710,$d_i= i/(1+i)$ +6711,$0.99$ +6712,"$\{E\subset\Omega\mid E\in\sigma(\C'), \C'\subset\sigma(\C),\text{ countable}\}$" +6713,$\tilde F^{-1}(\tilde p)=F^{-1}(p)$ +6714,"$\mathrm{ess\,sup}$" +6715,$\mathsf{E}_{\mathsf{Q}}[X_i \mid X] = \mathsf{E}[X_i \mid X]$ +6716,$R:=\bar P_{act}-\bar S$ +6717,"$\lambda=0.045, 0.0625, 0.085, 0.125,$" +6718,$\bar \zeta$ +6719,$\rho_{m_0}$ +6720,$\mathsf{EPD}$ +6721,$v\le 1$ +6722,$\sum_i \alpha_i(x)=1$ +6723,$n \ge 1$ +6724,"$(s_{j},g_{j})$" +6725,$=18\times 4 = 72$ +6726,$\mathscr F_1:=\sigma(I_1)$ +6727,$X(1)=X_1$ +6728,$\mathsf{E}[X_i/X\mid X>x]$ +6729,$g(s)=s$ +6730,$X_0+\epsilon Y$ +6731,$\mathsf{E}(U(Z))=\mathsf{E}(U(Z) \mid A) = \mathsf{E}(U(X))p + \mathsf{E}(U(Y))(1-p)$ +6732,$\sup(X\wedge a)=a$ +6733,$X_{i}^{j}$ +6734,$S(\omega)=\omega$ +6735,$\rho(X_n)=1$ +6736,"$150,000. Then the \$" +6737,$ corresponding to an extreme stable distribution with $ +6738,$a>10$ +6739,$C:=1-H$ +6740,$\bar P(a+y) - \bar P(a)$ +6741,$Z=(1-p)^{-1}1_{\tilde X>q_{\tilde X}(p)}$ +6742,$Pa$ +6743,$\bar P_0>\mathsf{E}[Y_{0}]$ +6744,$\mathscr{F}_s$ +6745,$\epsilon^C$ +6746,$\mathrm{Sh}(\eta_i)$ +6747,$q_{X_1+X_2}(p) \le q_{X_1}(p) + q_{X_2}(p)$ +6748,$s_0 > s_1 > s_1^2$ +6749,$\mathsf{Pr}r(L'= l)$ +6750,$e+l+r^*$ +6751,$(E\cap U) + x$ +6752,$g(x) = (x-\mu)^2$ +6753,$\displaystyle\int_0^\infty S(x)dx$ +6754,$g(S_X(X))$ +6755,$\mu_L=r_L +\pi$ +6756,$l_p>0$ +6757,$\bar G'(a)=\frac{d\bar G}{da}:=G(a)$ +6758,$-br-v$ +6759,$p(1-p)/p^2(\nu_p-l_p)^2$ +6760,$\mathsf{TCE}_p(X):=\mathsf{E}[X\mid X\ge q_l(p)] < \mathsf{E}[X\mid X\ge q_u(p)]$ +6761,$\rho(X)=r$ +6762,$\mathbf {X}$ +6763,"$c_1, c_2\ge 0$" +6764,$\mathsf{var}phi$ +6765,$\sum \alpha_i S\Delta (X\wedge a)$ +6766,$\mathsf{E}_{g}[X_i \mid X=x]=\mathsf{E}[X_i \mid X=x]$ +6767,$X\in L^0$ +6768,$r<0$ +6769,$p-\nu-il$ +6770,$P(X_i(a_{gc}))$ +6771,$c=0$ +6772,$m=m' - b/2$ +6773,$U(a)$ +6774,$(\mathsf{TVaR}_p - q(p))/(1-p)$ +6775,$h(1-p)=1-g(p)=1-\sqrt{0.9}=0.051$ +6776,$x \!\urcorner$ +6777,$n/2$ +6778,$\phi(\tilde r)^2 = \phi(r_1)\phi(r_2)$ +6779,$a_{i-1} < a_i < a_{i+1}$ +6780,$\rho_g(X)=$ +6781,${}^1S=S$ +6782,$nu$ +6783,"$X_0, X_1$" +6784,"$\mathsf{Pr}(A\cap B)= \int_B 1_A\,p\mathsf{Pr}$" +6785,$v_2=v_1<0$ +6786,$n\times r$ +6787,$X(t)=X(\mathbf{x})=(1-t)X_1 + tX_2$ +6788,$\mathsf{P}(X=1)$ +6789,$\rho^\star(X)$ +6790,$a=\infty$ +6791,$X_{i}^{( j )}$ +6792,"$\rho(X) = \rho(\rho(X, P_I), \mu)$" +6793,$p\le S(x^*)$ +6794,$\mathcal F = \{F_\alpha\mid \alpha<\omega_c\}$ +6795,$g'(S(s))f(s)$ +6796,$\mathsf{Pr}r(X_i=0)>0$ +6797,$\mu^p$ +6798,$\mathscr{G}amma(\alpha+1)/\mathscr{G}amma(\alpha)=\alpha$ +6799,"$P = 1 \, \text{GPa} = 1 \times 10^9 \, \text{Pascals}$" +6800,$\omega\in B_1\cup\dots\cup B_r$ +6801,$g(S(x))=u$ +6802,"$ϕ(s) = α^{-1}1_{[1-α, 1)}(s)$" +6803,$\ge 1$ +6804,$T_x=x$ +6805,$P = \mathsf{E}[X] + \pi \mathsf{E}[(X-\mathsf{E}[X])^+]$ +6806,"$T:(\Omega, \mathscr{F}) \to (M,\mathcal{B})$" +6807,$\mathbf {\Delta_{m}}$ +6808,"$(M,\mathcal{B}, \nu)$" +6809,"${}^{[<72,47]}$" +6810,$\beta=-\theta$ +6811,$\kappa_i(x)=\mathsf{E}[ X_i \mid X = x]$ +6812,$\theta$ +6813,$g(s) = 1 - (1 - s)^d$ +6814,$S(x_i-)-S(x_i) =\mathsf{Pr}r(X=x_i)$ +6815,"$\phi(x_1,...,x_n)$" +6816,"$(\s,4.5-\s)$" +6817,$x_3$ +6818,$1=P(a)+Q(a)$ +6819,$F = U - TS$ +6820,$\beta_i(x)/\alpha_i(x)$ +6821,$L^r$ +6822,$cv=0.287$ +6823,$g'(t)dt < dt$ +6824,$0 \le p_0 \le p_1 \le 1$ +6825,$X\circ T$ +6826,$i=\sqrt{-1}$ +6827,$\mathsf{E}[S_t]=e^{\mu t}$ +6828,$s=\mathit{EL}$ +6829,$s^{\star} := s_R + \frac{1-g_R}{\phi_R}$ +6830,$\rho(X) = \mathsf{E}_{\mathsf{Q}}[X] = \mathsf{E}_{\mathsf{Q}}[X\wedge a + (X-a)^+] = \mathsf{E}_{\mathsf{Q}}[X\wedge a] + \mathsf{E}_{\mathsf{Q}}[(X-a)^+] \le \rho(X\wedge a) + \rho((X-a)^+) = \rho(X)$ +6831,$X\wedge d$ +6832,$g(uv)=g(u)v\le g(u)g(v)$ +6833,$2aw=0.0035$ +6834,$\mathsf{E}[X\mid\mathscr{G}](ω)=\mathsf{E}[X]$ +6835,$W(t)$ +6836,"$\bar\delta,\bar\nu$" +6837,$\phi_R-\phi_{R+1}^u$ +6838,$\ll$ +6839,$T_2 := ((n+1)-pN)x_n$ +6840,$X(\mathbf{w})=\sum_i w_iX_i$ +6841,"$\mathbf{v}=(v_1,\ldots,v_n)$" +6842,$h_x$ +6843,$\le p$ +6844,$c_r^Y := g(0+)\Theta_0^Y + \sum_{i=1}^{R-1}\Delta_i s_i\Theta_i^Y$ +6845,$n+1$ +6846,"$1,453.05 \$" +6847,$O(\delta^2)$ +6848,$n\log_2(n)$ +6849,$\rho(X\wedge a)=\bar P(a)$ +6850,$a(X_i+X_j) < a(X_i)+a(X_j)$ +6851,$t\mapsto \rho(X+tY)$ +6852,$\theta_s >0.5$ +6853,$|f|$ +6854,"$\langle \cdot,\cdot\rangle:\mathcal{X}\times\mathcal{P}\to \mathbb{R}$" +6855,$p={{p}}$ +6856,$\zeta\in\mathscr{O}(\eta)$ +6857,"$f(0,0)=0$" +6858,"$\mathcal E_{X,r}\subset R_{X,r}$" +6859,$X$ +6860,$2\square^2 + 2\square$ +6861,$\mathsf{Pr}r(X\le x)=0$ +6862,$\sigma_i^2=\mathsf{var}((\mathsf P_{i+1} - \mathsf P_{i})X)$ +6863,$100 \times 100$ +6864,$\mathbf {\mathsf{VaR}_p(X_1+X_2)}$ +6865,$g(0+)=0$ +6866,"$(p, q(1-g^{-1}(1-p)))=(p, q(\hat p))=(p, \hat q(p))$" +6867,$Q=(P+P')/2$ +6868,$\int_0^1 x / \sqrt{x}dx=\int_0^1 \sqrt{x}dx=2/3$ +6869,$x^{-3}$ +6870,$S_X(x) \ge S_{X_1}(x)$ +6871,$\mathsf{VaR}_{0.995}$ +6872,$L(a)=\mathsf{E}(X\wedge a)$ +6873,$\mathcal{N}_{X}(X_i(a))$ +6874,$1.805$ +6875,"$P^T_S(\cdot\mid\cdot):\mathcal{A}\times M\to [0,1]$" +6876,$\nu+10\delta$ +6877,$n=8$ +6878,"$(X^∗_1, \dots, X^∗_n)$" +6879,$\rho(X_0+\epsilon Y)-\rho(X_0)$ +6880,$dF(X)$ +6881,$d(\alpha F+ (1-\alpha)G) = \alpha dF + (1-\alpha)dG$ +6882,$-1_{B_r}$ +6883,$p(1-\nu(p))=p\delta(p)$ +6884,$1+V^{\ast}(1) > V(2)$ +6885,$X=X_{k+1}$ +6886,$\rho:{ \mathcal{X} }\to\mathcal{B}bb R$ +6887,$u''<0$ +6888,$1-p=g(S(x))$ +6889,$0=x_0q(p)$ +6903,$P = S + R$ +6904,$E(X_{-1}\wedge a)$ +6905,$\pi=0$ +6906,$0.7 \le p < 0.8$ +6907,$A^k=(g^a)^k$ +6908,$p=F(a)=1-s$ +6909,"$g:[0,1]\to [0,1]$" +6910,$16\times 4=64$ +6911,$X_2=c_2+2Y$ +6912,$\mathscr F_1\subset \mathscr F$ +6913,$d\theta= dm/V(m)$ +6914,$s=0.1$ +6915,"$(x_i-x_j,y_i-y_j)$" +6916,$\frac{1}{2}\sech(\pi y/2)$ +6917,$p=\frac{\alpha-2}{\alpha-1}$ +6918,"$\forall A\ \forall p\ [\forall x\in A\ \exists !y\ \phi(x, y, p)\rightarrow\exists Y\ \forall x\in A\ \exists y\in Y\phi(x, y,p)]$" +6919,"$l=0,1,\dots,n/2+1$" +6920,$q(p)+y$ +6921,$\beta_{2}$ +6922,$g_{j} = g(s_{j})$ +6923,$\Omega\times \mathscr{F}$ +6924,$p\not=0.5$ +6925,$T\lambda$ +6926,$\mathsf{E}[Y_d]$ +6927,$\phi(s)\ge 0$ +6928,"$14,000 \$" +6929,$\mathsf{E}[X] = \mathsf{E}[Y]$ +6930,$\Delta_d=a_{d}'-a_{d}$ +6931,$x\ge a$ +6932,$k\ge 0$ +6933,$e$ +6934,$P=L + \iota Q = \nu L + \delta a=L(1+\rho)$ +6935,$v_f(\mathsf{E}_\mathsf{Q}[X_i] - \mathsf{E}_\mathsf{Q}[X_i/X(X-a)^+])$ +6936,$\rho(X) = \mathsf{E}(X) + V(X)$ +6937,$\rho_i(X_i) - \rho_i(F_i)$ +6938,$q_Y(U)$ +6939,$i=1\dots N$ +6940,$\mathcal{R}=\mathcal{R}^b$ +6941,$\mathsf{E}_Q(Y)=\tilde \rho(X)$ +6942,$p=1.995$ +6943,$g'''>0$ +6944,$X(x_i)=x_i$ +6945,$\alpha(1+fg/(1-g))$ +6946,$X_j^1$ +6947,$X\_{2}$ +6948,$A_0=\mathsf E[X]$ +6949,$\delta_p=1-\nu_p$ +6950,$X_\nu$ +6951,$A(\omega)$ +6952,$g'(s)=\alpha s^\alpha/s$ +6953,$\mathsf{E}\_\mathsf{Q}[X\_2]$ +6954,$j+1$ +6955,$\mathbf{T}_0$ +6956,$\tpx \mu_{x+t}$ +6957,$s<{s_equity2:.3g}$ +6958,"$t_1 \in [0, s_L]$" +6959,$x_{ij}$ +6960,$A^∗_i$ +6961,$\alpha=0$ +6962,$\rho=\mathsf{TVaR}_p$ +6963,$f(y;\mu)=c^*(y)e^{-d(y;\mu)/2}$ +6964,$\propto x^{-\alpha-1} e^{\theta x}$ +6965,$D\rho_X(X_2)=45.1801$ +6966,$g(S(x_B))-g(S(x_B-))$ +6967,$(\mu-\sigma^2/2)t$ +6968,$\rho(X_0)\ge \mathsf{E}[X_0 Z_\epsilon]$ +6969,"$p_1, p_2, \dots, p_n$" +6970,$q(p^*)=0.0$ +6971,$2^{256}\approx 10^{77}$ +6972,$f_{\mathbf{w}}$ +6973,$R_0(t)>P(0)$ +6974,$\mathsf{P}(\{X\in A\})$ +6975,$\mathsf{E}_Q[ \cdot ]$ +6976,$X_h$ +6977,$x\subset X\leftrightarrow \forall z(z\in x\rightarrow z\in X)$ +6978,$\approx 1$ +6979,$T = 2L / \sqrt{g d}$ +6980,$\mathsf{E}[X]+\mathsf{SD}(X) \le \mathsf{E}[Y]+\mathsf{SD}(Y)$ +6981,$1M less \$ +6982,$p=0$ +6983,$J(x)<\infty$ +6984,$Q^i=M^i/\iota$ +6985,$\mathsf{E}[Y\mid \mathcal F']$ +6986,$S_{\mathbf{v}}(x)$ +6987,$\rho_F(X)=\max_{\mathsf{Q} ~ F} \mathsf{E}_\mathsf{Q}[X]$ +6988,"$uv, u, v \in (s_1, s_0]$" +6989,$\rho(X)=\mathsf{E}_{\mathsf{Q}}[X\wedge a] + \mathsf{E}_{\mathsf{Q}}[(X-a)^+] \le \rho(X\wedge a) + \rho((X-a)^+)=\rho(X)$ +6990,$V_1 / V_2 = 12:1$ +6991,$\nabla g'$ +6992,$\hat p_s(\hat\theta_s)$ +6993,$(f)$ +6994,$\iota^*=0.125$ +6995,$x^2\mathsf{Var}[N]$ +6996,$m_1=am$ +6997,$\mathsf{Pr}(B=0)=1-p$ +6998,$1/beta$ +6999,$\mu+\mu^3/2+(\mu^2/2)\sqrt{2+\mu^2}$ +7000,$X:\Omega\to\mathcal{B}bb R$ +7001,$10^{11}$ +7002,$g(t) = h_R + \phi_R(t-s_R)$ +7003,$t_0^* > t_1^*$ +7004,"$[0, 1-p]$" +7005,$U/4$ +7006,$\mathsf{E}[X^2]$ +7007,$\mathbf S$ +7008,"$m=2,\dots,6$" +7009,$6 \times 10^7$ +7010,$2\square^2 + \square + 5$ +7011,$\mathsf{Pr}r(X > q_{\mathbf{v}}(p))=1-p$ +7012,$M=P-S$ +7013,$\alpha < 2$ +7014,"$\max(y, 0)$" +7015,"$\eta_{p,\alpha}$" +7016,$486 | \$ +7017,$\mathsf{Pr}(N=n)=p_n$ +7018,"$\mathsf{CP}(1,X)$" +7019,$\phi:M\to\mathbb{R}$ +7020,"$500,000 per claimant except that workers' compensation claims are paid in full; $" +7021,$10^{27}$ +7022,$\mathsf Q_t$ +7023,"$[0, s_1]$" +7024,$=1$ +7025,$\mathsf{E}[X_ih(X)]=\mathsf{E}[\kappa_i(X)h(X)]$ +7026,$\mathsf{E}[X_i \mid X=x]$ +7027,$\sigma=2.15$ +7028,"$T_{m_1}\in\mathcal R^c_{X,r'}$" +7029,$f(x)=|x|$ +7030,$\liminf \rho(-k_i 1_{A_i}) \ge \rho(0)=0$ +7031,$1/(1+r_f) = \mathsf{E}[p]$ +7032,$\mathsf{MONO}$ +7033,$X\Delta g(S)$ +7034,$\mathsf{VaR}\_p(X\_0)$ +7035,$\mathscr{G}\subseteq \mathscr{F}$ +7036,$Z'$ +7037,$K=B^a=A^b=g^{ab}\pmod p$ +7038,$\rho_t(X)\le \rho_t(Y)$ +7039,$X_i \ge \mathsf E[X_{i+1}\mid I_i]$ +7040,$\dfrac{(1-t)x_0+ta_1}{(1-t)x_0+tX_1}<1$ +7041,$\mathsf{VaR}_{0.7}(X)=$ +7042,"$[p,1]$" +7043,$x_2\leftrightarrow y_2$ +7044,$j=1$ +7045,$\rho(X_1) \ge D\rho_X(X_1)$ +7046,"$M=\max(X_1, \ldots ,X_N)$" +7047,$\int_0^s q_Z(1-t)dt\le g(s)$ +7048,$N(a)=\int_0^a F(x)dx=a-\mathsf{E}(X\wedge a)$ +7049,$g\Delta X \wedge a$ +7050,$g'\left (S(X)\right )$ +7051,$Q=T\mathsf{Pr}$ +7052,"$[s^L,s^U]\times([s_n,s^L]\cup [s^U,1])$" +7053,$p=0.75$ +7054,$g_{FS}(s*)=1$ +7055,$\lambda = \phi\lambda'$ +7056,"$S(1),S(2),\dots,S(A)$" +7057,$(1+\epsilon)x_1$ +7058,$\rho^x_h(s_R)=\rho^x_h(1)=\rho^o$ +7059,"$j=1,\dots r$" +7060,$P(a)+K(a)=a$ +7061,$\bar P(a)$ +7062,$(-a)\mathscr{G}amma(-a) = \mathscr{G}amma(1-a)$ +7063,$a-\bar P(a)$ +7064,$\bar\nu a$ +7065,"$\mathcal{M}_{X,r_X}$" +7066,$ag(S_{\mathsf{j}(a)})=(80)(0.5)=40$ +7067,"$[a, b]$" +7068,"$100,000 and an aggregate deductible limit of \$" +7069,"$[a,b] \subset [0,1]$" +7070,$\prec_3^*$ +7071,$R_1(0)$ +7072,$\lambda = \lambda_0+\lambda_1$ +7073,$g'\circ S_{X\wedge a}$ +7074,$\Delta X_k$ +7075,$\implies\mathsf{FATOU}$ +7076,$\epsilon^2$ +7077,$X_n(\omega)\to 0$ +7078,$a_Y \le \rho(Y)$ +7079,"$\bar P_{0,t}:=\rho(Y_{0,t})$" +7080,"$g(s)=\min(g_1(s), g_2(s))$" +7081,$\theta=-\mu^{-1}$ +7082,$A_k = A_0 + kN$ +7083,"$\mathrm{ess\,sup}(X)$" +7084,$L$ +7085,$x\mapsto x^c$ +7086,"$x_c, x_n$" +7087,$X(x)$ +7088,$\rho(X)=\int_0^1 q(s)\phi(s)ds=\int_0^1 q(s)g'(1-s)ds$ +7089,$\mathsf{Pr}r(X=y_j)$ +7090,$\mathsf{E}[X_1]=\mu$ +7091,$\omega_1=1$ +7092,$p_i/(1 - \alpha)$ +7093,$X_1=P_1X$ +7094,$g(0) = 0$ +7095,$\mathsf{var}phi (2 X) = \mathsf{var}phi(X + X) = 2 \mathsf{var}phi(X)$ +7096,$\mathsf{E}[X\mid \mathcal F_{\tau}]$ +7097,$e^{-2\pi i}=-1$ +7098,$a_Y$ +7099,$X^{\tau_n}$ +7100,"$(X,\mathscr{F},\mathsf{Pr})$" +7101,$E[G]=1$ +7102,$S_t \ge 0$ +7103,$\rho^e_g$ +7104,"$(\Omega,\mathscr F, \mathsf P)$" +7105,$W_t=\zeta_1Z_1+\dots +\zeta_tZ_t$ +7106,$\mathsf{S}$ +7107,$r:=r_p=r_c>i$ +7108,$\rho(x+X) = x + \rho(X)$ +7109,"$r_2 = 3852972862.741849 \approx 3,852,972,862.7$" +7110,$g(s)^2t_1^*$ +7128,$=A_y =$ +7129,$r_i = (P_i - \mathsf{P}[X_i]) / P_Q$ +7130,$m_Y$ +7131,$m_i$ +7132,$s_i=1-p_i$ +7133,$\mathsf{E}[Z_1]=\mathsf{E}[Y]$ +7134,$Z\circ\tau$ +7135,$0 \le \alpha \le 1$ +7136,$D_t \ge 0$ +7137,$L_0^{500}(x)$ +7138,$\hat\rho(X)=g(s/\omega_I)g(\omega_I)s$ +7172,$S_1$ +7173,"$t\in[0,0.7]$" +7174,${}^{[>45]}$ +7175,$m(y)$ +7176,$\sum_i q_iX_i$ +7177,$c(\{i\})=c(i)$ +7178,"$X_{t-d,d}$" +7179,$g(s)=s^b$ +7180,$T=T_B\circ T_A$ +7181,$y=q_h(p)$ +7182,$n\ge N$ +7183,"$1 million in punitive damages from Steyn, $" +7184,"$(A, a)$" +7185,$X_{2}$ +7186,$f(p)=E[f(X)]$ +7187,$1-q$ +7188,$F_1^{-1}(\omega_1)+F_2^{-1}(\omega_2)$ +7189,$\mathsf{Var}(X)=\mathsf{E}[X^2]-\mathsf{E}[X]^2$ +7190,"$\mathsf{E}[e^{\theta X_t}] = e^{t\,\kappa_X(\theta)}$" +7191,$\mathbf {X_1(a)}$ +7192,$R_1(t) \ge \mathsf{E}[X_1]$ +7193,$\int X=0$ +7194,$o_p$ +7195,$k=\rho(X)-\pi(X)$ +7196,$A=\alpha(X)$ +7197,$0\le b\le 1$ +7198,$p=0.458$ +7199,$\lambda < 1$ +7200,$L_p/L_q$ +7201,$\mathsf{Q}_2$ +7202,"$\mathsf{cov}(X_i,\sum_j X_j)=\mathsf{cov}(X_i,X_i)=\mathsf{Var}(X_i)>0$" +7203,$0\le p\le 1$ +7204,$M_t$ +7205,$\mathsf{E}_{QQ'}[X_i(a)] \ne \mathsf{E}_{QQ}[X_i(a)]$ +7206,$RM$ +7207,$s\mapsto 1-g(1-s)$ +7208,$\rho(X)= (1+r_f)^{-1}\mathsf{E}_Q[X]$ +7209,$1.05/1.3 = 0.808$ +7210,"$400,000 Treaty Retention: \$" +7211,${}^{[>104]}$ +7212,$0\leq f \leq 1$ +7213,"$(valu1.south east)+(\boundpad,-\boundpad)$" +7214,"$\lambda p_{k,n}\approx \lambda f(k/n)/n$" +7215,$x\mapsto xX$ +7216,$s_R$ +7217,$P/l-1 =\rho= \iota Q / l = \iota(C/l + g)$ +7218,$\rho(X)=\int_0^\infty x g'(S(x))f(x)dx$ +7219,$2^1+1\rightarrow 3^1+1-1=3^1 \rightarrow 4^1-1=3 \rightarrow 2 \rightarrow 1 \rightarrow 0$ +7220,$\bar M(a):=\int_0^a M(x)dx$ +7221,$v_i=\mathsf{Pr}hi^{-1}(i/(2m+1))$ +7222,"$\mathrm{ess\,sup}(X)=1$" +7223,$\mathsf{Var}^+(X) = \int_{\mathsf{E}[X]}^\infty (x-\mathsf{E}[X])^2 f(x)dx$ +7224,"$[0, \epsilon_1]$" +7225,"$[-\infty, 2]$" +7226,$\mathsf{E} X + c\mathsf{E}[((X-\tau)^+)^p]^{1/p}$ +7227,"$EL = E[X] = \int S(x)\,dx = \int xf(x)\,dx$" +7228,"$X_{t-1,2}$" +7229,$\alpha_1(99)=0.1$ +7230,$P_k=g(S_k)=\nu_k S_k+\delta_k$ +7231,$S(x_1)(x_2-x_1)$ +7232,$\rho(X)\ $ +7233,$(g(s)-s) / (1-g(s))$ +7234,"$X,\, X_i\in L^\infty$" +7235,$X_t$ +7236,$\mu=\tau(\theta)$ +7237,"$\{s_L,\dots,s_m=1\}$" +7238,$1-\exp(-q(p)/\mu)=p$ +7239,$c(1)-c(\mathsf{var}nothing)=c(1)$ +7240,$Y_n\uparrow 0$ +7241,$\bar Y$ +7242,$\kappa_i(X_k)$ +7243,$\sqrt{s^2}$ +7244,$f(1)=1$ +7245,"$81,858,610 in return premiums**, which is netted against premiums ceded (reinsurance) in the statements of income and $" +7246,$X(\omega)=1-\omega$ +7247,$\mathsf{Pr}r(X ≥ x_0) = p$ +7248,$400 to over $ +7249,$ which is an extreme stable with Lévy distribution $ +7250,$\mathsf{E}[|X_1|]<\infty$ +7251,$X\preceq_m Y$ +7252,$D\rho_X(X_1)=6.2085$ +7253,$\mathbf {a_1'}$ +7254,$\mathsf{Pr}r(\omega)$ +7255,"$\alpha_i(a) = \dfrac{\sum_{j:X_j>a} (X_{i,j}/X_j)p_j}{\sum_{j:X_j>a} p_j}$" +7256,"$\mathsf{E}[Y]/\text{mean}(\mathcal E_{X,r}(Y))$" +7257,$\sup_{\alpha \in A} \mathsf{E}[|X_\alpha|^p] < \infty$ +7258,$a_0=1000$ +7259,$\mathbf {a}$ +7260,$\mathsf{E}[X^k-Y^k]=\int x^k\mu_X(dx)-\int y^k\mu_Y(dy)=\int x^k\nu(x)$ +7261,$\mathsf{E}[Y] = 50.4$ +7262,$N'/w-\alpha a\le Mg$ +7263,$\mathsf{E}[v^T] \ge v^{\mathsf{E}[T]}$ +7264,$B_r$ +7265,$\Lambda = \dfrac{E( r_{U} ) - r_{f}}{\sigma_{U} \sigma_{r_{U}}}$ +7266,$\int_0^1 \mathsf{TVaR}_p(X)m(dp)$ +7267,"$s_R,s_{R+1},s_m$" +7268,$\rho(X\wedge a)=0.909$ +7269,$\partial f_{\bar x}/\partial x_i$ +7270,$B=\Omega\setminus C$ +7271,$p\nu(p)=p((\nu(p)-l(p))+l(p)) = \nu^*\sqrt{pq} + v(p-\sqrt{pq})$ +7272,$\mathcal F_{\tau}$ +7273,$\rho(X) = a = \mathsf{E}[X | A] = ES$ +7274,$\subset$ +7275,$\alpha=$ +7276,"$j=i^\star+1,\dots,m$" +7277,$\Delta X_k-P_k$ +7278,$A=X_1+\cdots +X_N$ +7279,$F_m\succ_m F_0$ +7280,$V^*$ +7281,$\omega=0.\omega_1\omega_2\dots$ +7282,$\int_0^1$ +7283,$_{b}$ +7284,$g=\mathsf{TVaR}_p$ +7285,$X\le b$ +7286,$Y_\nu=Z_\nu/\nu$ +7287,"$500,000 excess of \$" +7288,$5 trillion business. Property casualty insurers write $ +7289,"$(1,\infty)$" +7290,$\mathsf{E}_{\mathsf Q}[X]$ +7291,$X=\sum_i X_i$ +7292,"$1,000 | \$" +7293,$\sigma=2.5$ +7294,$(\cdot)^i$ +7295,$\bar M_i(a) = \bar P_i(a) - \mathsf{E}[X_i(a)]$ +7296,$\mathsf{E}[X_1h(X)]$ +7297,"$, @Pichler2015a, 6.1. @Dentcheva2010 (DPR) goes to great lengths to prove represented by transforms (AVaR to spectral transform) with $" +7298,$X>Y$ +7299,"$\mathit{ED}(\mu, \sigma^2)$" +7300,${X}_p=\mathsf{E}[|X|^p]^{1/p}$ +7301,"$\min(X,a)$" +7302,$\mathit{ROL} = \exp( \alpha_{0} \log( \mathit{AEL} ) + \alpha_{1} \log( \mathit{AEL} )^{2} )$ +7303,$\alpha=1/\nu^2$ +7304,$=\displaystyle\int_0^\infty x f(x)dx$ +7305,$<$ +7306,$\rho(X \circ T)=\rho(X)$ +7307,$\mu_{rU} = M/K = 0.133$ +7308,$\|Y\|_{\sigma} = \rho_\sigma(Y)$ +7309,$\mathsf{E}[(a-X)^+]$ +7310,$\sigma^2>0$ +7311,$g^1=g^2=g$ +7312,$V(m)=m^2$ +7313,$\delta = \iota /(1+\iota)$ +7314,$m\left(1+\dfrac{m}{p}\right)\left(1+\dfrac{a+1}{a}\dfrac{m}{p}\right)$ +7315,$s_m$ +7316,$S(u)$ +7317,$g(s) = d + (1-d)h(s)$ +7318,$\Delta P$ +7319,$dG/dF=g'(S(x))$ +7320,$\mu=0.1$ +7321,$m=0.25$ +7322,$\mathscr{G}=\sigma(Y)$ +7323,$p=(\alpha+2)/(\alpha+1)=3$ +7324,$G_i'$ +7325,$m(s)=\beta(\beta-1)s^{\beta-2}(1-s)$ +7326,$r=0.1$ +7327,$\delta p$ +7328,$EL(x)=S(x)$ +7329,$\le 1/N$ +7330,$\text{AVaR}_\alpha(X)$ +7331,$N \mid G$ +7332,$5/6$ +7333,$\rho(Y)=T_{m_2}(Y)-v(m_2)$ +7334,"$g\in D_n^*=\{ g \mid (-1)^{k+1} g^{(k)} \ge 0, k=1,\dots,n-1, (-1)^n g^{(n-1)}\text{ non-increasing} \}$" +7335,$\alpha_i(k) = \mathsf{E}_p[\kappa_i(X)/X\mid X > X_k]$ +7336,$\beta=\alpha/\mu$ +7337,$g(s)=\mathsf{Pr}hi(\mathsf{Pr}hi^{-1}(s)+\lambda)$ +7338,$p_- < p_0 < p_+$ +7339,"$242,000 (= \$" +7340,$X > X_{k+1}$ +7341,$X_i^y$ +7342,$q_{j} = g_{j} - g_{j + 1}$ +7343,$\rho(X^{\oplus N})$ +7344,$0.5L_{250}^{500}(x)+0.75L_{500}^{750}+L_{750}^{1000}$ +7345,$h=2$ +7346,"$(Bob) + (0,-3.5)$" +7347,$p_{j+}$ +7348,$Y\circ T_i$ +7349,$p_0>0$ +7350,"$(0,0,\dots,0,10)$" +7351,$1 < \alpha < 2$ +7352,$g(s)=m(s)+s$ +7353,$n(n-1)/2$ +7354,"$\{1,\dots,n \}$" +7355,$t+1$ +7356,$1-\alpha$ +7357,$\int_0^a g(S(x))dx$ +7358,$\mathsf{E}[X]=\mathsf{E}[X;q(p)] + (1-p)\mathsf{TVaR}_p(X)$ +7359,$F_I$ +7360,$E(X_{0}(a))$ +7361,$X=m + (n-m)H$ +7362,$\mathsf{E}[X_ih(X)]$ +7363,$\alpha_i'(x)<0$ +7364,$p(a) = S(a) + \delta F(a)$ +7365,$\alpha_i(x):=\mathsf{E}\left[\dfrac{X_i}{X}\mid X > x\right]$ +7366,$\eta=0$ +7367,$aY_i/Y$ +7368,$\beta\gamma/(\beta+\gamma)$ +7369,$Z_{\tilde X}$ +7370,$p<0.1$ +7371,"$[9750, 10550]$" +7372,$2\mathsf{VaR}_p(X_1) - \mathsf{VaR}_p(X)$ +7373,$t=t_1^*$ +7374,$\mathsf{E}[Z\mid X]$ +7375,$Le^{-\delta T_x}1_{T_x244]}$ +7390,$p\approx -\log(1-p)$ +7391,"$(\Omega, \mathsf P, \mathscr F)$" +7392,"$\bar Q_{0,0}:=a_{0,0}-\bar P_{0,0}$" +7393,"$(D.south east)+(0.1, 0.05)$" +7394,$\rho_g(X \wedge a)$ +7395,$t<\tau$ +7396,$\theta \pm 3$ +7397,$0 < r \le 1$ +7398,$\mathop\square\rho_i$ +7399,$=v_f \mathsf{E}_\mathsf{Q}\left[\dfrac{X_i}{X}(X\wedge a)\right]$ +7400,$\mathsf{TVaR}_{p}$ +7401,$b=0.3$ +7402,$\mu=0.107$ +7403,$(L+F)/(1-v-\pi)$ +7404,"$t\in (0,1)$" +7405,$C_i = (V_i/V) C$ +7406,$\mathrm{CV}(Y_t)=\mathrm{CV}(Y_1)/\sqrt{t}\to 0$ +7407,$\mathsf E[T_s T_t]$ +7408,$\lim_{s \downarrow 0} s/g(s) = \lim_{s \downarrow 0}1/g'(s)$ +7409,$\rho(X)=53.565$ +7410,"$\mathsf{Tw}(\lambda, \alpha, \beta)$" +7411,$\rho(X)\le\rho(Y)$ +7412,$0 < s \le 0.2$ +7413,$p=1-1/n$ +7414,$\beta_Q=(a/Q)\beta_A + (P/Q)\beta_L$ +7415,$\mathsf{E}(X_iX_i \mid X)\not=\mathsf{E}(X_i \mathsf{E}(X_i\mid X)\mid X)=\mathsf{E}(X_i\mid X)^2$ +7416,$p=29$ +7417,$c_x$ +7418,${}^nS^{-1}_X(q)\le {}^nS_Y(q)$ +7419,"$n=0.5,1,\dots,7$" +7420,$\sum_x x p_x$ +7421,$pqV(X-Y)=pV(X)+qV(Y)-V(pX+qY)\ge0$ +7422,"$\min(x+y^c, a+bz)$" +7423,$|e^{isx}|=1$ +7424,$p\mapsto g'(1-p)$ +7425,$\pi=1.2497$ +7426,$q$ +7427,$1-K_s = P_s = g(s)$ +7428,$g(s^2)=g(s_1)\ge g(s)^2=g(\sqrt{s_1})^2$ +7429,$12.5 billion ($ +7430,$\rho(X):= \int_0^1 q(p)\phi(p)dp$ +7431,$\mathbf {\alpha_1}$ +7432,$g(S(x))$ +7433,$\hat \theta_s = c/n$ +7434,$\mathsf{var}(\sum C_i)=\sum (m_i v_i)^2 = n(mv)^2$ +7435,$g(s)=(20s)\wedge 1$ +7436,$F(x)=\mathsf{P}(X\le x)$ +7437,$\mathsf{VaR}_{0.7}(X)=2.439 > 2 \times 1.204=2.408$ +7438,"$g(0-)f(\mathrm{ess\,sup}(X))$" +7439,$\rho(X\wedge a)=\mathsf{E}_\mathsf{Q}[X\wedge a]$ +7440,$\approx 10^{-5}$ +7441,$\lambda X_1$ +7442,$\rho(U)=1$ +7443,$s\sqrt{1+n^{-1}+(x-\bar x)^2/((n-1)s_x^2)}$ +7444,$r = d / v$ +7445,$p=3/4$ +7446,$\text{VaR}_p$ +7447,$x\mapsto (x-d)_+^{n-1}$ +7448,$s_0$ +7449,$(1-g(s))(1-q)$ +7450,$g(S(x))>g(x)$ +7451,$1-t> (1-t)^2 > s_0$ +7452,$R_t$ +7453,$\mathsf{VaR}_p$ +7454,$\sigma > 1$ +7455,$<\int_0^1 x^{\alpha}dx$ +7456,$I$ +7457,${}^{[>4]}$ +7458,$F_\sigma$ +7459,"$\mathbf{j, p, S, \kappa_1, \Delta X, \Delta(X\wedge a)}$" +7460,$10^{18}$ +7461,$ and gamma severity. Distributions with $ +7462,$\mathsf{Pr}r(X_0\le 0)=1$ +7463,$\mathsf{TVaR}_{0.8}(X)=8.5$ +7464,$E_\mathsf{Q}$ +7465,$P_k = g(S_k)$ +7466,"$a_{0,2}$" +7467,$P_g\{X=M\}=g(0+)>0$ +7468,$S(x) + dF(x) + (\delta^*-d)\sqrt{S(x)F(x)}>1$ +7469,$\bar P(a) = \bar S(a) +\bar\delta(a) \bar F(a)$ +7470,$X_0 + \epsilon Y$ +7471,$Y=W+Q$ +7472,$y=$ +7473,$E[X\_{2}(a)]$ +7474,$c\ge 0$ +7475,$r_D=0$ +7476,$0.98u(52)>0.98 (63/61)u(50)>u(50)$ +7477,$\zeta\circ T\in\mathcal{M}$ +7478,"$\mathsf{MON,\ NORM}$" +7479,"$G(t, 1/\nu)$" +7480,$4\times 10^{19}$ +7481,$\int xf(x)dx$ +7482,"$Z \sim \mathrm{DM}^*(\theta, \lambda)$" +7483,$(c(S\cup \{i\})-c(S))$ +7484,$s^{*} = \max\{ s(j)|X^{< j >} > A\}$ +7485,$X=q$ +7486,$\rho(Y) \le \sup_m b_m < b_{m_0}$ +7487,$g:\mathcal{B}bb R_+\to\mathcal{B}bb R\cup\{\infty\}$ +7488,$AK$ +7489,$\mathsf{Pr}r(X\ge x_0)=p_-$ +7490,"$\mathbf {S\,\Delta X}$" +7491,$(y-\mu)/V(\mu)$ +7492,$Q=\sum_j Q_j$ +7493,$L_\sigma^*:=\{ Z\in L_1\mid \| Z\|_\sigma^*< \infty \}$ +7494,$s=S(x)$ +7495,$\downarrow\downarrow$ +7496,$827.41 and \$ +7497,$G_\delta\subset\R$ +7498,${}_tE_x=e^{-\delta t}{}_tp_x$ +7499,"$(0,\infty)$" +7500,"$1 between any of the tranches, then $" +7501,$\tilde x$ +7502,$l(y;\mu)=\mu y - \dfrac{\mu^2}{2}$ +7503,$E_{\mathsf{Q_X}}[X_i(a)]$ +7504,$|^F$ +7505,$p(\omega_1)+\cdots +p(\omega_1)$ +7506,$(s+\iota) / (1+\iota)$ +7507,$X_n(\omega)\to X(\omega)$ +7508,"$\mathcal{M}_{X,r}$" +7509,$c>0$ +7510,$p_2 = p_1 / 2$ +7511,$\mathit{EL}$ +7512,$A_t = t$ +7513,$j=2$ +7514,$2\times 10^{30}$ +7515,$N_t - A_t = -t$ +7516,$P^T(T=t\mid t)=\mathsf{Pr}(\{t\})=0$ +7517,"$\mathsf{Pr}(A)=\mathsf{Pr}(A\cap\Omega)=\int_\Omega 1_B\,d\mathsf{Pr}=\mathsf{Pr}(B)$" +7518,${}^{[<33]}$ +7519,$Z_0$ +7520,"$s_i, i=0,\dots,R$" +7521,"$([0, 1], \tF,\mathsf{Pr})$" +7522,$_p$ +7523,$(1-p)^{-1}\mathsf{E}[X_i1_{X\ge x_p}(X)]$ +7524,$y_4$ +7525,$0 \le p<1$ +7526,$R^2=0.886$ +7527,$1-\delta=\nu$ +7528,$s=\exp(-a/b)$ +7529,$\zeta_{\bar x}$ +7530,$Z_k=q_k/p_k<1$ +7531,$p(1-p)/(p\nu_p)^2$ +7532,$Q=a-\rho(X)$ +7533,$M=\inf\{ x\mid S(x)=0\}$ +7534,$RA$ +7535,$\beta_i(a)g(S(a))=\mathsf{E}_{\mathsf{Q}}[(X_i/X) \mid X>a]g(S(a))=\mathsf{E}_{\mathsf{Q}}[(X_i/X) 1_{X>a}]$ +7536,"$\{ \omega\mid p(\omega, A)=1_A(\omega),\ \forall A\in\mathscr{G} \}$" +7537,$\mathsf{Pr}r(Y_m > y) = 1 - (1 - \mathsf{Pr}r(X > y))^n$ +7538,$q_1(t)=t$ +7539,$\mathsf{E}[X_iX]$ +7540,$0 on the \$ +7541,$t\ge T$ +7542,$\tau'(\theta)=\kappa''(\theta)=\mathsf{Var}(Y_\theta)>0$ +7543,"$Y_{1,2}$" +7544,$\mathsf{FAT}$ +7545,$X \le a$ +7546,$E[X_i \mid X]$ +7547,$\lambda_2/\lambda$ +7548,$F(x)=\mathsf{Pr}r(X\le x)$ +7549,$X_n=-e^{-nx}$ +7550,$\mu_{rU}$ +7551,$1\mapsto 1$ +7552,"$m_{p_i,p_j}:=w_{p_i,p_j}\delta_{p_i} + (i-w_{p_i,p_j})\delta_{p_j}$" +7553,"$f:(L,\mathcal{A})\to(M, \mathcal{B})$" +7554,$R_1=C_1$ +7555,$\bar M(a) = \bar P(a) - \mathsf{E}[X\wedge a]$ +7556,$R_x:=T_x-x$ +7557,$=64 \times 4 = 256$ +7558,$a=9532.0$ +7559,$\displaystyle\int_\Omega X(\omega)\mathsf{Pr}(\omega)$ +7560,"$500, with the other options being \$" +7561,$\mathsf{Pr}hi(-d^*)>0$ +7562,$(m_1-m_0)/s_1$ +7563,$P_i = L_i + \iota K_i$ +7564,$\rho(X)\le b$ +7565,"$(s_{j}, g_{j})$" +7566,"$[f'_-(x_0), f'_+(x_0)]$" +7567,$c_r := g(0+)\Theta_0^X + \sum_{i=1}^{R-1}\Delta_i s_i\Theta_i^X$ +7568,$\mathsf Q$ +7569,"$\mathsf{CP}(\lambda, B)$" +7570,$600 million; Allstate is expecting $ +7571,$Q_k=\Delta X_k-P_k$ +7572,$\omega=.\omega_1\omega_2\dots$ +7573,$\mathsf{E}[X \mid U]$ +7574,$\mathsf P=\mathsf P_0$ +7575,"$\mathsf{PH, SA, CX, COH}$" +7576,$s/\sqrt{(n-1)s_x^2}$ +7577,$\exists c>0: \left\Vert K_\delta\right\Vert_{L^1}\le c\ \forall \delta>0$ +7578,$\bar S(a):=\mathsf{E}(X\wedge a)$ +7579,$-u''(w)/u'(w)$ +7580,$\hat{\tilde p}=1-g^{-1}(1-[1-g(1-p)])=p$ +7581,$C_i = m_i - X_i$ +7582,"$1,\dots,m$" +7583,$\rho(A_0) \le \rho(A_0) + \mathsf E[A] \le \rho(A)$ +7584,$\mathsf{E}[X\mid X \ge x]\ge x$ +7585,$X_t:=X\mid\mathscr F_t$ +7586,$s^2$ +7587,"$ (range.south)+(0, -1) $" +7588,$(8)(0.25)+(10)(0.25)=4.5$ +7589,$\rho_g(X)=\mathsf{E}[q(U)\phi(U)]$ +7590,$P = 3.103$ +7591,$1+v_s$ +7592,$R_2(t)\approx \mathsf{E}[X_1]$ +7593,$M(x)=g(s)-s$ +7594,$uv<1-s_0$ +7595,$0< m\le 1$ +7596,$\Delta X_k=P_k+Q_k$ +7597,$m^2(1+m)$ +7598,$\phi_i = 1/n$ +7599,$K-1$ +7600,$M:=\max(X)$ +7601,$ and also that $ +7602,$W'_t=\zeta_{t+1}Z_{t+1}+\cdots + \zeta_nZ_n$ +7603,$\mathit{BEL}_s = \sum_{i=0}^{T-1} v^{T-i} \mathsf P_{s} X_{s-i}$ +7604,$(\rho_t)_t$ +7605,$\mathsf{E}[\zeta \tilde X]$ +7606,"$\iota, \iota(x), \iota(p)$" +7607,$\sum \Delta X_K=a$ +7608,$Q(\omega) = q(\omega)P(\omega)/(1+r)$ +7609,$X:\mathbb{R}\to\mathbb{R}$ +7610,$X=m$ +7611,$g(s)=d + O(s)$ +7612,$x/c$ +7613,$U_s$ +7614,$c_h(1-\alpha)$ +7615,$v-\nu^{\star}=(\iota^{\star}-i)/v\nu^{\star}$ +7616,$\bar{\mathbf P}$ +7617,$\{X = q_X(p) \}$ +7618,${}^{[<12]}$ +7619,$1./16=0.0625$ +7620,$\mu()$ +7621,"$k=1,2,\dots,m$" +7622,$R=L^*+R^*$ +7623,$(1-t)X_0 + tX_1$ +7624,$X\ge Y$ +7625,$0.4/44.4 = 0.0090$ +7626,$(p-\nu-il(p))/(\nu-l(p))$ +7627,$(y-\mu)/\sqrt{V(\mu)}$ +7628,$\mathcal A=\mathcal A_\rho$ +7629,$p<0.5$ +7630,$dF$ +7631,$\phi'(s)\ge 0$ +7632,$D_s = X_s - X_{s-1}$ +7633,$gn$ +7634,$g_2(s)=s^{0.5}$ +7635,$-k$ +7636,$L \propto M^{3.5}$ +7637,$(1-t)\mathsf{E}[X_1]$ +7638,$mX_p}$ +7642,$M_i := \beta_ig-\alpha_iS$ +7643,"$\mathsf{Tw}_{1.05}(2, 5)$" +7644,$A\in \mathscr{F}$ +7645,$\rho(X)=\sup_{\mathsf Q\in\mathcal Q} \mathsf{E}_\mathsf{Q}[X]$ +7646,$\{ X>x \}$ +7647,$f(Y_t)=\log(Y_t)$ +7648,$\Delta X_k=X_{k+1}-X_k$ +7649,"$(t,g(t))$" +7650,$\mathscr F_1=\sigma(N)$ +7651,$\lambda_1/\lambda$ +7652,"$g(s)=s^{{{port.dists[""ph""].shape:.3f}}}$" +7653,$0\le t\le 1$ +7654,$g^{m+ra} = g^m (g^a)^r = g^m A^r$ +7655,$f(y;\mu)=\mu e^{-y/\mu}$ +7656,$P(1)$ +7657,$f=0$ +7658,"$(\exp(X),\exp(Y))$" +7659,$y^{\ast}:=\min(y)$ +7660,$\rho(X)=\sup\{\mathsf{E}[hX] \mid h \in \mathscr P \}$ +7661,$S_Y$ +7662,"$\sum_i h_{i, \epsilon}=h_0$" +7663,$L_\sigma$ +7664,"$(\mathsf{E}(X_i)-\mathsf{E}(X_{i,2}(a))/\mathsf{E}(X_i)$" +7665,$\mathsf{E}[X]=\infty$ +7666,$\rho_g(X\wedge a)=(\bar L + ra)/(1+r)$ +7667,$0.41$ +7668,"$\mathbf X = (X_1, \dots, X_n)$" +7669,$LR_{TVaR}$ +7670,$\mathsf{E}_Q[\dfrac{X_i}{X}(X\wedge A)] + \delta A \mathsf{E}_Q[X_i/X\mid X > a]$ +7671,$\alpha_Y = \alpha_X=\alpha$ +7672,$P(\{\omega_2\})=2/3$ +7673,"$. The bracketed term in the middle expression is the objective, expected monetary future value claims, accounting for the rate of growth of liabilities, and $" +7674,$g'\circ S_X$ +7675,$g(x)$ +7676,$x < y$ +7677,$9+1=10+0$ +7678,$S_i(x)=\alpha_i(x)S(x)$ +7679,$g(s) / (1-g(s))$ +7680,$\mathsf{MONETARY}$ +7681,"$\displaystyle\int_0^{1-g(S(a))} \mathsf{E}[X_i\mid X=q(1-g^{-1}(1-p))]\,dp + a\beta_i(a)g(S(a))$" +7682,$q y) = 1 - (1 - \mathsf{Pr}r(X > y))^n$ +7694,$X_0$ +7695,$f(P)=\mathsf{E}[f(X)]$ +7696,$N = G$ +7697,${}^{[>169]}$ +7698,$\phi_m^o$ +7699,$\check g(1-t)^2-\check g((1-t)^2)= 1-2kt+k^2t^2 - (1-2kt+kt^2)= kt^2(k-1)>0$ +7700,$t 0$ +7708,$Z\in L^1$ +7709,$\int_{-\infty}^\infty$ +7710,"$1,579.14 \$" +7711,"$0, 1/p$" +7712,"$\mathcal R^c_{X,r}$" +7713,"$\displaystyle\int_0^{F(a)} \mathsf{E}[X_i\mid X=q(p)]\,dp + a\alpha_i(a)S(a)$" +7714,$R_i>C_i$ +7715,$R_0(t)\le P(0)$ +7716,$z_p^{(2)}=z_{p/2}$ +7717,$g^{-1}(s)$ +7718,$g(S(x))>S(x)$ +7719,$\alpha(1-\alpha)(1-s)^{\alpha-1}$ +7720,$\mathcal V(X)=\mathsf{E}[X]+c\mathsf{E}[X^2]$ +7721,$g(0+) \gt 0$ +7722,$-1.2403$ +7723,$\rho(X+Y) = \rho(\lambda(X/\lambda) + (1-\lambda)(Y/(1-\lambda))))$ +7724,$g(t)=0$ +7725,$\delta = (P-L) / (a-L)$ +7726,$\mathbb{E}[|X_i|]$ +7727,$SdX$ +7728,$A(X\wedge a)$ +7729,$R_2(t) \ge \mathsf{E}[X_2]$ +7730,$1-\beta_i(x)g(S(x))$ +7731,"$C/n=Bin(\theta,n)/n$" +7732,$Q(a)=\nu F(a)$ +7733,"$[0, 0.25]$" +7734,"$\mathcal Q =\{ \mathsf Q \mid \mathsf Q\ll \mathsf P,\ \alpha(\mathsf Q)=0 \}$" +7735,$f(y;\theta)=c(y)e^{y\theta-\kappa(\theta)}$ +7736,$\alpha^2 S$ +7737,$P=Pg^{ak}/g^{ak}$ +7738,$\mathsf{Pr}r(X=0)=p$ +7739,"$i=0,\dots,m$" +7740,$m(a)=S(a) + \delta F(a)$ +7741,$\partial Y/\partial X_i$ +7742,$\rho_g(X)=\mathsf{E}[X]$ +7743,$\hat \theta<\theta_r$ +7744,$\mathsf{E}_g[X_i(a)]$ +7745,$\mathsf{Pr}(\cdot\mid\mathcal{A})$ +7746,$\mathsf{Q}'=\mathsf Q\psi$ +7747,$R_0(t)\ge P(0)$ +7748,$0s$ +7770,$250$ +7771,$(a_2 - (X-a_1)^+)^+ \leftrightarrow$ +7772,$EL(a)$ +7773,$A \pm \epsilon$ +7774,$\mathsf{E}_Q[0]=0$ +7775,$(g-S)dX$ +7776,$p<0.01$ +7777,"$b\in (-1,1)$" +7778,"$F,f,P,M,Q$" +7779,$s\le s^*$ +7780,$X_{\mathsf j(a)+1}>a$ +7781,$1-X$ +7782,$\mathsf{j}(a)=4$ +7783,$\{X_t=x\}$ +7784,$s'$ +7785,$\mathsf{E}[Y^n]$ +7786,$h>0$ +7787,$\mathsf{Pr}(\bigcup_i B_i \mid\mathscr{G})_{\omega_0} = \sum\mathsf{Pr}(B_i\mid\mathscr{G})_{\omega_0}$ +7788,$v(A\cup B)\le v(A)+v(B)$ +7789,$\|\cdot \|_\rho=\rho(|\cdot |)$ +7790,$\displaystyle\int_0^\infty g(S(x))dx$ +7791,"$c = 0.5,1.0,\dots,2.5$" +7792,$\phi(s)=(1-p)^{-1}$ +7793,$a_{0}$ +7794,$\rho_m(Y)\le b_m$ +7795,$P_1(t) = \mathsf{E}[\tilde X_1 g'S_t]\le \sup_{\mathsf Q} \mathsf{E}_Q[\tilde X_1] = \rho(\tilde X_1)$ +7796,$Q_i$ +7797,$(Z)$ +7798,$c(x)=\rho(\sum_i x_iX_i)$ +7799,"$\mathsf{CP}(\lambda, B) \sim \mathrm{Poisson}(\lambda p)$" +7800,$I/a + U/P > 0$ +7801,$K_0(t)=\kappa(t)$ +7802,$p_s\mapsto \bar p_s$ +7803,$X=G\circ F(\bar x)$ +7804,$\rho\ge \mathsf{VaR}$ +7805,$\rho_{\mathsf{Dual}}$ +7806,$l\ge 0$ +7807,$\rho(\mathsf{E}[X_2\mid X_1])\le \rho(X_2)$ +7808,$\delta < 0$ +7809,"$79,807,699 | 11.6% | $" +7810,$\mathsf{E}[ X_i]= M^{-1} \sum_{j}X_{i}^{( j )}$ +7811,$P=L + d(a-L)$ +7812,$g'(s-)$ +7813,$\omega > 1/n$ +7814,$p=0.05$ +7815,$\bar M_i(a)>0$ +7816,"$(x)^- = \max(-x, 0)$" +7817,$s\beta(x) g(S(x))$ +7828,$\mu\to\infty$ +7829,$\mathsf{cv}(X)=\sqrt{e^{\sigma^2}-1}$ +7830,$g(s)=ks^\alpha(1-s)^\beta$ +7831,$u'''\ge 0$ +7832,$Q_k = \Delta X_k - P_k$ +7833,$r_h<0$ +7834,"$(1-w_{p_i,p_j})\mathsf{TVaR}_{p_i}(Y) + w_{p_i,p_j}\mathsf{TVaR}_{p_j}(Y)$" +7835,"$Y_i\sim N(\mu, \sigma^2 / w_i)$" +7836,$p = 1-1/200 = 0.995$ +7837,$\mathsf{E}[X]=c$ +7838,$\mathsf{Pr}r(\cdot\mid N)$ +7839,$P^T_S(\cdot\mid\cdot):\mathcal{A}\times M\to\mathbb{R}$ +7840,$A + \mathsf E[B] \rho(B - \mathsf E[B]) = A + \rho(B)$ +7841,$t_1^*-\epsilon$ +7842,$p\nu(p)$ +7843,"$12,500 at poiicy inception: \$" +7844,"$\max(x,0)$" +7845,"$1 billion in debt equity has met a similar response, Mr. Butler said. He cited the case of Chase Manhattan Bank. It was willing to put up $" +7846,$\mathsf{Var}(L(\nu))=\nu\mathsf{Var}(L(1))$ +7847,$\mathsf{VaR}_{0.95}(X)=3395$ +7848,"$\mathrm{Tw}_p(\mu, \sigma^2)$" +7849,$a_i=\mathsf{E}_\mathsf{Q}[X_i]$ +7850,$q=q_j$ +7851,$\sigma\sqrt{t}$ +7852,$\psi_i(a)=\mathsf{E}(X_i/Y \mid Y>a)$ +7853,$r_{O} = m/(1 - m).$ +7854,$Z=(1-p)^{-1}1_A$ +7855,$W=99$ +7856,$\mu = \delta_\alpha$ +7857,$a=100$ +7858,$f^*=(L^t)^+m$ +7859,$r/(1+r)$ +7860,"$\{x_1,...,x_n\mid X < \max(X)-\epsilon\}$" +7861,"$a_{X,r}(Y)$" +7862,$X(\omega)$ +7863,$1/(1+r) = 0.893$ +7864,"$^{\,9}$" +7865,$p'0$ +7896,$1-\tilde p=g(1-p)$ +7897,$dt^2$ +7898,$Z=z(X)$ +7899,$\rho(X+X_i)=\rho(X)+\rho(X_i)$ +7900,$X=98$ +7901,$X\Delta S$ +7902,$B\subset E$ +7903,$\mu_*(E_0)=0$ +7904,$n^2$ +7905,$\sigma/\mu$ +7906,$\alpha_i(a)S(a)=\mathsf{E}[(X_i/X)1_{X>a}]$ +7907,$\phi'(p)$ +7908,"$835,979,663 payable in installments over the life of the contracts. The minimum payments remaining under these contracts as of December 31, 1997 are $" +7909,$P_0(0)=P(0)$ +7910,$g(s)\ge s$ +7911,$r_0=0.01$ +7912,$F_0 = P_{act}-P = I-R$ +7913,${\mathsf{Q}}$ +7914,$\{X=a\}$ +7915,"$P(t)=F(a(t), t)$" +7916,${}^nS^{-1}(t) = \displaystyle\int_0^t {}^{n-1}S^{-1}(p)dp$ +7917,"$968,000 - (80%)(\$" +7918,$\mathsf{VaR}_p>2000$ +7919,$(\beta_i g(S))'(x)=-\mathsf{E}[X_i\mid X=x]g'(S(x))f(x)/x=-\kappa_i(x)g'(S(x))f(x) / x$ +7920,$x v^2st = \check g(s)\check g(t)$ +7927,$1_B$ +7928,$B_\cdot$ +7929,$0.8 \times 1.2 = 24/25$ +7930,$-g''(t) = w \delta_{\alpha_1}/\alpha_1 + (1-w) \delta_{\alpha_2}/\alpha_2$ +7931,$1-g_\tau(s)$ +7932,$\lambda\in\mathbb R^+$ +7933,"$h(x)=\sup_{s\in[0,1]} g(s)-sx$" +7934,$\rho(0)=\rho(0\cdot X)=0\rho(X)=0$ +7935,$iR$ +7936,$g(s) \ge 1$ +7937,$q(p^*)={port.q(pstar)}$ +7938,$\mathsf{E}_g(X\wedge a)$ +7939,$m_0$ +7940,$\mathsf{E}[X\_1]$ +7941,"$(1-S(x),x)=(p,q(p))$" +7942,$\mathsf{E}[XM]$ +7943,$\rho(X)=\log\mathsf{E}(\exp(\alpha X))/\alpha$ +7944,$a_{2}$ +7945,$e^{\mu+\sigma^2/2}\mathsf{Pr}hi(-d+\sigma))=\mathsf{E}[X1_{\{X>a\}}]$ +7946,$m^2+bm+c$ +7947,$s = s_l (1 - s) + s_u s$ +7948,$s_1$ +7949,$X > x$ +7950,$N=2^{256}=10^{77}$ +7951,$v\pi = \iota K$ +7952,$x_s$ +7953,$\mathsf{Pr}r(X > x)$ +7954,$a_i=a_{i+1}=\dots=a_{i+l}$ +7955,$T'(p)$ +7956,$R_1(t) \le \frac{1-t}{t}P(0) + P(1)$ +7957,$\rho(X) = \rho(\mathsf{E}[X | A]1_A + E[X | A^c] 1_{A^c})$ +7958,$s_2=1$ +7959,"$160,000 + \$" +7960,$\mathsf{E}[\cdot\mid Y]$ +7961,$\mu = \kappa'(\theta)$ +7962,$h(p)=1-g(1-p)=1-(1-p)^{1/3}$ +7963,$\mathsf{E}(W\mid X\ge 100) = 99/4=19.8$ +7964,"$\rho\in \mathcal R^b_{X,r}$" +7965,$\mathsf{E}[X_i]=100$ +7966,"$\Omega=\{x_1,\dots,x_n\}$" +7967,$\mathbf {\alpha_1S\Delta X}$ +7968,$\int_\Omega \zeta=1$ +7969,"$0,1$" +7970,$\dfrac{X_1}{a_1} \dfrac{ta_1+(1-t)x0}{tX_1+(1-t)x0}\ge 1$ +7971,$\theta_s<0.5$ +7972,$\mathscr{O}(\zeta)\subset\mathcal{A}$ +7973,$\mathsf{E}_{\mathsf{Q}}[X\wedge a]$ +7974,$P=\nu L +\delta a$ +7975,$u''' \ge 0$ +7976,"$(\nodespc/2, -\nodespc/2%)$" +7977,$v+d=1$ +7978,$\eta=0.49$ +7979,$X+Z$ +7980,$\mathsf{Pr}r(X=x_i)=\mathsf{Pr}r(X>x_{i-1})-\mathsf{Pr}r(X>x_i)=S(x_{i-1})-S(x_i)$ +7981,${}^{[>78]}$ +7982,$g(S(x))=\exp(-bH(x))$ +7983,$\mathsf{E}[X\mid \mathscr{G}](\omega)$ +7984,$0.9$ +7985,$\ell(I_k)$ +7986,$\rho_t(X) \le \mathsf E[\rho_{t+1}(X)\mid \mathcal F_t]$ +7987,$e^x=m$ +7988,$160.57 \$ +7989,$g(0)=0$ +7990,$\rho(X)\le-\rho(-X)\le 0$ +7991,$M^i = P^i-L^i$ +7992,"$3,000 \$" +7993,$g/(1-g)$ +7994,"$k+1,\dots,n$" +7995,$\sum_i \zeta_i^2 = 1$ +7996,$\alpha\to 0$ +7997,$\mathsf{E}[(X-\mu)^2]$ +7998,"$d=1,\dots,N$" +7999,$U_j$ +8000,$K = (B)^{a} = g^{ba}$ +8001,"$242,000 = \$" +8002,$Mg+F_2>Mg$ +8003,$X\wedge a=\sum X_i(a)$ +8004,$\phi(p)=g'(1-p)$ +8005,"$Cov(X,Y)\le\sqrt{V(X)V(Y)}$" +8006,$\bar P(a)=\rho_g(L_0^a(X))$ +8007,$\mathcal{N}_X(X_i(a))$ +8008,$\dfrac{\partial l}{\partial \mu}=\dfrac{y-\mu}{V(\mu)}$ +8009,"$\delta_{p_1},\delta_{p_2}$" +8010,"$\mathsf{E}(\min(X,a))=\mathsf{E}(X\wedge a)$" +8011,$E[s|W=t]$ +8012,"$\rho^E(t_1,t_2)(X)$" +8013,$S\mid \mathcal{B}$ +8014,$\bar F'(x) = F(x)$ +8015,"$1.4 billion in excess of the CEA' s claims paying ability, which may not exceed an amount equal to the $" +8016,$F^{(-2)}(p)$ +8017,$y_{i}$ +8018,$Y_t$ +8019,$X_2=c$ +8020,$n\mathsf{Pr}r(Y > y_c)$ +8021,$\rho_1(X\wedge a)<\rho_2(X\wedge a)$ +8022,"${}^{[<331,10]}$" +8023,$A+B$ +8024,$a_i = \mathsf{VaR}_p(X) - \mathsf{VaR}_p(\sum_{j\not=i} X_j))$ +8025,"$1,000,000,000) plus costs of issuance and sale of those revenue bonds or other debt and amounts paid or payable to bond issuers and providers of credit support and letters of credit for, and interest on, those revenue bonds or other debt, regardless of the frequency or severity of earthquake losses at any and all times subsequent to the creation of the authority. Once the authority has levied policy surcharges in an amount equal to the sum calculated pursuant to paragraph (3) of subdivision (a) of Section 10089.23, and in no event more than one billion dollars ($" +8026,$\mathscr F_1$ +8027,$t_i$ +8028,"$245,099.97 = \$" +8029,$q_2$ +8030,$z(X)$ +8031,$\nu_p=(1+\rho_p)^{-1}$ +8032,$\mathsf{CTE}_p$ +8033,$M = \beta g(S)-\alpha S$ +8034,"$c(y)=\binom(n, k)\delta_k$" +8035,$\tilde\Theta$ +8036,$g = s^{0.4}$ +8037,$\nabla a = \nabla P + \nabla Q$ +8038,$\forall t\in M$ +8039,$\emptyset = \forall x(x!=x)$ +8040,$k\cdot R\cdot O\cdot L$ +8041,"$X(x,-x)\equiv 0$" +8042,$x_j < \mathsf{TVaR}_{p_j}$ +8043,$\Delta G = \Delta H - T\Delta S$ +8044,$a/x$ +8045,$M^1$ +8046,"$a \in_{R} \{2,\dots,p-2\}$" +8047,$g(S(x)) > S(x)$ +8048,$T(x)=\{2x\}$ +8049,$\mathsf E[X\mid I](\omega)=\mathsf E[X\mid I=I(\omega)]$ +8050,$X^y:=0$ +8051,$\mathsf E[X\mid I=\iota_i]$ +8052,$\mathit{ROL} = \exp( \alpha_{0} + \alpha_{1} \log(\mathit{AEL}) )$ +8053,$T_n = \sum_{i=1}^{n} T(X_i)$ +8054,"$ Finally, since we are interested in the left-shift $" +8055,$\mathsf{TVaR}_p(X)= \sum_i X_iZ_i / 10$ +8056,"$\mathsf P(\{ \omega\mid X(\omega)=X(\omega_0), \omega \le \omega_0 \})$" +8057,$\mathsf{Pr}r(X < x)\le p\le\mathsf{Pr}r(X\le x)=F(x)$ +8058,$\sqrt{x_2/\lambda}$ +8059,$p-p\nu_p = p\delta_p$ +8060,$t x \right]$ +8074,$Z_1=q_Z(F_X(X))$ +8075,$s(X)=s(X;\mu)$ +8076,$p^-$ +8077,$\rho(X_i\wedge a_i)$ +8078,$\lim_{t\to 0}a(X_1; X+tX_1)=a(X_1;X)$ +8079,"$(x,y)\mapsto (x,y)$" +8080,$P+Q-L=L+M+Q-L=M+Q$ +8081,$n = 1$ +8082,$ROL = a + b\ \mathit{EL} + c \ C(t)$ +8083,"$p_i \in [0,1]$" +8084,$j \ge 0.99M$ +8085,$\mathbb{Q}=\pi_Y(\R)$ +8086,"$\exists\, s_0\in(0,1):\ g(s)=1 \ \forall s\ge s_0$" +8087,$<\alpha$ +8088,$\beta_i(t)g(S(t)) = \mathsf{E}[(X_i / X) g'(S(X)) 1_{X > t}] = \mathsf{E}_g[(X_i / X) 1_{X > t}]$ +8089,$\lambda(p=1)=0$ +8090,$\tilde \rho(X)=\inf\{ \alpha \mid X+\alpha \in\mathcal{A} \}$ +8091,$X_k=X_0+k$ +8092,$X_t=X-t\bar X \le X$ +8093,$1<\alpha<2$ +8094,$\mathsf{TVaR}_{0.9}$ +8095,$Y=Y(\mathbf{X})$ +8096,$X= \sum_iX_i$ +8097,$\rho(\cdot\mid \mathscr F_1)$ +8098,$a=b^c$ +8099,$b \urcorner$ +8100,$T=t$ +8101,$\omega\mapsto \psi=F(X(\omega))$ +8102,$\mu + h\sigma$ +8103,$b + v_i / r$ +8104,$\xi(\omega)=\xi(X \mid \mathscr{G})(\omega)$ +8105,$X_n=1/n$ +8106,$\nu(0.5)=1/(1+\iota^*)$ +8107,$\Delta_{2}$ +8108,$a(X)=\mu+4\sigma$ +8109,$L_0^a(X)=X\wedge a$ +8110,"$P:\mathscr{F}\times\Omega\to[0,1]$" +8111,"$25,000) for individual condominium units of one hundred thirty-five thousand dollars ($" +8112,$\kappa_1(X)$ +8113,$f=(1-p)^{-1}1_A$ +8114,$p=0.98$ +8115,$\alpha \ge s_0 g'(s_0)/g(s_0)$ +8116,$g:\Omega\times\Omega\to\mathbb{R}^2$ +8117,"$\{f' \in L_q \mid f'=1+f-\mathsf{E} f,\ \|f\|_q\le c \}$" +8118,$q = 1-p$ +8119,$c=\mathsf{Var}(G)=\nu^2$ +8120,$q_X(U)$ +8121,$\mathsf{E}[X_i \mid X=q(p)]$ +8122,$t_1 > t_2$ +8123,$\mathsf{E}[X_m\mid X_{m+n}=x]=mt/(m+n)$ +8124,"$\mathsf{cov}(N, Z_0)<0$" +8125,$\hat y_i$ +8126,$N=20$ +8127,$R_1(1)=P_1(1)$ +8128,"$\rho(X) = \max\,\{ \mathsf{E}[\zeta X] \mid \zeta\in\mathcal{M} \} = \int q_\zeta(s)q_X(s)ds$" +8129,$\mathsf P_I^g$ +8130,$s<0.5$ +8131,$\nu(p)-l(p)= \nu^*\sqrt{(1-p)/p}$ +8132,"$m=m_{p_i,p_j}$" +8133,$1=\delta+\nu$ +8134,$X_j=x$ +8135,$g'\left (S_{X\wedge a}(X\wedge a)\right )$ +8136,$p:=1-s$ +8137,$V(\phi)$ +8138,$\mathsf{E}[1/X_i \mid X>a]$ +8139,$n/(n-1)=1/p$ +8140,"$2,000,000,000), regardless of the frequency or severity of earthquake losses at any and all times subsequent to the creation of the authority. Once a participating insurer has paid amounts equal to its residential earthquake insurance market share percentage multiplied by two billion dollars ($" +8141,$a=\mathsf{E}_\mathsf{Q}[X]$ +8142,$a(f + (1-f)/q)$ +8143,$P(A\cup B)\le P(A)+P(B)$ +8144,$P(\{\omega_1\})=1/3$ +8145,"$(s_{R+1},g_o)$" +8146,$1_\omega(\omega')=1$ +8147,$-U$ +8148,$E_0+a_2$ +8149,$dx\to 0$ +8150,$g((1-t)^2)=(1-k)+k(1-2t+t^2)=1-2kt+kt^2$ +8151,$>P(1)$ +8152,$\mathit{LGD}$ +8153,$\rho'(x)=U'(-x)$ +8154,$P=\mathsf E_q[.]$ +8155,"$\omega\in[0,1]$" +8156,$\delta\downarrow 0$ +8157,$n^y=0$ +8158,$L_a^\infty$ +8159,$n=N-1$ +8160,"$\mathsf{Tw}^*(\lambda, \alpha, \beta)$" +8161,$X_2=2$ +8162,$\times T^4 \propto 4\pi R^2T^4$ +8163,$\mu\{ l<1\}=0$ +8164,$k(t):=\log(M(t))$ +8165,$\mathsf{E}[X\mid X>2000]-2000=\mathsf{TVaR}_{F(2000)}(X)-2000=624$ +8166,$\kappa_i(x)\approx x -\sum_{j\not=i} \mathsf{E}[X_j]$ +8167,$h$ +8168,"$f(\cdot, t)$" +8169,"$c(1,2,3)-c(2,3)$" +8170,"$425,642,634 | 10.7% | $" +8171,$t-dt$ +8172,$0\le X_n\le 1$ +8173,$-1<-\alpha<0$ +8174,"$\rho(X) = \int (vS(x) +d)\, dx = v\mathsf{E}[X] + da$" +8175,$D/L=\mathsf{E}[A\wedge L]/\mathsf{E}[L]$ +8176,$1-v^{n-2}$ +8177,$\mathsf{E}|X|<\infty$ +8178,$l(p) = v(1-\sqrt{(1-p)/p})$ +8179,$A(c)=c$ +8180,$\kappa'(\theta)=\left(\dfrac{\theta}{\alpha-1}\right)^{\alpha-1}$ +8181,$. By comparison $ +8182,${}^{[>2]}$ +8183,$P_k = g(S_k)\Delta X_k$ +8184,$\omega_i$ +8185,$\lim_{\mu\to 0} V'(\mu)=\delta$ +8186,"$t\in[t_1^*,t_1^{**}]$" +8187,"$j=1,\dots, d$" +8188,"$\int g(S(x))\,dx$" +8189,$\mathsf{Pr}r(X>a)=S(a)$ +8190,$\rho(X) = n\rho(X/n)$ +8191,$\rho(X) \ge \rho(Y)$ +8192,$\mathsf{Pr}r(S_T > a)=\mathsf{Pr}r(X_T > a/S_0)=1-\mathsf{Pr}hi\left([\log(a/S_0)-(r-\sigma^2/2)T]/\sigma\sqrt{T} \right)=\mathsf{Pr}hi(d^*-\sigma\sqrt{T})$ +8193,$x^{\alpha}e^{\theta x}/x$ +8194,$\rho:\mathcal{S}\to \mathbb{R}$ +8195,$X a]$ +8222,$\sigma^2 t$ +8223,$\mathsf P_I$ +8224,$a\ge \sup(X)$ +8225,$\tilde U$ +8226,$A_1=X$ +8227,$u_j$ +8228,$X^2$ +8229,$V(a) = 1_{X > a}$ +8230,$\rho(X+Y)\ge\rho(X)+l(Y)$ +8231,$ rather than 1. Since $ +8232,$1 in a layer with loss probability $ +8233,"$(s,1)$" +8234,$L_d^{d+l}$ +8235,$\nu^{-1}\mathsf{E}[\nu(X)]$ +8236,$\mathsf{Pr}r(|X_n(\omega)-X(\omega)|>\epsilon)\to 0$ +8237,$\mathscr{P}=\{\mathsf{Q} \mid d\mathsf{Q}/d\mathsf{P} \le 1/(1-p) \}$ +8238,$0\mathsf{E} Z)$ +8317,$\theta = \displaystyle\int\dfrac{dm}{V(m)}$ +8318,$s=q(1-g^{-1}(1-\tilde p))$ +8319,$+1$ +8320,$0.4\times 0.6 \times 0.3 \times 0.48 = 0.03456$ +8321,$1/(24\times 60\times 60) = 0.000011574074074\dots$ +8322,$\le 1/(1-p)$ +8323,$\omega\ge 0.4$ +8324,$U=X\wedge A$ +8325,"$\mathsf{E}[\mathsf{CP}(\lambda, X)]=\lambda\mathsf{E}[X]$" +8326,$\rho(X)=\mathsf{E}_{\mathsf{Q}}[X\wedge a] + \mathsf{E}_{\mathsf{Q}}[(X-a)^+] \le \rho(X\wedge a) + \rho((X-a)^+=\rho(X)$ +8327,$\alpha=0.5$ +8328,$t_1=0.544$ +8329,$(1-s)\phi'(s)$ +8330,$\mathsf{P}(X=X_j)=S_{j-1}-S_j$ +8331,$k(h):=\log\mathsf{E}[e^{hX}]$ +8332,"$\int_B \mathsf{Pr}(\bigcup_i A_i\mid\mathscr{G})\,d\mathsf{Pr}$" +8333,$f_n$ +8334,$E[X\wedge a]$ +8335,$P_i(x)=\beta_i(x)g(S(x))$ +8336,$\bar x + t\bar h$ +8337,$\rho(A_0) + k \mathsf E[N]$ +8338,$1+r_f$ +8339,$\mathsf{P}(\{\omega\})$ +8340,"$457,989,704 | | $" +8341,"$\dfrac{\partial\rho}{\partial P} = \dfrac{0.4^2 P}{\rho(P,R,a)}$" +8342,$\theta = -1/\mu$ +8343,$X(\omega)=\omega$ +8344,"$(E,\mathsf{E}E)$" +8345,$g(s)=s(1-s)$ +8346,"$A,B,C,D$" +8347,$1-r_0$ +8348,$0\le k < 2^m$ +8349,$P=g(s)$ +8350,$p>p^*$ +8351,"$406,387,744 | | $" +8352,$d(a-L)$ +8353,$s_{R+1} < s^{\star}$ +8354,$t=w$ +8355,"$0,1,2,\ldots$" +8356,"$\mathcal E_{X,r}\subset \mathcal R_{X,r}$" +8357,$Mg$ +8358,$G=g$ +8359,$f\le 0$ +8360,$\mathcal F'$ +8361,$\Delta(v)$ +8362,"$\mathcal R^x_{X, r}$" +8363,$X=NF(\bar x)$ +8364,$|X|=X_++X_-$ +8365,$O(dt^2)$ +8366,$d+vs<1$ +8367,"$s_0, s_1, s_2$" +8368,$k_1 > k_0$ +8369,"$\mathsf{Pr},\mu,P^T_S(\cdot\mid\cdot)$" +8370,$p=1-s$ +8371,$C > 0$ +8372,$\mathsf P_t$ +8373,"$\mathcal{S}, \succeq$" +8374,"${}^{[<42,81]}$" +8375,$X\wedge a(X)$ +8376,$\rho(0) = \rho(0+0)\le \rho(0)+\rho(0)$ +8377,$Z=\sigma(U)$ +8378,$\log(\hat g(s)) - \log(g(s))$ +8379,$2 billion at $ +8380,$p_i=i/(N+1)$ +8381,$NPV = 0$ +8382,$Z(a')=g(S_X(a))/S_X(a))$ +8383,$\mathsf E[Y'\mid X']=X'$ +8384,$\alpha(t) = t$ +8385,$\mathsf{E}[r(X)]=\mu$ +8386,$\rho(X \wedge a)$ +8387,$E=\tau=0$ +8388,$\mathsf{Pr}r(X\ge x)\ge 1-p\ge \mathsf{Pr}r(X> x)$ +8389,$ and becomes more concentrated in the interior of $ +8390,"$[t_2,1]$" +8391,$T=y$ +8392,$0.5 < t_0 < t_1$ +8393,$v(AB) + v(ABCD) = 3/2 > v(ABC) + v(BCD) = 4/3$ +8394,"$p=0.01,0.004$" +8395,$M^1=\beta^1 g(S)-\alpha^1 S$ +8396,$z(p^+-p^-) + (1-p^+) / (1-p)=1$ +8397,$p_\beta$ +8398,$\mathsf{TVaR}_p(X) := (1-p)^{-1}(T_1+T_2)/N$ +8399,$X=8$ +8400,$\Xi( s ) = p - \log(s)$ +8401,$\mathsf{E}[X_i \mid X = x]$ +8402,$IJW=E_g[X_a(a)]$ +8403,$\mathsf E[X\mid \mathscr F]$ +8404,${\mathcal{X}}$ +8405,$V(m)=\sum_{n\ge 1} a_nm^n$ +8406,$\rho_{CCoC}$ +8407,$r_c\le r_i$ +8408,$\hat\rho(A_0)\ge \rho(A_0)$ +8409,"$e(fT, y) = f(y)$" +8410,$M_F^+$ +8411,$x_2(S(x_1)-S(x_2))=x_2\mathsf{P}(X=x_2)$ +8412,$y_5$ +8413,$\rho_g(X)>\rho_g(Y)$ +8414,"$(X-a)^+=\max(X-a, 0)$" +8415,$1=P(x) + Q(x)$ +8416,$cr=l(1-e) + e$ +8417,$Q^2$ +8418,$i\not=j$ +8419,$\mathcal Q=\{\mathsf Q\mid \alpha(\mathsf Q)=0 \}$ +8420,$200K/\$ +8421,$\beta_i(t)/\alpha_i(t)$ +8422,$n+1=N$ +8423,"$\forall \omega\in\Omega,\ A\to P(A, \omega)$" +8424,$r = (g(s)-s)/(1-g(s))$ +8425,$g=2\nu^4/(1-f)+3c+1$ +8426,$d(y;\mu)=(y-\mu)^2$ +8427,$\mathsf{Pr}r(X_i>\bar q(s))=s$ +8428,$s<1$ +8429,$\mathsf{E}(R(\Theta))$ +8430,$o(dt)$ +8431,$P=\alpha +F_1 /a = \alpha + (N'/w - \alpha a)/ a = N'/aw$ +8432,"$D=\{ (x,y) \mid x=y \}$" +8433,"$g(\mathsf{Pr}(H,T))=1$" +8434,"$S(x)=d/dx(\mathsf{E}[1_{(0, x]}(X)])$" +8435,$\mu=\lambda\alpha/\beta$ +8436,$U \Delta p$ +8450,"$(0,1)$" +8451,$\{ r_i \}$ +8452,$q^-(s)=\mathsf{VaR}_s(X)$ +8453,$\rho(X)=\int_0^1 q(p) \phi(p) dp$ +8454,"$f_{X_t},f_{T_x}$" +8455,$\DeltaP$ +8456,$X_N$ +8457,"$2,000,000,000), regardless of the frequency or severity of earthquake losses at any and all times subsequent to the creation of the authority. Once a participating insurer has paid, pursuant to this section, amounts equal to its percentage share of the authoritys total gross written premium, multiplied by two billion dollars ($" +8458,$M_G(\zeta):=\mathsf{E}(e^{\zeta G})$ +8459,"$(\Omega, \mathcal{F}, P)$" +8460,$\sum e_i^2 / (n-2) = \sum e_i^2 / df$ +8461,$P=S+M$ +8462,$p_1>0$ +8463,$0.725$ +8464,$2\times 10^{20}$ +8465,$f_Y(y)=\nu f_Z(\nu y)$ +8466,$\mu=-\alpha/h$ +8467,"$(S,\S)$" +8468,$L=\mathsf E_p$ +8469,$\rho(X)\le \rho(\lambda X)/\lambda$ +8470,$-\alpha < -2$ +8471,"$\displaystyle\int HX \,d\mathsf{Pr}$" +8472,$L(\nu)=\sum_{1\le i\le \nu} X_i$ +8473,$\rho_{t+1}(X)$ +8474,$p_0 =e^{-\lambda}$ +8475,$x$ +8476,$\mathsf{E}[X_1g'(S(X))]$ +8477,$\mathsf{j}(a)=6$ +8478,$0\le s\le \epsilon$ +8479,$\rho(W_1\wedge a_0)$ +8480,$s=t=s^\star$ +8481,$g(s)=100s \wedge 1$ +8482,$s' > 8.75$ +8483,$(1-p)(1-\alpha)=(p-1)(\alpha-1)= -1$ +8484,"$i=R,\dots,m$" +8485,$\hat\rho(X)=\rho(X)$ +8486,$A=P+Q$ +8487,$-1.466$ +8488,$\mathbf {\mathsf{P}(X_1)}$ +8489,"$X:(\Omega,\mathscr{F})\to(S,\S)$" +8490,"$\mu^t P^x_t f(x,t)$" +8491,$x_5$ +8492,$\le\rho(tX_1 \dfrac{X_t\wedge a_t}{X_t})$ +8493,$\nu(M\setminus f(L))=0$ +8494,$\rho_{Wang}$ +8495,$\mathrm{PQ}$ +8496,$j(x)=\alpha x^{-\alpha-1}$ +8497,$\rho(A_k)\le\hat\rho(A_0) + k\rho(N)=\hat\rho(A_k)$ +8498,$Z=\mathsf{E} Z$ +8499,$\rho_{\mathsf{CCoC}}$ +8500,"$p_1,p_2,p_3>0$" +8501,$s/(1-s)$ +8502,$\tilde\rho_T=\rho_T$ +8503,$S_{k-1} - S_k = p_k$ +8504,"$1-e^{-\lambda S(\mathsf{PML}_{n, \lambda})}=1/n$" +8505,$>$ +8506,$c_h>c=\mathsf{VaR}$ +8507,$\{ X>a \}$ +8508,$y\theta-\kappa(\theta)$ +8509,"$\{q_k:k=0,\dots,n\}$" +8510,$F(x)=\mathsf{Pr}(X\le x)$ +8511,$Q^1$ +8512,"$(s_L,h_L)$" +8513,$E[s|t]$ +8514,"$\{0,\dots,m\}$" +8515,$\tau(\theta)=\mu$ +8516,"$100,000 | \$" +8517,$r=\rho_m(X)$ +8518,$X(\omega) / P$ +8519,$\bar P^a(\mathbf{v})$ +8520,$\rho(X)=\max_{\mathsf Z\in\mathscr{Z}} \mathsf{E}[XZ]$ +8521,$\{X=q_X(p)\}$ +8522,${p^*}$ +8523,$W_{t}$ +8524,$\mu=19.005$ +8525,$750k xs \$ +8526,$\le 89$ +8527,"$3,279,059, implying a risk load of \$" +8528,"$1,513.59 \$" +8529,$y-m = m(x-\log(m))$ +8530,$t_1=0$ +8531,$X=5)$ +8532,$\mathbf {f}$ +8533,$P_1(t)/t$ +8534,$\mathsf{E}[X]=\int xdF(x)$ +8535,$\theta=\log(\mu/(n-\mu))$ +8536,$X_2=0.3 + 0.7X'_2$ +8537,$L_X(v)=l(v)$ +8538,$\mathsf{E}(L) = q(p)$ +8539,$p\delta -q\nu=p-\nu$ +8540,"$\rho \in\mathcal R_{X,r}$" +8541,$\text{ш}$ +8542,$q^-(U(\omega))$ +8543,$X_4^i$ +8544,$x_{(j)}-x_{(j-1)}$ +8545,"$[0,a)$" +8546,$m=\lfloor n/2\rfloor$ +8547,$\Delta\mu$ +8548,$u=v=s_0$ +8549,$K_\theta(t)= -\log\left(1 + \dfrac{t}{\theta} \right)$ +8550,$100 - 10^5$ +8551,$\phi_Q=1-\phi_W$ +8552,$dy/y = kdx / kx=dx/x$ +8553,$(X+Y-x-y)_+\le (X-x)_+ (Y-y)_+$ +8554,$p^\ast = 0.48732$ +8555,$s_L=0$ +8556,"$\displaystyle\int_{T^{-1}(B)} \mathsf{E}[X\mid T]\,d\mathsf{Pr}$" +8557,"$\mu=0,\sigma=1.75$" +8558,$M = M^+ - M^-\ge 0$ +8559,$\alpha(X_u) = \mathsf{E}[X\mid X > F_u^{-1}(p)]$ +8560,$\mathsf{Var}(\sum C_i)=\sum (m_i v_i)^2 = n(mv)^2$ +8561,$a-L$ +8562,$\mu+\sigma^2/2$ +8563,$Q_1\Delta X$ +8564,"$\mathcal R_{X,r}$" +8565,"$[0,0] \succeq [\epsilon, \epsilon]$" +8566,$\mathbf {s}$ +8567,$\mathscr{P}=\{ dQ/dP\le 1/\alpha\}$ +8568,$\kappa(\theta)=\log\int e^{\theta y} c(y)dy$ +8569,$F=(X-a)^+$ +8570,$g(0+):=\lim_{s\downarrow 0}g(s) >0$ +8571,$x=e^{\mu + y\sigma}$ +8572,$mX$ +8573,$\Delta_L$ +8574,"$\alpha>1,0\le\beta\le 1$" +8575,$f(s)=\alpha(1-\alpha)(1-s)^{\alpha-1}$ +8576,$\pi(p)=p$ +8577,$P(X_{-1}(a_{gc}))=9094.25$ +8578,$p=1-g^{-1}(1-\tilde p)$ +8579,$\bar x\mapsto \sum_i F_i(\bar x)$ +8580,$S\approx \mathsf{E}[X]$ +8581,$\theta>0.5$ +8582,"$P(A, ω) = 1_A(ω)$" +8583,"$(0,t_1]$" +8584,$y_1$ +8585,"$x~\text{Unif}[0,1]$" +8586,$X\le 0\implies\rho(X)\le 0$ +8587,$\rho(Y)=\rho_{m_Y}(Y)$ +8588,$\mathsf{Pr}r(Y>y)$ +8589,$s<1-p$ +8590,$X\_1$ +8591,$t_2-\epsilon/2$ +8592,$X_j$ +8593,$\lambda=\sigma^2$ +8594,"$(x_A,g(S(x_A)))$" +8595,"$P = E[Xg'S(X)] = \int gS(x)\,dx = \int xg'S(x)f(x)\,dx$" +8596,$-\log(0.3)=1.204$ +8597,$P=80$ +8598,$\iff$ +8599,$V(\mu)=\mu^p$ +8600,"$i=1,2,3$" +8601,"$t\in[t_*, 1]$" +8602,$xE_g[X_i/X | X > x]g(S(x))$ +8603,$p=1-s_j$ +8604,$P_1(t) = E[tX_1 \dfrac{X_t\wedge a_t}{X_t} g'S_t(X_t)]$ +8605,$\omega\in \Omega$ +8606,$\sum c_i^2$ +8607,"$742,716,047 | 19.1% | $" +8608,$\rho(X)>\mathsf{E}[X]$ +8609,"$ if one or other of the lines is thicker tailed. This is confirmed experimetentally. Looking at the plot, this function will increase and then decrease with $" +8610,$\beta>1$ +8611,"$\mathsf P(X(u_1, u_2, U_3) > x) = \mathsf P(X > x\mid\mathscr F_2)(u_1, u_2)$" +8612,$fC + g$ +8613,"$\Omega=[0,1]$" +8614,$S^i$ +8615,$\alpha = c+1$ +8616,$\mathsf{E}[X]=27.5$ +8617,$_{ro}$ +8618,$ABC$ +8619,$\rho E/(1-\tau) - rA$ +8620,"$0, 8, 10$" +8621,$F(q(p))=p$ +8622,$=E(X_i / X)$ +8623,"$\mathbf{x}=(x_1,\dots,x_n)$" +8624,$t_10.5$ +8658,$(1-t)\mathsf{E}[X_0]$ +8659,$\rho=0.6$ +8660,$P = \mathsf{E}^Q[L] + \delta A$ +8661,$\mathcal{R}=\mathcal{R}^h$ +8662,$v_3$ +8663,$\rho(L) = q(p)>q(p)$ +8664,$\mathcal{C} = \{ A \in \Sigma : \mu(A) = \nu(A) \}$ +8665,${}^{[<47]}$ +8666,$\lambda dt\to 0$ +8667,$\mathsf{TVaR}_{p^\star}(X)=P$ +8668,$\{\tau_n\}$ +8669,"$8,000 at policy inception and the inflow of \$" +8670,$\lim_{\epsilon \downarrow 0} (f(x+\epsilon)-f(x))/\epsilon$ +8671,$\mu={mu:.5f}$ +8672,$\C$ +8673,$\mathsf{CP}_1$ +8674,$(1-{}_b\bar V)$ +8675,"$s\in (0,1)$" +8676,$32$ +8677,"$1,600 in the first year, an extra cash inflow of \$" +8678,"$\forall A\forall p[\forall x\in A\exists !y\phi(x, y, p)\rightarrow\exists Y\forall x\in A\exists y\in Y\phi(x, y,p)]$" +8679,$Q=\sum_k Q_k > \sum_k \Delta X_k - P_k = a - P = Q$ +8680,$\mathscr{F}F$ +8681,$p=23$ +8682,$X^i(a)$ +8683,$\int_{a_0}^{a_1}$ +8684,"$\Omega=\{ \omega_H, \omega_C \}$" +8685,$a=1/c$ +8686,$K = \mathsf{E}[\exp (\lambda x)]^{-1}$ +8687,$\rho_g(X)=51.156$ +8688,$\rho(-1_{A^c})=0$ +8689,$l_2$ +8690,"$\inf\,\Omega=0$" +8691,$\frac{1}{2}$ +8692,$E[X_2 | X]$ +8693,"$(Alice)+(0,-3)$" +8694,$\kappa_i(x) = \mathsf{E}[X_i \mid X=x]=\mathsf{E}_{\mathsf Q}[X_i \mid X=x]$ +8695,$g\ge 0$ +8696,$X(u)$ +8697,$y_0$ +8698,$\displaystyle\int_0^1\phi(s)ds=\displaystyle\int_0^1\displaystyle\int_{1-s}^1\dfrac{\mu(dt)}{t}ds = \displaystyle\int_0^1\displaystyle\int_{1-t}^1ds\dfrac{\mu(dt)}{t}=\displaystyle\int_0^1\mu(dt)=1$ +8699,$U:=X\wedge A\le X$ +8700,$r-1$ +8701,$= \rho(B(s_l)) (1 - s) + \rho(B(s_u)) s$ +8702,$\mathit{NPV}_1 = \bar Q - \bar Q = 0$ +8703,$|x_i-x_j|<1/k$ +8704,$C_i=c_i$ +8705,$\mathsf{Pr}r(L=l)$ +8706,$t=m/n$ +8707,"$\omega_1,\omega_2\in\Omega$" +8708,$X\wedge a$ +8709,$\bar P(a)= (1-e^{-a\alpha\beta})/(\alpha\beta)$ +8710,"$132,500 | 0 | 0 | \$" +8711,$f(y;\theta)=c(y) e^{\theta y - \theta^2/2}$ +8712,$C_2(0) = \mathsf{E}[X_2]$ +8713,$X(\mathbf{x})=\sum_i x_i X_i$ +8714,$T_X\in\mathscr{S}(X)$ +8715,$\rho_{\theta^\star}(X)=P$ +8716,$\rho(X) \ge \mathsf{E}[X]$ +8717,$t_1^*$ +8718,$5000. So the expected loss for the policy is \$ +8719,$X_t = X_0 e^{(\mu - \frac{1}{2} \sigma^2)t + \sigma W_t}$ +8720,"$(\omega',\omega'')$" +8721,$-\rho$ +8722,$\mathsf{Pr}(B\mid\mathscr{G})$ +8723,$\mathsf{E}[T]$ +8724,$E_2$ +8725,$\mathcal R$ +8726,$s>0.15$ +8727,$p\ge 0$ +8728,$\int_0^x (x-y)^{n-1}dG(y)$ +8729,$X_{-3}$ +8730,$\mathsf{E}_g[X\wedge a]$ +8731,$\mathsf{I}$ +8732,$A_{x+b}$ +8733,$9+1=10+0=10$ +8734,$1 <\alpha < 2$ +8735,$\mathsf E[W_2]=\mathsf E[e^{\bar s \tau}]$ +8736,$g'(t)=1-r_0$ +8737,$P(x)=g(S(x))=g(s)$ +8738,$m_3=0$ +8739,$X'=0$ +8740,$g(S(x))=1$ +8741,"$(\nodespc/2, -\nodespc/2)$" +8742,$K=\{1\}$ +8743,"$t=1,2,...,\tau$" +8744,$\lim_{n \to \infty} E[|X_n - X_\infty|] = 0$ +8745,$\lambda e^{-\lambda}$ +8746,$K_\theta(t)= \sqrt{-2\theta}\left(1-\sqrt{1+\dfrac{t}{\theta}} \right)$ +8747,$\mathit{NPV}_{\infty} = a_xF_0$ +8748,$S_t$ +8749,$\mathbf{X}\times\mathbb{R}$ +8750,$X_i\ge 0$ +8751,$P=\mathit{EL} + r(a - P)$ +8752,$N_t - A_t$ +8753,$\mathsf x\mathsf{VaR}$ +8754,$x\mapsto g(s)+g'(s)(x-s)$ +8755,$h > 0$ +8756,$\sigma=0.125$ +8757,$\mathsf{E}(X) = \displaystyle\int_0^\infty xf(x)dx = \displaystyle\int_0^1 q(p)dp$ +8758,$\mathsf{E}[X\mid\theta]$ +8759,$\pi_\sigma(L)$ +8760,$8.75=10.5 / 1.2$ +8761,$\mathsf P(G)$ +8762,$P+Q-S=S+M+Q-S=M+Q$ +8763,$X_p=^d Y_p$ +8764,$\mathsf Q(A)=0$ +8765,$\Delta_j$ +8766,$\alpha_{1} \not= 0$ +8767,$U_i = X_i$ +8768,$l(y;\mu)=y\log(\mu) + (1-y)\log(1-\mu)$ +8769,$\sum (y_i-\hat y)^2$ +8770,$\mathsf{VaR}(X+Y)$ +8771,$0 < a < b < 1$ +8772,$S(x)=\mathsf{Pr}r(X > x)$ +8773,$1/2$ +8774,$\mathsf{E}[X \mid X > q(p)]$ +8775,$\mathbb R$ +8776,$P_1(t)$ +8777,$(F((k-1/2)b)-F((k+1/2)b)) / b$ +8778,$\phi(s)=\mathsf{E}[e^{isY}]$ +8779,"$\beta(X,M)=\mathsf{cov}(X,M)\sigma_M^2$" +8780,$\mathsf{Pr}(\omega)$ +8781,$ds(t)/dt$ +8782,$f(x) \ge f(x_0) + f'(x_0)(x-x_0)$ +8783,$ which is an extreme stable with Lévy distribution $ +8784,$p={p:.3f}$ +8785,"$^{\,7}$" +8786,"$T:(\Omega, \mathscr{F})\to(E,\mathsf{E}E)$" +8787,$\mathsf{E}[Y\cdot Z_1]=\mathsf{E}[g\circ X\circ T_X^{-1}\cdot Z]=\mathsf{E}[Y\cdot Z]=\mathsf{E}[Y\cdot \tilde Z]$ +8788,$P=L/(1+R_L)$ +8789,$se(pred)$ +8790,$v=1-d=1/(1+r)$ +8791,$F(a)$ +8792,$Pa=kg /ms^2=N/m^2$ +8793,$5.14\times 10^{19}$ +8794,$C=\mathrm{conv}S$ +8795,$\nu=0.870$ +8796,"$d, r>0$" +8797,$\mathsf{Pr}(\cdot\mid \mathscr{G})(ω)$ +8798,$= 10^{1+6+12}=10^{19}$ +8799,$g^k$ +8800,$t = g(s)$ +8801,"$(m,\sigma^2)$" +8802,"$^{\,6}$" +8803,$X_n(\omega)=0$ +8804,$\log(2)$ +8805,"$1,779.21 \$" +8806,"$(0,1]$" +8807,$\sum x_iX_i$ +8808,"$\mathbf{x}=(x_1,\ldots,x_n)$" +8809,"$\rho(\mathsf E[X\mid I], \mu)$" +8810,$L_d^{d+l}(X)$ +8811,$g_n \ge s_n$ +8812,$\sigma^2=\lambda=1$ +8813,$\delta+\nu=1$ +8814,$s_R < 1$ +8815,$\mathsf{TVaR}_\pi(X) := (1-\pi)^{-1}(T_1+T_2)$ +8816,$R_0(t)<\mathsf{E}[X_0]$ +8817,$\mathbf{v}$ +8818,$X = X\wedge a + (X - a)^+$ +8819,"$=\mathsf{E}(X_{i,2}(a))$" +8820,$1_{X > x}$ +8821,"$Y_{t,d=0}$" +8822,$D\rho_X(X_1)$ +8823,$g(uv) = k_1 uv = \left( \displaystyle\frac{1 - w}{s_0} + \frac{w}{s_1} \right) uv$ +8824,$\rho(\lambda X)=\lambda\rho(X)$ +8825,$S(x) = x\cup \{x\}$ +8826,$\mu_\sigma$ +8827,$X=g(Z)$ +8828,$n\times 1$ +8829,"$(Alice)+(0,-1.75)$" +8830,$\beta=n-c+1$ +8831,$X+{shift}$ +8832,"$4,171.00 \$" +8833,$z_i = \zeta$ +8834,$\Lambda\dfrac{\mu_{U}}{\sigma_U} = \dfrac{E( r_{U} ) - r_{f}}{\sigma_{r_{U}}} \left(\dfrac{\mu_{U}}{\sigma_{U}}\right)$ +8835,$\mathsf{Q}(\Omega_a) >0$ +8836,$\phi(r_1 + r_2)$ +8837,$1-U^2$ +8838,$A \in \mathcal{M}$ +8839,$\alpha_i$ +8840,"$700,000,000), any policy issued or renewed on or after that date shall provide, less any applicable deductible, not less than two thousand five hundred dollars ($" +8841,$P=l + \iota Q$ +8842,$g(s)=s^{0.75}$ +8843,$Y_1$ +8844,$v_f(\mathsf{E}_Q[X_i] - \mathsf{E}_Q[X_i/X(X-A)^+])$ +8845,$\kappa(\theta)=-\log(\cos\theta)$ +8846,$yS(a)=y\times 1=y=$ +8847,$\mathsf{TVaR}_0(X)=\mathsf{E}[X]$ +8848,$a_{2}'$ +8849,$a_t>a_1$ +8850,$\mathsf{TVaR}_{1-c\epsilon}(X) = \mathsf{VaR}_{1-\epsilon}(X)$ +8851,$P\ge (\mathsf{E}[X] + \rho A)/(1 + \rho)$ +8852,"$(0,\omega]$" +8853,$\delta=\dfrac{\iota}{1+\iota}=\dfrac{M}{a}$ +8854,$S<0$ +8855,$pl_p$ +8856,$P=\rho(U)$ +8857,$\dot P(t)=R_1(t)-R_0(t)$ +8858,$\mathbf {Z_5}$ +8859,$ but if $ +8860,$(X-a)1_{X>a}$ +8861,$A=g^a\mod p$ +8862,$5M and an aggregate limit on the deductible of \$ +8863,$s_{R+1} \ge s^\star$ +8864,$m_Y(·)$ +8865,$\rho_{m_Y}(X)=r$ +8866,$R_x$ +8867,$R \propto M^{0.8}$ +8868,$\sum_{i\in I}(D_i-N_I)$ +8869,$p\downarrow 1$ +8870,$v_f=1/(1+r_f)$ +8871,"$\mathsf{MON,TI,CX}$" +8872,$\bar F(a)=\int_0^a F(x)dx$ +8873,$\beta=\infty$ +8874,$R_i r_i$ +8875,$\mathsf{Pr}\vert_\mathscr{F} = λ$ +8876,$\mathsf{E}[Xe^{hX}]/\mathsf{E}[e^{hX}]$ +8877,$2n$ +8878,$\nabla\zeta$ +8879,$d^2 1$ +8884,$\mathsf{TVaR}_{p^*}(X_1)+\mathsf{TVaR}_{p^*}(X_2)=80$ +8885,$1-p=g(1-\hat p)$ +8886,$g(p)/p$ +8887,$s(x;\mu)=(x-\mu)/\sigma^2$ +8888,$\rho(X_g)-\rho(X_n)=51.1560-49.8986=1.2574$ +8889,$23$ +8890,$10^8$ +8891,$k\mapsto k\rho(-X)$ +8892,$X_\infty$ +8893,"$S(x) + d\,F(x) + (\delta^*-d)\sqrt{S(x)F(x)}>1$" +8894,$\Omega\to\mathbb{R}$ +8895,$x=200$ +8896,"$\langle \zeta, G \rangle$" +8897,$\rho(X)=\sup \{ \mathsf{E}[XZ] \mid \text{distribution of }Z\preceq\text{ distribution of } g'(1-s) \}$ +8898,$\mathsf{E}_{\mathsf{Q}'}[X_a]<\rho(X)$ +8899,$\tilde R=R-\delta A$ +8900,$\mathbf {g(s)}$ +8901,$\mathsf{TVaR}_\beta\mathbin{\square}\mathsf{TVaR}_\gamma = \mathsf{TVaR}_\gamma$ +8902,$\rho_{m_i}(X)>\rho(X)-2^{-i}$ +8903,"$(rep.east) + (1.5, 0.5)$" +8904,${}^{[<91]}$ +8905,$X_{-1}=x$ +8906,$p_0 = p^\ast = p_1$ +8907,"$\mathsf{P}(\omega', \omega''_0)=\mathsf{P}(\omega',\{\text{any }\omega''\})\mathsf{P}(\{0,1\}, \omega''_0)$" +8908,$c_i=c_j$ +8909,$g(0+) > 0$ +8910,$\mathbf{P_i} \in \mathbb{R}^2$ +8911,$\rho(A+B)64.5>63.5=\rho(A)+\rho(B)$ +8912,$2.35 \times 10^{-4}$ +8913,$\mathrm{P}$ +8914,"$\mathsf{VaR}_{0.995}=64,861$" +8915,$1/P$ +8916,$\uparrow$ +8917,$\nu\otimes P$ +8918,$0 \zeta$ +8931,"$P_i,M_i, Q_i$" +8932,$\mathsf{var}phi(0) = 0$ +8933,$h(0)=0$ +8934,$EL_a =\mathsf{Pr}r(Y>a)=1-\exp(-\lambda S(x))$ +8935,$\sum_t \zeta_t^2 = 1$ +8936,$q<1$ +8937,"$a=10,20,40,50,60$" +8938,$VaR_{0.98}((1+x_1) X_1 + (1+x_2) X_2)$ +8939,$B(-X)$ +8940,$\rho(A_k) \le \rho(A_0) + k \rho(N)$ +8941,$g_{\min}(s):=\min_i (g_i(s)$ +8942,"$\mathsf{cov}(X_0, g'(S(X(t)))$" +8943,$X(S(U))$ +8944,"$\int S(x)\,dx$" +8945,"$F(\omega, \cdot)$" +8946,$g(S(u)$ +8947,$d$ +8948,$a - P$ +8949,$X=k$ +8950,$g\in \nabla\rho(X)$ +8951,$\mathsf{Pr}(A>x) \approx \mathsf{Pr}(M>x) \approx \lambda \mathsf{Pr}(X>x)$ +8952,$d=i/(1+i)=iv=1-v$ +8953,"$(X-a)^+ = \max(0,X-a)$" +8954,$\mathsf{Q}(A)=\mathsf{E}_\mathsf{Q}[1_A]$ +8955,$X_1>a_1$ +8956,$C_1(t) < C_2(t)$ +8957,$b =\sigma_s$ +8958,"$\rho(X)=\int xg'(S(x))f(x)\,dx$" +8959,"$r_1,r_2$" +8960,$\lim_{\mu\to 0}V'(\mu)=\delta$ +8961,$\mathbf {Z_\mathit{lin}}$ +8962,$s_L$ +8963,$X=F^{-1}(U)$ +8964,$X_1 = \mathsf E[A+B\mid A] = A + \mathsf E[B]$ +8965,$\mathsf{E}_g[X]$ +8966,$Q=1-g$ +8967,$l(y;\mu)=y\theta(\mu)-\kappa(\theta(\mu))$ +8968,$p(p+k)^{k-1}\frac{\delta_k}{k!}$ +8969,$\bar P(a+da)-\bar P(a)$ +8970,$A-P-D$ +8971,$u_10$ +8992,$g'(1-p)=\phi(p)$ +8993,$age^2$ +8994,$\rho(X-P)=\rho(X)-P$ +8995,$t>t^* = 0.544$ +8996,$Z(\omega)\mathsf{P}(\omega)$ +8997,$P^T(T\not=t\mid t)=0$ +8998,$\mathcal{A} = \{ X \mid \rho(X)\le 0 \}$ +8999,$\sigma_T$ +9000,$\mathsf{E}[X] = \mathsf{E}[\mathsf{E}[X\mid Y]]$ +9001,$\partial l/\partial \beta_i=0$ +9002,$p_s=E(R(\Theta_s)$ +9003,$\rho_a(X)>2\rho_a(X_1)$ +9004,$f=1/b$ +9005,$P=L + \delta (a-L)$ +9006,$\mathsf{E}[X_i(x)]$ +9007,$\mathsf{TVaR}_p(X)=\mathsf{TCE}_p(X)=\mathsf{E}[X\mid X \ge \mathsf{VaR}_p(X)]$ +9008,$\delta = \delta(p) = 1-\nu(p)$ +9009,$\lim_n \mathsf{E}_{\mathsf{Q}_n}(X)=\rho(X)$ +9010,$X=X(I)$ +9011,$X^a$ +9012,$k\DeltaR$ +9013,$n-2=n-k-1$ +9014,$g'=0$ +9015,$g(s)=\dfrac{r_{occ}+s(1+r_{use})}{1+r_{occ}+r_{use}s}$ +9016,"$F_\alpha - \bigcup_{\beta<\alpha} \{p_\beta, q_\beta\}$" +9017,$(1+\epsilon)v_1$ +9018,$U_0$ +9019,$\iota\alpha(X)=\iota a$ +9020,$G_1=R_1-1$ +9021,$p_j-p_{j-}$ +9022,$\mathsf{E}_{\mathsf{Q}'}[X_a]=\mathsf{E}_{\mathsf{Q}}[X_a\psi^{-1}]=\rho(X_a)$ +9023,"$(fun3.north west)+(-\smlspc,\smlspc)$" +9024,$-\gamma|\theta|^\alpha$ +9025,$\mathsf{E}[X_\theta]=\infty$ +9026,$\mathsf{VaR}_p(X) = q_l(p)$ +9027,"$\psi_{X, m}(u)$" +9028,$m=$ +9029,$\text{OEP}(y \text{ year occurrence PML}))=1/y$ +9030,$X+tX_1$ +9031,"$(fun1a.south east)+(\smlspc,-\smlspc)$" +9032,"$\{(\mathcal{A}_y, P_y)_{y\in Y} \}$" +9033,$\mathsf P$ +9034,$X_1=0$ +9035,$\mathsf{P}(X=1)=0.6$ +9036,"$v'>0, v''>0$" +9037,$A_1 = \mathsf E[X_1] = \mathsf E[X \mid \mathscr F_1]$ +9038,$\exists x[\forall z(z=\emptyset)\rightarrow z\in x \wedge \forall x\in x\forall z(z=S(x)\rightarrow z\in x)]$ +9039,$d(y\mu) > 0$ +9040,$=(1-\alpha)\mathsf{TVaR}_\alpha(X)$ +9041,$\rho(I)\rho(X)=g(s)\rho(X)$ +9042,"$\mathscr{G}amma(S,T)=\{ (S(\omega), T(\omega)\mid \omega\in\Omega \}$" +9043,$S\to Y$ +9044,$\chi^2(n)$ +9045,$\nu(p)=1/(1+\rho(p))$ +9046,$\delta=1-\nu$ +9047,$\mathscr{G}_Y=\sigma(Y)$ +9048,$1-\mathit{ROL}$ +9049,"$\mathrm{St}(\alpha, \beta, 1, 0)=\mathrm{St}(\alpha, \beta)$" +9050,$H(x)$ +9051,$\mathsf{Pr}hi(\mathsf{Pr}hi^{-1}(s) + \lambda)$ +9052,"$[-1,1]$" +9053,$q_j$ +9054,"$(fun1.north west)+(-\medspc,\medspc)$" +9055,$g(s)=(s+\iota)/(1+\iota)$ +9056,$g(0.05)=0.05\nu + \delta=0.1364$ +9057,"${x}\times A\subset [0,1]^2$" +9058,$\kappa(\theta)=-\alpha\log(-\theta)$ +9059,$\tilde M_i(a) = \bar P_i(a) - \mathsf{E}[X_i(a)]$ +9060,$c(x;\lambda)$ +9061,$x^\alpha e^{-\beta x}/x$ +9062,$\Omega=C^\circ$ +9063,"$\bar L, \bar P, \bar M$" +9064,$s \ge s^\star$ +9065,"$\Lambda_{a,b}$" +9066,"$a_2=8.5, y_2=10.5$" +9067,"$(X_{1,j},\dots,X_{m,j})$" +9068,$g'(S(x))=0$ +9069,$\bar\delta(a)=\bar\iota(a)/(1+\bar\iota(a))$ +9070,$i^{-a}=e^{-a\log i}=e^{-ia\pi/2}$ +9071,$(x-y)^n$ +9072,$L_a^{a+da}=L_0^{a+da}-L_0^a$ +9073,$a_i=i$ +9074,$A\subset \{ Z=0 \}$ +9075,$\alpha\ge A(n)=\sum_s n_s(1-g(s))$ +9076,$\beta(x)\le \alpha(x)$ +9077,$Y=X^2$ +9078,$1/S(x)$ +9079,$\rho(X)=\mathsf{E}[f_X X]$ +9080,$\kappa_i'(x)\uparrow 1$ +9081,$m=0.02$ +9082,$j \ge L$ +9083,$c^*(y):=e^{l(y;y)}=c(y)e^{y\tau^{-1}(y)-\kappa(\tau^{-1}(y))}$ +9084,$F(x_0)=p_+$ +9085,$\mathsf{Pr}(A\mid\mathscr{G}) = \mathsf{E}[1_A\mid\mathscr{G}]$ +9086,$X \le 0$ +9087,$J(x)/J(0)$ +9088,"$Q,\iota,M$" +9089,$X_i(v_i)=v_iX_i(1)$ +9090,$CV(L(1))/\sqrt{\nu}$ +9091,"$P(\omega, B)=\mathsf{Pr}(B\mid\mathscr{G})(\omega)$" +9092,"$X_{t,3}$" +9093,$1+1/3+1/5+\cdots$ +9094,$\mathbf p$ +9095,$X=\sum_j X_j$ +9096,"$\int_\Omega f_0\,d\mu=\alpha$" +9097,$X(a)$ +9098,$g_{ROC}$ +9099,$\bar P_i(a):=\rho_{X\wedge a}(X_i(a))$ +9100,$\nu(dx)$ +9101,$CC = \mathrm{PV}_r(C)+ \mathrm{PV}_r(C)+\pi$ +9102,$1_{X < q(1-s)}-(1-g)$ +9103,"$b\in[-1,1]$" +9104,"$S=\{0,1,2,\dots\}$" +9105,"$X_1=1+cos(X_3), X_2=1-cos(X_3)$" +9106,"$(ckey1.north west)+(-\boundpad,\boundpad)$" +9107,$\le c$ +9108,$M[G]$ +9109,$\lambda^2\sigma=\lambda$ +9110,$k_r$ +9111,$d(y;\mu)=|y-\mu|$ +9112,$\int_0^1 xj(x)dx=\infty$ +9113,$\mathsf{VaR}_1$ +9114,$t=t_2$ +9115,$g(s)=\mathsf{Pr}hi(Z(s)+\lambda)$ +9116,"$t\not\in[t_*,t^*]$" +9117,$T_t$ +9118,"$p\in (0,1)$" +9119,$H:\mathcal X\to\mathbb R$ +9120,$r=0$ +9121,$a-X=P+Q-X$ +9122,$\mathsf{E}_Q(Y\mid X)\mathsf{E}(Z\mid X) = \mathsf{E}(YZ \mid X)$ +9123,$S_{X\wedge a}$ +9124,$w=\sum_i w_i$ +9125,$t=U_X(s)$ +9126,$\alpha(\mathsf{Q})$ +9127,"$2*(1,1)$" +9128,$\rho(T)\ge T$ +9129,$<\mathsf{E}[X_0\wedge a(0)]$ +9130,$p\mathsf{E}[X\mid X0$ +9140,$X(p)=q(T(p))$ +9141,$X_t=X_{t+1}$ +9142,$m_X(s) \to \mathsf E[X]$ +9143,$P + \rho_i(F_i)$ +9144,$p\neq 2$ +9145,"$\{0, 9, 10\}$" +9146,$\rho(X+c) = \rho(X)+c$ +9147,$\sigma_\mu(\alpha) = \int_0^\alpha \frac{1}{1-p}\mu(dp)$ +9148,"$\kappa_{i}(x) = \dfrac{\sum_{j:X_{j} = x} X_{i,j} p_j}{\sum_{j:X_{j} = x}p_j}$" +9149,$x\in\mathbb{R}$ +9150,$\lim_{y\downarrow x} f(y)$ +9151,$age$ +9152,$(\alpha-1)(1-p)=1$ +9153,"$X_{t-2,1}$" +9154,$0 < w < 1$ +9155,$g(s)=s^a$ +9156,$\rho-\iota g>0$ +9157,$\mathsf{COM}$ +9158,"$i=1,\dots, N$" +9159,$\sqrt{FS}\gg S$ +9160,$y_i$ +9161,$\rho(-X)>-\rho(X)$ +9162,$P'<\rho(W_1\wedge a_1)$ +9163,$\mathsf{E}[hY]$ +9164,$g(X+Y)=g(X)+g(Y)$ +9165,$f'_+(x)=\lim_{h\downarrow 0} (f(x+h)-f(x))/h$ +9166,$100 rate and one \$ +9167,$U$ +9168,$\mathsf{TVaR}_{1}$ +9169,"$\displaystyle\int_0^a \alpha_i(x)S(x)\,dx$" +9170,$\mathbf {X_{2c}}$ +9171,$2(Mg+\alpha a)w$ +9172,$12.318 / 260.81 = 4.7\%$ +9173,$\eta\nu$ +9174,$p=0.283$ +9175,"$[0,\theta]$" +9176,$\mathsf{E}[X]$ +9177,$M^i = P^i - S^i$ +9178,$a_i + b_i\ \mathit{EL}$ +9179,$\mathscr{G}\subset \mathscr{F}$ +9180,$D\rho(\cdot)$ +9181,$Q^1=M^1/\iota$ +9182,"$P(X_1+X_2)=M(X_1+X_2, \psi(X_1+X_2))=$" +9183,$m'(1) \to -1$ +9184,$\mathsf{E}[p] \le 1$ +9185,$h\in \nabla\rho(X)$ +9186,"$(-\infty,0]$" +9187,$g(S_6)\Delta X'_6$ +9188,$\rho''(x)=-U''(x)>0$ +9189,$p_1\not=p_2 \ge p^*$ +9190,$s>{s:3g}$ +9191,$x=z$ +9192,$A_k=A_0 + kN$ +9193,$\mathbf {\mathcal Q}$ +9194,$K_\theta(t)=\kappa(\theta)\left[ \left( 1 + \dfrac{t}{\theta} \right)^\alpha -1 \right]$ +9195,$k = 3.3 s^{0.82}$ +9196,$ is the total return on invested assets and $ +9197,$b\approx 0.95$ +9198,$n^y$ +9199,$t=1-g(1)=0$ +9200,$\mathsf j(a)$ +9201,$\mathcal{W}$ +9202,"$\mu_0 : \mathcal{A} \to [0, \infty]$" +9203,$\nu(F(x))F(x) = \nu(p)p$ +9204,$p=0.999$ +9205,$\Delta_n\subset\mathbb{R}^{n+1}$ +9206,"$r_o,r_K$" +9207,$\alpha:{\mathcal{M}}\to\mathcal{B}bb R\cup\{\infty\}$ +9208,$a_1=2272$ +9209,$\rho(W_0\wedge a_0)=\bar P_0 +\bar P'$ +9210,$(E\cap U)$ +9211,"$f(s, \cdot)$" +9212,$X\wedge a=a=90$ +9213,$Y\preceq Z$ +9214,$g(1-s)$ +9215,$G_\nu$ +9216,$y^*-x^* < \epsilon$ +9217,$g(s)=s^{0.8}$ +9218,$m\ge n$ +9219,$M_\odot \approx 1.989 \times 10^{30} \ \text{kg}$ +9220,$\xi:\Omega\to \mathbb{R}$ +9221,$\approx 45\%$ +9222,$p=0.967$ +9223,$\theta_s=\hat\theta_s+\hat\sigma_s Z_s$ +9224,"$(1,1)$" +9225,$\mathsf{E}[X_1Z]$ +9226,$L=\mathsf{E}_p[.]$ +9227,$\rho= \pm 1$ +9228,"$ are better. By a shaping argument, assume $" +9229,$A_1=\mathsf E[X\mid \mathscr F_1]$ +9230,$C=B+1$ +9231,$f(x+t)\ge f(x) + st$ +9232,$p\neq 1$ +9233,$\lambda_1$ +9234,$h_2\le w$ +9235,$\phi(r1) \phi(r2)$ +9236,"$\rho=\mathsf{biTVaR}_{1-s,1-t}^w(s,t)$" +9237,$k\mathsf B(s)$ +9238,$\rho_g(X)$ +9239,$\rho(X)=\mathsf{E}_{\mathsf{Q}}[X]=\mathsf{E}_{\mathsf{Q}}[\sum_i X_i]=\sum_i \mathsf{E}_{\mathsf{Q}}[X_i]$ +9240,"$1,553.08, which is \$" +9241,$\phi$ +9242,$g^{ak}=(g^a)^k$ +9243,$0<\mathsf{E}[X_i]\le 1$ +9244,$m_i\to m\in\mathcal{M}$ +9245,$x\not= y$ +9246,$\frac{dy}{dt}$ +9247,$ which is a gamma distribution with shape $ +9248,$d_f = r_f / (1+r_f)$ +9249,"$\rho(X)= \sup_\zeta \langle \zeta, X \rangle$" +9250,$\sum\mathsf{E}[C_i^2]=\sum m_i(1+v_i^2)$ +9251,$A\subset \{X_a=a\}$ +9252,$P(dx)$ +9253,$X_0=C_1 + \cdots + C_N$ +9254,$\{X\in L^\infty \mid \rho(X)\le c \}$ +9255,$X_{t-1}$ +9256,$\alpha_1\ge \beta_1$ +9257,$\sum_{i} X_i(a) = X\wedge a$ +9258,$A=X_1 + \cdots + X_N$ +9259,$\tilde R_i$ +9260,$D_1\supset D_2\supset \cdots \supset D_\infty$ +9261,$\rho(X\mid \mathscr F_1)$ +9262,$q(1-g^{-1}(1-p))$ +9263,$x\!\urcorner$ +9264,$n - 1$ +9265,$\bold x$ +9266,$X=100$ +9267,$\theta=\log(\mu/(n+\mu))$ +9268,$\alpha<-2$ +9269,$N(m)$ +9270,$a/Q = 1 + R/Q$ +9271,$X=\displaystyle\sum_i X_i$ +9272,$\{\mathsf{P} \}$ +9273,$(a_1 - X)^+ \leftrightarrow$ +9274,$U=U_0 - i_\mathrm{a/tax}\mathrm{PHSF}$ +9275,"$t\in(0,1]$" +9276,"$\rho_0(\rho_1(X))=\rho_{U_1}(\rho_{U_2}(X(U_1,U_2)))$" +9277,$\mathbf {X_{1}(a)}$ +9278,"${}^{[<29,62]}$" +9279,"$(x_1-\epsilon,x_1]$" +9280,$\phi_{m}^e$ +9281,$\ge a$ +9282,$R^i=L^i/P^i$ +9283,$\mathcal{A}_2$ +9284,$1+\bar\zeta_t^2$ +9285,$P=\mathsf{E}[X]$ +9286,$m\left(1+\dfrac{m^2}{p^2}\right)$ +9287,$\rho_1(X)(u_1) = X(u_1) = X$ +9288,$Z\in \mathcal Q$ +9289,$N$ +9290,$r={base_roe:.3f}$ +9291,$\mathcal F_1\subseteq \mathcal F$ +9292,$\kappa(\theta)<\infty$ +9293,$\phi(s)=\int_0^s \mu(dt)/(1-t)=\int_{1-s}^1 \mu(dt)/t$ +9294,$da=da(\mathbf{x})$ +9295,$\omega_I \ge s$ +9296,$\mathsf{E}[X_i/X ; X > a]$ +9297,$0.125$ +9298,$\omega \in \Omega$ +9299,$10^{-6}$ +9300,$\mathsf{E}[X_i | X] / X$ +9301,$f(y;\mu)=$ +9302,$Z\circ T\in \mathcal Q$ +9303,"$\mathbf {X\,\Delta S}$" +9304,$\mathsf{E}[X] + \pi\mathsf{Var}^+(X)$ +9305,"$(x,-x)$" +9306,$\sup X=\mathsf{E}[XZ]=\int XZ$ +9307,$\bar Q(a)=a-\bar P(a)$ +9308,$\nu(x)$ +9309,$0.2 < s < 1$ +9310,$\mathsf{E}[X\wedge a] = (1-e^{-a\beta})/\beta$ +9311,$m\in\mathbb{R}$ +9312,$m/V(m)$ +9313,"$z_1, \dots, z_t$" +9314,"$\mathsf{biTVaR}_{p_0,p_1}^w(X)=\mathsf{TVaR}_{p^\ast}(X)$" +9315,$S(x)=p$ +9316,$Mg(0+)$ +9317,$l(y;y)$ +9318,"$x=2, M=1.5,\sigma=0.75, K=6$" +9319,$ where $ +9320,"$. As usual, reduce to premium of 1 per unit time by adjusting the time period. $" +9321,$a \ge 0$ +9322,$LR_{\mathsf{CCoC}}$ +9323,$-(1-s)g''(1-s)$ +9324,$\mathsf{E}[WX] \le \rho(X)$ +9325,$\partial a/\partial x_1=3x_1/a$ +9326,$s_{R} < s_{R+1} < s_{R+2} < s_{m}$ +9327,"$\nu p\,da=\nu F(a)\,da$" +9328,$x+dx$ +9329,$E_g[X\wedge a]$ +9330,$d\tilde p =g'(1-p)dp$ +9331,$\lambda\sigma$ +9332,$x_6^1+x_6^2=10+1=11=x_6$ +9333,$=g(s)$ +9334,"$\mathsf E[X_iZ]=\mathsf E[X_i] + \mathsf{Cov}(X_i, Z)$" +9335,$q \in D$ +9336,$M(a)=g(S(a)) - S(a)$ +9337,$V_X(m)$ +9338,$\mathsf{TVaR}_{p^\ast}$ +9339,$h = 1$ +9340,$\kappa'(\tau^{-1}(\mu))=\tau(\tau^{-1}(\mu))=\mu$ +9341,$P-D$ +9342,$\rho_\phi(X)=\displaystyle\int_0^1 q(p)\phi(p)dp=\displaystyle\int_0^\infty g(S(x))dx=\rho_{(g)}(X)$ +9343,"$100,000 / \$" +9344,$a_{ro}:=\mathit{VaR}_{p}(X_{-1})={{a_x0}}$ +9345,$f(x;\theta)=c(x)e^{\theta x - \kappa(\theta)}$ +9346,$a_{ro}$ +9347,$\rho^*(\zeta-1)$ +9348,$Y(\omega)=1$ +9349,$\rho_1 \ne \rho_2$ +9350,"$[0,0] \succeq [-k -k]$" +9351,$p = 0.5$ +9352,$\mathcal{Q}$ +9353,$\mathscr{E}$ +9354,$A$ +9355,$x_0=q^-(p_0)$ +9356,$E[u_j(W_j - X_j)]$ +9357,$\mathsf{TI}$ +9358,$\nu_B(A)=\mathsf{Pr}(B\cap T^{-1}(A))$ +9359,$m(s)$ +9360,$P_{act}$ +9361,$a = a(W) = \mathsf{E}[W]+4\sigma(W)$ +9362,$10^6 - 10^9$ +9363,$\mu=1$ +9364,${}_1F_1$ +9365,$S(X_j)>0$ +9366,"$t^\star \in [0,1]$" +9367,"$16,800 | | $" +9368,$\displaystyle\int_0^1 X(1-g^{-1}(1-\tilde p))d\tilde p$ +9369,$X>A$ +9370,$dX$ +9371,$M_i(x)+Q_i(x)=\alpha_i(x)F(x)$ +9372,$\rho_{m_Y}(X) = q < r$ +9373,$-iv^{n-1}$ +9374,$t/(1-t)$ +9375,$\hat\rho_{\mathcal F_1}$ +9376,$\mathsf{Pr}r(X > \mathsf{VaR}_p(X))$ +9377,"$\Omega=\{0,1,2,\dots \}$" +9378,$v/\sqrt{n}$ +9379,$1-\pi$ +9380,$1/x$ +9381,$\mathsf{Pr}(\mathsf{CP}=n)=\sum_{k\ge n}\mathsf{Pr}(\mathsf{CP}=n\mid N=k)\mathsf{Pr}(N=k)$ +9382,$\mathsf{E}(X) = E(X_i \mid X\le a)F(a) + =E(X_i \mid X > a)S(a)$ +9383,$X^{\oplus n}=X_1 + \cdots + X_n$ +9384,$\mu_*(M\cap E)=\mu_*(M'\cap E)=0$ +9385,$A_1 \subseteq A_2 \subseteq \cdots$ +9386,$\hat q$ +9387,$\rho_g(-X)=-\rho_{g^*}(X)$ +9388,$z_p^{(2)} \le 2z_{p}$ +9389,$NT$ +9390,$\dfrac{\partial }{\partial v} g(S_v(x)) = g'(S(x))\dfrac{\partial S}{\partial v}$ +9391,$a(w_1X_1+w_2X_2;X)=w_1a(X_1;X)+w_2a(X_2;X)$ +9392,$2^{\aleph_0}$ +9393,$1 < a < 2$ +9394,$X_n\downarrow 0$ +9395,$k_1 >0$ +9396,$\mathsf E[A] \le \mathsf E[\rho(X^{\oplus N})] \le \rho(A)$ +9397,$T_1-1$ +9398,"$p=0.01, 0.02, \dots, 0.99$" +9399,"$(s,\iota)$" +9400,$0 < g' \le 1$ +9401,$f(\omega)\ge 0$ +9402,$0\le \alpha<1$ +9403,"$m\ge 1, n\ge 0$" +9404,$\nu(p) = v-(v-\nu^*)\sqrt{(1-p)/p}$ +9405,"$X=[0,1]^2$" +9406,$\mathbf{U}$ +9407,"$\int |X_n(\omega) - X(\omega)| \,\mathsf{Pr}(d\omega)\to 0$" +9408,$\sigma^2\mu^p=\lambda\alpha(\alpha+1)/\beta^2$ +9409,$d\downarrow 0$ +9410,$\nabla\partial\rho(Z)$ +9411,$g^mA^R=g^{m+Ra}$ +9412,"$\sigma=2.0,3.0$" +9413,$\ge 0.95$ +9414,$s>{s0:.3g}$ +9415,$u_l>0$ +9416,"$1M. But in fact, the true expected loss is \$" +9417,$R\to\infty$ +9418,$\mathsf{Var}(X_i)>0$ +9419,$18.582.08 *X* 0.018055** = **\$ +9420,$2^2\rightarrow 3^3-1=2\times 3^2 + 2\times 3 + 2 = 26$ +9421,$\hat\rho(Y)$ +9422,$Xm1=X_{-1}$ +9423,$\tilde\rho$ +9424,$a=0$ +9425,$N=0$ +9426,$1+\gamma$ +9427,"$u,v\in[0,1]$" +9428,$\rho(U)=\mathsf{E}_\mathsf Q[U]$ +9429,$r_iC_i(t)$ +9468,$se(\hat\beta)$ +9469,$\alpha_iSdX$ +9470,$\beta_i(x) \ne \alpha_i(x)$ +9471,$\mathscr{A}$ +9472,$1 - g(s)$ +9473,$l_s$ +9474,$n \times 3$ +9475,$\rho_1(X)>\rho_2(X)$ +9476,$\kappa_i(X)=\mathsf{E}[X_i\mid X]$ +9477,$\rho(-k_1 1_{A_1}) = k_1 \rho(-1_{A_1}) < c$ +9478,$\mathsf{Pr}(\{\omega \mid X_n(\omega)\to X(\omega) \})=1$ +9479,$dt/t$ +9480,$\alpha=d$ +9481,$(X_t)$ +9482,$\alpha(X)=a$ +9483,$p'\ge p$ +9484,$\Delta X'$ +9485,$x\mapsto x^{1/2}$ +9486,$\mathsf{E}[X_i\wedge a_i]$ +9487,$\min_{\eta\in \mathbb{R}} \eta + \alpha \mathsf{E}[(X-\eta)^+] -\beta\mathsf{E}](X-\eta)^-]$ +9488,"$U_1,\dots,U_T$" +9489,$n_s\ge 0$ +9490,$g(s)=1-(1-s)^3$ +9491,$p\not=1/2$ +9492,$\mathsf{P}[(X(x)-a(x))^+]$ +9493,$\hat \theta=0.5$ +9494,$\mathsf{E}_Q[X_i(x_i)]=x_i\mathsf{E}_Q[X_i(1)]$ +9495,$v^b{}_bq_x\bar a_{x+b} /\bar a_x=v^b{}_bq_x(1-{}_b\bar V)$ +9496,$P_X$ +9497,$a=90$ +9498,$p_a<1$ +9499,$g(S_n)=g(0)=0$ +9500,$S_j=S_{j-1}-p_j$ +9501,$\mathsf{Var}^+(X)$ +9502,$\omega_I=s$ +9503,$1 — 1_B$ +9504,$=a$ +9505,$P_{x+b}-P_x > 0$ +9506,$g(s)=20s\wedge 1$ +9507,$0<\alpha<2$ +9508,$\mathsf E[X_1]=\mathsf E[X]$ +9509,$45𝐾𝐾 + 3 × (\$ +9510,"$\int_a^\infty g(S(x))\,dx$" +9511,$u = g(S(x))$ +9512,$\mathsf{E}[X_d]$ +9513,$\mathbf {X_{2}}$ +9514,"$\{s_j: j=L,\dots,R\}$" +9515,$50) and $ +9516,$p_i a)$ +9532,$a_1$ +9533,$\rho(X)>-\rho(-X)$ +9534,$1/10$ +9535,"$1,479.11 \$" +9536,$t=0.5$ +9537,$p^y := p^y \prod_{i=0}^{n^y} w_i^y$ +9538,$\mathrm{Pr}_{rn}\{P_{act}>P\}$ +9539,$g(s) = r_0 + (1-r_0)s$ +9540,$1000e^{\mu}$ +9541,"$\rho(Y)\le b_{X,r}(Y)$" +9542,"$Y\sim\mathrm{Ga}(\mu,\alpha)$" +9543,$10^{23}$ +9544,"$a_i=\rho(X_i, p^*)$" +9545,$0 \ge \rho(Y-X) \ge \rho(Y) - \rho(X)$ +9546,$l'>l=0.49$ +9547,"$A,B$" +9548,$1-r$ +9549,$X-a$ +9550,$\epsilon^i$ +9551,$Q:\Omega\times \mathsf{E}E\to\mathbb{R}$ +9552,$0 < \mu < \lambda$ +9553,$\mathcal X\times M$ +9554,$X(\mathbf{x})(\omega)=q_\omega(\mathbf{x})$ +9555,$>q(p)$ +9556,$\alpha(X_u) = \text{E}[X\mid X > F_u^{-1}(p)]$ +9557,"$(0+, g(0+))$" +9558,$\zeta-\zeta_\epsilon$ +9559,$dW_t$ +9560,$\mathsf{E}_Q(X) =\mathsf{E}(\theta X /\mathsf{E}(\theta))$ +9561,"$(0.5,1]$" +9562,$\bar Q(a):=\int_0^a Q(x)dx$ +9563,$dS=-f(x)dx$ +9564,$Y=\mathsf{E}[Z\mid\mathcal G]$ +9565,$\rho^\star$ +9566,$l'-l$ +9567,$\rho(X)=\mathsf{E}[X]/r$ +9568,$\zeta\in\partial(X)$ +9569,$\rho(T)$ +9570,$x_{95} < x_{96} < x_{97} < x_{98} < x_{99} < x_{100}$ +9571,"$ In general, define $" +9572,$P(a) = L(a) + \iota (a-P(a)) = \nu L(a) + \delta a$ +9573,$\mathsf{E}[Z]\ge 1$ +9574,$\R$ +9575,$N_2$ +9576,$t\downarrow 0$ +9577,$g(0.1)=\sqrt{0.1}=0.316$ +9578,$g(s) = 1 - (1-s)^3$ +9579,$\sum_j \mathsf{TVaR}_{p_j}(X)m_j$ +9580,$\mathscr{O}(\zeta)=\{\zeta T \mid T\in MPT\}$ +9581,$R_1(t)\le P(1)$ +9582,$n-4$ +9583,$\rho(1_A) = 1$ +9584,$h(x):=f(x)/S(x)$ +9585,$\bar p_s$ +9586,$(a-X)$ +9587,$13.8$ +9588,"$h(t)=\int_0^t F_Z^{-1}(1-u)\,du$" +9589,"$\kappa_\nu(\theta)=\nu\,\kappa(\theta)$" +9590,"$[-0.03,0.03]$" +9591,$\mathsf{E}(X) = \mathsf{E}(X\mid X \le a)F(a) + \mathsf{E}(X\mid X > a)S(a)$ +9592,$s > s_1$ +9593,$g'(S_{X}(X))$ +9594,$D_t = (X_{t-1} + \epsilon_t)^2 - X_{t-1}^2 - 1 = 2X_{t-1}\epsilon_t + \epsilon_t^2 - 1$ +9595,$\alpha=\mathsf{E}[X \mid X > F_u^{-1}(p)]$ +9596,"$\mathsf{E}[1_{(0, x]}(X)]$" +9597,"$\rho(X,a)=\int_0^a S(x) + \delta(F(x))F(x)dx$" +9598,$\mathsf{TVaR}_p(X)=1=\mathsf{E}_\mathsf Q[X]$ +9599,$S(y_j-)-S(y_j)$ +9600,"$a_{0,t}' := a_{0,t-1}-X_{0,t}$" +9601,$\mathsf{Pr}hi_i(a)/a$ +9602,$q_{\cdot}(\mathbf{x})$ +9603,$ is the cumulant generating function for a CP with expected frequency $ +9604,$\mathsf{E}[X^k]\le \mathsf{E}[Y^k]$ +9605,$\mathsf{Pr}si^{-1}(t)=\log(-\log(t))$ +9606,$\mathsf{E}[p]\not=1$ +9607,$Sdx$ +9608,$h(x)$ +9609,$g(s)g(t)-g(st)$ +9610,$0.475$ +9611,"$g(s)=\min(1, s / (1-p))$" +9612,$0\in\Theta_p$ +9613,$S_k$ +9614,$\alpha_i(x)<\kappa_i(x)/t$ +9615,"$(\cdot)^u,(\cdot)^l$" +9616,"$(ckey2.north west)+(-\boundpad,\boundpad)$" +9617,$dS$ +9618,$\bar M=\bar P-\bar S$ +9619,$\rho(T)=76.11$ +9620,$\bar R(a)$ +9621,$\bar q_{X_1+X_2}(s) \approx \bar q(s/2)$ +9622,$p=1.0005$ +9623,$g'(1)=\alpha < 1$ +9624,$E[YZ]$ +9625,$\mathsf{E}[X_i]$ +9626,$\rho=\mathcal{A}VaR$ +9627,$X_2=X-X_1$ +9628,$\rho(X\wedge \alpha(X)) = \displaystyle\int_0^{\alpha(X)} g(S(x))dx$ +9629,$xg(S(x))\vert_0^a$ +9630,$5 \times 10^9$ +9631,$\mathsf{E}_Q=\mathsf{E}$ +9632,"$s,p$" +9633,$\kappa_i(x)=\mathsf{E}[X_i\mid X=x]$ +9634,$\kappa'(\theta)=e^\theta=\mu$ +9635,$\mathcal{C}$ +9636,$x_n\downarrow 0$ +9637,$X^{\oplus n} -\mathsf E[X] = X^{\oplus n-1} + (X'-\mathsf E[X])$ +9638,"$\mathsf{CP}(\lambda, \hat X)$" +9639,$q + 2pq + 3p^2q+\cdots=q(1+2p+3p^2+\cdots)=1/q$ +9640,$0=\rho(0)$ +9641,$\mathsf{E}_Q[0]=\mathsf{E}_Q[0+0]=\mathsf{E}_Q[0]+\mathsf{E}_Q[0]$ +9642,"$P(\mathscr{G}amma\cap Z, \omega)$" +9643,$0\le x\le 200$ +9644,$M_1\Delta X$ +9645,$g'(t)=αt^{α-1}$ +9646,$\hat\theta$ +9647,$P = S + M$ +9648,$\rho(X_1+X_2)\le \rho(X_1)+\rho(X_2)\le 0$ +9649,$X_1=X+b$ +9650,"$(0.2, 0.304)$" +9651,$\rho(X)=\mathsf{E}_\mathsf{Q}[X]$ +9652,$\sigma^2$ +9653,$X_1=18$ +9654,$A=\sum_n 1_{N=n}X^{\oplus n}$ +9655,$\rho((X-a)^+)$ +9656,$\mathsf E[e^{r_2Z_2}]$ +9657,$iv^{n-1}$ +9658,$L_p$ +9659,$SM$ +9660,$\alpha e^{-\beta x}/x$ +9661,$x_{(j)}$ +9662,$g'(s_1) \ge (1-g(s_1))/(1-s_1)$ +9663,$f(x_i):=\mathsf{Pr}r(X=x_i)=\mathsf{Pr}r(X>x_{i-1})-\mathsf{Pr}r(X>x_i)=S(x_{i-1})-S(x_i)$ +9664,$t=2/3$ +9665,"$\int fdP = \int dQ/dP\, dP = \int dQ =1$" +9666,$d=r/(1+r)=1-v$ +9667,"$A,B\subset \Omega$" +9668,"$\mathbf{v}=(v_1,v_2)$" +9669,$100 while companies C and D request approval of \$ +9670,$p_3 = p_1 / 4$ +9671,"$500,000, \$" +9672,$\int g(S(x))dx$ +9673,$\Delta^e$ +9674,$q_X(U)=X(T(U))$ +9675,$S/M$ +9676,$\mu-\sigma^2/2$ +9677,$\mathsf{E}_Q(X) =\mathsf{E}[\theta X /\mathsf{E}[\theta]]$ +9678,$Q=S$ +9679,$\mathsf{E}(X_i / X)$ +9680,$\iota(s)=(1-s)/(1-1)=\infty$ +9681,$j^{th}$ +9682,$0 < a-1 < 1$ +9683,$\mathsf{Pr}r(L\ge 270)=77.1\%$ +9684,$t=t_0$ +9685,$\beta_2(x)>\alpha_2(x)$ +9686,"$f'_\omega (\bar x, h)$" +9687,$g^a$ +9688,$\mathsf P(f^{-1}(A))=\mathsf{Pr}r(A)$ +9689,$\mathsf{E}\_\mathsf{Q}[X\_1]$ +9690,"$ x = 0, 1, 2, \ldots $" +9691,$(N+2)$ +9692,$\Delta_j =g'(s_j-)-g'(s_j+)=\phi((1-s_j)+)-\phi((1-s_j)-)$ +9693,$L_0^{a-Y}$ +9694,$\mathit{LOSS}=\mathit{LR}\ \mathit{PREM}$ +9695,"$\bar P^a(t):=\bar P^a(1-t, t)$" +9696,$p_\alpha$ +9697,$\mathsf{E}(X)=\int S(x)dx$ +9698,"$(0,t]$" +9699,$y(t_{n+1})$ +9700,$\tau=0+d$ +9701,$s < s_m$ +9702,$\bar a_x = (1-\bar A_x)/\delta$ +9703,"$750,000/\$" +9704,$\mathsf{TVaR}_p(X) := X_{N-1}$ +9705,$10^3$ +9706,$\mathsf{j}(a) = \max\{ j:X_j < a \}$ +9707,$Y^S$ +9708,$\mathsf{VaR}_{0.99}$ +9709,$s=1$ +9710,$C_v$ +9711,$x_{\min{}}=0$ +9712,$\mathsf{Pr}r(X\ge x_j)=1-p_{ j- } > 1-p_j$ +9713,$(1-\mathrm{Gamma})/\lambda$ +9714,$U_s)$ +9715,$\le$ +9716,$S^{-1}(g_i)$ +9717,$g'(S_X)$ +9718,$\bar Q_{0}=a_{0}-\bar P_{0}$ +9719,$x_0x_1/x_t^2>0$ +9720,"$(X+Z, Y)$" +9721,$N_i=N_i(x_i)$ +9722,$21$ +9723,$\bar G'(a)=\frac{d\bar G}{da}=G(a)$ +9724,$\nu(p)<1$ +9725,$\mathsf{E}(L)$ +9726,$s_{(i)}$ +9727,$T-1$ +9728,$\mathsf{TVaR}_{0.75}(X_2)=90$ +9729,"$x_{2,2}$" +9730,$a_{1}'$ +9731,$p={pbe:3g}$ +9732,$dF(x)=f(x)dx$ +9733,$\alpha X$ +9734,$\hat\rho(A)<\rho(A)$ +9735,$S(R)$ +9736,$Q=A-P$ +9737,$J(0)<\infty$ +9738,$\mathsf{E}[Z_A]=1$ +9739,$p^g$ +9740,$\rho_h(X):=\mathsf{E}[X_h]$ +9741,$\mathsf{E} X + c{(X-\tau)^+}_p$ +9742,$1+\epsilon$ +9743,"$u,v>1-s_0$" +9744,$-d g(S(x))/dx$ +9745,$\mathsf{Pr}r(\{\omega \mid X_n(\omega)\to X(\omega) \})=1$ +9746,$-X_i$ +9747,$A_T=X$ +9748,$P(1)=P_1(1)=R_1(1)$ +9749,$\rho(X)\ge -\rho(-X)$ +9750,$\mathrm{MV}$ +9751,$\zeta_X$ +9752,"$(s_{R}, g_{R})$" +9753,$v+l$ +9754,$kS = m + Ra$ +9755,$\hat\rho(X)\ge \rho(X)$ +9756,$\{X(\mathbf{v}) = q_{\mathbf{v}}(p)\}$ +9757,"$X,X_1,X_2,\dots$" +9758,$\rho(X\wedge a)$ +9759,$\mathsf{VaR}_p(X)=F^{-1}(p)=q (p)$ +9760,$0=\rho(0)=\rho(X-X)\le \rho(X) + \rho(-X)$ +9761,$E(X_i/X \mid X)$ +9762,$3\times 3$ +9763,$X - X_1$ +9764,$\mathsf{E}(Z \mid \mathcal{G})=Z$ +9765,$(y-\mu)^2$ +9766,$\lambda=1/m$ +9767,$d^2$ +9768,$\kappa_{T_x}$ +9769,$X(x)=x$ +9770,"$335,222,202 | 20.3% | $" +9771,$E(X\wedge a)$ +9772,$0.5 < t_* < t^*$ +9773,$\mu_x = A+Bc^x$ +9774,$(c)$ +9775,$S_t=a_0 + (1+c)\mu t - X_t$ +9776,$P=(1-\delta)L + \delta a = L + \delta (a-L)$ +9777,"$a_1,\dots,a_n$" +9778,$F_Z^{-1}(U)\in\mathscr{P}$ +9779,$g(0-)$ +9780,$A\subseteq \mathbb{R}^N$ +9781,$\mathsf{E} X + c\mathsf{E}[((X-\mathsf{E} X)^+)^p]^{1/p}$ +9782,$h(y) = -\log y$ +9783,$\{ r \}$ +9784,$1\le p\le 2$ +9785,$C_2(0)>\mathsf{E}[X_2]$ +9786,$\mathsf{E}[X] = \displaystyle\int_\Omega X(\omega)\mathsf{Pr}r(d\omega)$ +9787,$a\mathsf{E}_Q[X] + \mathsf{E}_Q[Y]$ +9788,$c_k-G\le 0$ +9789,$a=P+Q=EL+M+Q$ +9790,$\alpha\beta$ +9791,$\rho(X) < \infty$ +9792,$X(\omega)=\exp(10 + 2\mathsf{Pr}hi^{-1}(\omega))$ +9793,"$[-2\pi, 2\pi]$" +9794,$\iota_R$ +9795,$g(s)-s = d + vs - s = d - s(1 - v) = d(1 -s)$ +9796,"$(-1,0)$" +9797,$g(x)=x$ +9798,$L_{\sigma_1}\subset L_{\sigma_2}$ +9799,$\mathsf{E}[X_i(1) \mid X(\mathbf{x}) = q_p(\mathbf{x}) ]$ +9800,$S(x)dx$ +9801,"$\mu^*(E)=\inf\left\{ \sum_{n\ge 1} \bar\mu(E_n) \mid E_n\in \bar S,\ E\subset\bigcup_n E_n \right\}$" +9802,$S(x)\le s^*$ +9803,$g(S(x))=q(\tilde p)\phi(\tilde p)$ +9804,$\{T=t\}=\{t\}$ +9805,$X_\alpha$ +9806,$\bar P_{40}=6908.82$ +9807,$D_i-N_i > 0$ +9808,$\mathsf{Pr}r(I=1)=s$ +9809,$5$ +9810,$f=(1-p)^{-1}1_{W}$ +9811,$0.3$ +9812,$a_{1}$ +9813,$\{X_t\}$ +9814,$\mu(A) = \nu(f(A))$ +9815,"$[\alpha,1)$" +9816,$x_i=q(u_i)=F^{-1}(u_i)$ +9817,$\rho(X) - (-\rho(-X))=\rho(X)+\rho(-X)$ +9818,$1050\times 1.05^{-4} - 950=326$ +9819,$\epsilon(\mathsf{E}_q(X_1)-t)$ +9820,$P(A|B) = \frac{P(A \cap B)}{P(B)}$ +9821,$0.1005$ +9822,$\rho(X+Y)\le\rho(X) + \rho(Y)$ +9823,$\mathsf{E}[U]\approx \mathsf{E}[X]$ +9824,$A\in \mathcal{A}$ +9825,$\rho(X+x)=\rho(X)-x$ +9826,$\hat s$ +9827,$E[X_2|X=10]$ +9828,$R_t(t)$ +9829,$P^T_S(\cdot\mid\cdot)$ +9830,$M_X(t)$ +9831,$-2<\alpha<0$ +9832,$S(x)=(1+x)^{-\alpha}$ +9833,$\rho^\star(X) = \rho(A) + \rho(B)$ +9834,$PV=\mathrm{PV}_r(C)+\pi$ +9835,$\mathsf E[F_i]$ +9836,$\mathbb{Q}$ +9837,"$f_0, f_{1/2}, f_1$" +9838,$e^{-\lambda\nu}$ +9839,$\dfrac{X \wedge a}{X}$ +9840,$g(s) \ge h(s)$ +9841,$\rho(X)\ge\rho(X+Y)\ge \rho(X)+\mathsf{E}[YZ]$ +9842,$\frac{m^2}{p}(1+\frac{m}{p})$ +9843,$\alpha_i'(x)>0$ +9844,$\rho(Y)\ge \rho_{m_0}(Y)\ge a_{m_0}$ +9845,"$100,000 per-occurrence deductible and a \$" +9846,$\partial P/\partial x_i$ +9847,"$v=0,\dots,538$" +9848,"$Q_{i,j} = M_{i,j}/\iota_j$" +9849,$\bar R$ +9850,$\mathsf{E}_\mathsf{P}$ +9851,$g(S(x)) = S(x) + \delta(F(x))F(x)$ +9852,$u'''>0$ +9853,$U:\mathcal{S}\to \mathbb{R}$ +9854,$wrd$ +9855,$\alpha(\mathsf Q)$ +9856,$X=X_1+ (X-X_1)$ +9857,${}^{[<280]}$ +9858,$Z_j$ +9859,$\int_1^\infty x^{-2}dx=1$ +9860,"$[0.37, 0.55]$" +9861,$\mathsf{E}[X]=3$ +9862,$\propto$ +9863,$\mathsf{Pr}r(A\cup B)=\mathsf{Pr}r(A)+\mathsf{Pr}r(B)$ +9864,$\sigma=0.15$ +9865,$1_{X < a}$ +9866,$\mathsf{Pr}hi^{-1}$ +9867,$g'(0)>1$ +9868,$r'$ +9869,$(\mathsf{E}_q(X_1)-s)/\mathsf{E}_q(X_1)$ +9870,$\mathbf {t+1}$ +9871,$U_X = F(X-) + V(F(X) - F(X-))$ +9872,"$, which he describes as the standard way to obtain the $" +9873,$=\mathsf{E}(X_i/X \mid X > a)$ +9874,$X_i(\alpha)$ +9875,$N=1$ +9876,$\alpha_i'(x)=0$ +9877,$1/9=0.11\dot 1$ +9878,$\mathsf{E}[e^{sX_1}]=\mathsf{E}[e^{sX_{1/n}}]^n$ +9879,$p_1\not= p_2\le p^*$ +9880,$Y_\lambda$ +9881,$\sigma_t$ +9882,"$\psi=1_\mathscr{G}amma(S,T)$" +9883,$\tpx=\exp(-\int_0^t \mu_{x+s}ds)$ +9884,$\sigma_1=1$ +9885,$\mathsf{E} X+\lambda\sigma(X)$ +9886,"$b_{X,r}(Y)$" +9887,$S(x)=0.1$ +9888,$d=(\log(a/S_0)-(r-\sigma^2/2)t)/\sigma\sqrt{t}$ +9889,$g(\mathsf{Pr}(H))=0.71=g(\mathsf{Pr}(T))$ +9890,$r'=r+v(m_1)$ +9891,$\frac{1}{2}1_\mathscr{G}amma(\omega)$ +9892,$v$ +9893,$\{X>q_X(p) \}$ +9894,$\bar P_i = \mathsf{E}[\kappa_i(X)g'(S(X))]$ +9895,$\bar P_d=\mathsf{E}[Y_{d}]+\lambda\sigma(Y_{d})$ +9896,$s>s^\ast$ +9897,$Z_1=Z_2$ +9898,$\delta=\iota\nu$ +9899,$\mathsf{E}[X_2]$ +9900,$\Lambda \mu_{U} = \ \frac{\mathsf{E}[ r_{U} ] - r_{f}}{\sigma_{r_{U}}} \frac{\mu_{U}}{\sigma_{U}}$ +9901,"$\mathsf{TI,\ MON}$" +9902,$P=\nu L + \delta a$ +9903,$E[X]$ +9904,$-1.345$ +9905,"$(\mathsf{Pr}_S,\sigma(T))$" +9906,$g''(s)\le 0$ +9907,$bY$ +9908,"$P_c, P_n$" +9909,$g(s) =$ +9910,$\mathsf{E}_{\mathsf Q}[X]=\mathsf{E}[XZ]$ +9911,$\mathsf{E}_{\mathsf{Q}'}[X]<\rho(X)$ +9912,$\mathsf{E}(X_i \mid X=x)$ +9913,$\lambda_x\uparrow \infty$ +9914,$\bar\zeta_i^2 = \zeta_{i+1}^2+\cdots+\zeta_T^2$ +9915,$\mathsf{Pr}r(X a]$ +9932,$a(x)=xa(1)$ +9933,$-\partial g(S(x))/\partial x$ +9934,$U_X$ +9935,$x_2=0$ +9936,$F(b)-F(a)$ +9937,"$\bar P_i = \sum_{j} X_{i,j}\Delta g(S_j)$" +9938,$\delta \ge 0$ +9939,"$\mathcal{M} = \{ \zeta \mid \|\zeta\|_q\le c, \zeta\ge 0 \}$" +9940,$0\le p_1 \le p^* \le p_2\le 1$ +9941,$g(s)=1$ +9942,$P=\rho_{Wang}(X)$ +9943,$A=\{X(\omega) > x\}$ +9944,$\sigma_d = \mu_d/5$ +9945,$Z(y_j)$ +9946,"$[x-\delta, x+\delta]$" +9947,$\rho_{\alpha m_1 + (1-\alpha m_2)}(X) = \alpha\rho_{m_1}(X) + (1-\alpha)\rho_{m_2}(X)$ +9948,$\forall\omega\in\Omega$ +9949,$100$ +9950,$C_k$ +9951,$\mathsf P(X=\max(X))=0$ +9952,$\frac{p_n(a)}{n!}$ +9953,"$\mathcal{M}_{X,r_X}=\{m \in\mathcal{M} \mid \rho_m(X) = r_X \}$" +9954,$X_{-1}$ +9955,$-\rho(X)\le \rho(-X)$ +9956,"$f(x, \cdot)\in L_p(\Omega, \mathcal{F}, \mathcal{P})$" +9957,$\pi'(\sqrt k)=0$ +9958,$m_i\ge0$ +9959,"$Y_{0,0}:=\sum_{d>0} X_{0,d}$" +9960,$i>0$ +9961,$P_t\{T\not=t\}=0$ +9962,$M=\Omega$ +9963,$Z_c(3900)$ +9964,$g_0=0$ +9965,$\rho_c(X)=\mathsf{E}[X]+c\sigma(X)$ +9966,$X_k^2$ +9967,$a=0.75$ +9968,$\rho(X_n(t))$ +9969,"$\mathsf{Pr}r(\mathsf{CP}_n\le x)\to\mathsf{Pr}r(\mathsf{CP}(\lambda, X)\le x)$" +9970,$\mathsf{Var}(\pi)=\bar p/(\nu_p-l_p)^2$ +9971,$\pi_h^R$ +9972,$q_X(p)$ +9973,$X_i(a)=\dfrac{X_i}{X}(X\wedge a)$ +9974,$y\mapsto P_y(E)$ +9975,$s^\star$ +9976,$x_0 < \mathsf{TVaR}_{p_0}$ +9977,$g^1(s_{R+1})$ +9978,"$\mathsf{CP}_2(\lambda, (\mu/\lambda)X)$" +9979,$\beta_i(x)=\mathsf{E}_\mathsf{Q}\left[ \dfrac{X_i}{X}\mid X > x\right]$ +9980,$\lambda\rho(X)$ +9981,$r=50$ +9982,"$^{\,3}$" +9983,$t_* q(p)]$ +9999,"$\mathscr{G}=\sigma(A_1,\dots,A_r)$" +10000,$\rho_e^s$ +10001,$X_k - X_{k-1}$ +10002,$e^{x}$ +10003,$\mathsf{E}[Z_i\mid X] \ne \mathsf{E}[Z_j \mid X]$ +10004,$B'=\bigcup_i G_i'$ +10005,$X\_2$ +10006,$F(x)=u$ +10007,$x<\alpha$ +10008,$m(A)$ +10009,$\sigma(T)$ +10010,$F(x):=\mathsf{P}(X\le x)$ +10011,$R_1(t)$ +10012,$\mathsf E[X] = \mathsf E[Y]$ +10013,$g(0.001)$ +10014,$\nu = 1/\lambda$ +10015,"$g(0)=0, g(1)=1$" +10016,"$\mathrm{inf}\,S=0$" +10017,$\eta\ge$ +10018,"$1,2,3,\dots$" +10019,$\mathsf{Pr}(A_i)>0$ +10020,$n\times 12$ +10021,$b=0.53$ +10022,$c\approx -\sigma^2u''(w)/u'(w)$ +10023,$M-N$ +10024,$1/4 < s\le 1$ +10025,"$4,000 | \$" +10026,$\bar Q(a)$ +10027,$\bar P_i$ +10028,"$R_{X,r}$" +10029,$t_0^*=0.296$ +10030,"$\mathsf{E}[XZ] = \mathsf{cov}(X,Z) \le \sigma(X)\sigma(Z)\le \sigma(X)$" +10031,$\nu=1/(1+\rho)$ +10032,$ is not differentiable at 0 because it has a cusp). In the case $ +10033,"$\{1,2 \}$" +10034,$\rho^u(s)$ +10035,$c\le 1$ +10036,$A_Y = 2.155$ +10037,$X=U+D$ +10038,$[1;1]$ +10039,"$X_1=(0,0,0,0,0,0,2,4,8,0)$" +10040,$ for $ +10041,$g(s)(1-q)$ +10042,$g'(s) = rs^{r-1}$ +10043,$L_a^{a+y}$ +10044,$-kρ(-X)$ +10045,$g\in \mathcal{W}$ +10046,$X(\lambda\mathbf{v})$ +10047,$\mathsf{E}_{\mathsf{Q}}[X]=\infty$ +10048,$\mu(U)>1/k$ +10049,$\beta_i(t)/\alpha_i(t)> 1 > g(S(t)) / S(t)$ +10050,$R = P - U = 0$ +10051,$S_{X_i}$ +10052,"$Y_{1,1}$" +10053,$CV(G) = SD(G') = \nu$ +10054,$\lim_{\epsilon \downarrow 0} (f(x-\epsilon)-f(x))/\epsilon$ +10055,$\{\mathscr{F}_t\}_{t \geq 0}$ +10056,$1200/1800=0.667$ +10057,$1/\nu$ +10058,$\mathsf E[\bar s]$ +10059,$E(X^k)=E(Y^k)$ +10060,$a=a(\mathbf{x})$ +10061,$1=\mathsf Q(\Omega)\not=\sum_n \mathsf Q(\{n\})=0$ +10062,$\mathsf P_tU$ +10063,$b > 0$ +10064,"$3,000 | \$" +10065,$R_S$ +10066,$q(p)=\inf\{x \mid F(x)\ge p \}$ +10067,$x_{i+1}$ +10068,$\gamma = P/a$ +10069,$a=f=1$ +10070,$E\cap F=(E\cup F) \setminus (E\triangle F)$ +10071,$AH$ +10072,$\bar P_i(x)$ +10073,$\zeta_t\to\zeta$ +10074,$\frac{1}{2}mv^2 = \frac{3}{2}kT$ +10075,$\mathsf{E}[Xe^{\pi Z}]/\mathsf{E}[e^{\pi Z}]$ +10076,$(1 + \mu^2)^{3/2}$ +10077,$T_{R-1}$ +10078,$g_k(s)=1-(1-s)^k$ +10079,$S_{\tilde X}$ +10080,$1=P+Q=g+h=A+B$ +10081,"$n=1,p=0.8$" +10082,$F_X(x)\ge F_Y(x)$ +10083,$I_1 =t^ae^{ia\pi/2}\mathscr{G}amma(-a)$ +10084,$w_i$ +10085,$\mathsf{E}(X_i \mid X \ge a)$ +10086,$V(m)=m$ +10087,$W=W_t+W'_t$ +10088,"$\mathcal F_1=\sigma(I_1,\dots,I_n)$" +10089,$F(x)=p$ +10090,$\hat S(x)$ +10091,$R_2(t) > C_2(t)$ +10092,$(1-m)(1+\frac{1-m}{a})$ +10093,$\int_{k/n}^{(k+1)/n} j(x)dx$ +10094,$\mu(\{p_1\})=w$ +10095,$\bar P_1 + \bar P_2=\rho(X)$ +10096,$\theta=0$ +10097,$\mathsf Q_k$ +10098,$\nu F(a)$ +10099,$0\le \mathsf{Pr}r(E)\le 1$ +10100,"$\mathsf{MON,TI,PH,SA}$" +10101,$\bar P_{act}$ +10102,$R_0(t) a_1] \ge ta_1$ +10146,$\psi(0)=1-\mathsf{Pr}r(Y=0)=1-\mathsf{Pr}r(M=0)=\frac{1}{1+r}$ +10147,$Le^{-\delta n}1_{T_x\ge n}$ +10148,$p>1$ +10149,$P(X_{-1}(a))=\bar P^a_0$ +10150,$0.5 < t_0^* < t_1^*$ +10151,$b_i$ +10152,$g(s)=A(1_{U < s})$ +10153,$X=\sum_i X^i$ +10154,"$I=0, \dots, T-1$" +10155,$f(y;\theta=$ +10156,$\{ \omega \mid \mathsf{Pr}(B\mid\mathscr{G})(\omega) < 0 \}\in\mathscr{G}$ +10157,$P=\mathsf{E}_q[.]$ +10158,$P - L = R + S$ +10159,$q<0$ +10160,$\sigma=1.5$ +10161,$(1-s)^{-1/2}/4$ +10162,"$0\le k, h < n$" +10163,"$(p,q(1-g^{-1}(1-p)))$" +10164,$\rho_i(X)$ +10165,$ρ(X)$ +10166,$n=1000$ +10167,$\theta=\log(\mu)$ +10168,"$(1,2)$" +10169,$\mathsf{E}(X) + c\mathsf{E}(| X-\mathsf{E}(X) |^p)^{1/p}$ +10170,"$\mathcal X^\perp = \{X\in\mathcal X\mid \exists U\text{ uniform[0,1] rv independent of } X\}$" +10171,$\mathsf{E}_\mathsf{Q}\left[\dfrac{X_i}{X}(X\wedge a)\right] + \tau a \mathsf{E}_\mathsf{Q}[X_i/X\mid X > a]$ +10172,$y-\mu$ +10173,$\kappa(\theta)=-\log(-\mu)$ +10174,$\mathsf{TVaR}_{p_2}(X)\ge r$ +10175,$(\delta(t-1) + \delta(t+1))/2$ +10176,$\phi\mu \ll \nu$ +10177,$\rho(X)\le c$ +10178,$t=2$ +10179,$Q\in \partial\rho(X)$ +10180,$B_2 \succ A_2$ +10181,$\kappa(\theta)=n\log(1+e^\theta)$ +10182,$x^{-\alpha}$ +10183,"$\mathsf{CP}_n:=\mathsf{CP}(J_n(0), X_n)$" +10184,$1+r^*=(1+r)(1+\tau)$ +10185,$\mathsf{Pr}r(X\ge q(p))>1-p$ +10186,$\rho_{m'}(X)=r$ +10187,"$3,000,000,000), regardless of the frequency or severity of earthquake losses at any and all times subsequent to the creation of the authority. Once a participating insurer has paid, pursuant to this section, amounts equal to the percentage share of the authoritys total gross written premium attributable to that participating insurers sales of authority insurance policies, as of April 30 of the immediately preceding year or the most recent full year for which premium data not more than one year old are available, multiplied by three billion dollars ($" +10188,$\mu_*$ +10189,"$(B, Km)$" +10190,$F(t)$ +10191,$q_{X_i}(p)=\mathsf{Pr}hi^{-1}(p)$ +10192,"$\mathbf {X'\,\Delta g(S)}$" +10193,$d=(\log(a/S_0)-(r-\sigma^2/2)T)/\sigma\sqrt{T}$ +10194,$t=T$ +10195,$q^-_X(0.95)$ +10196,$\zeta$ +10197,$a=kP+Q$ +10198,$\mathsf{E}[X\mid T]$ +10199,$p_{i^*}\le p^* \le p_{i^*+1}$ +10200,$g'(s)\ge 1$ +10201,$t_*=0.296$ +10202,$\rho(X)=\mathsf{E}[X] + c\mathsf{Var}(X)$ +10203,$(y-m)^2$ +10204,"$(Bob) + (0,-4)$" +10205,$1 assets: $ +10206,"$\alpha\in (0, 2]$" +10207,$D\rho_X(X_i) \ge \mathsf{E}[X_i]$ +10208,$S_t(a(t))=1-p$ +10209,$F_X^{-1}(V)=q_X(V)$ +10210,$\rho GF$ +10211,$D_m\subset D_n$ +10212,$q^+$ +10213,$\Delta X_j=X_{j+1} - X_j$ +10214,"$\delta_{p_i},\delta_{p_j}$" +10215,$k_1 u \cdot [s_1(k_1 - k_0) + k_0 v]$ +10216,$\{X=x\}$ +10217,$g(p)=1-\tilde p(1-p)=1-\tilde F(\mathsf{Pr}hi^{-1}(1-p))=1-\mathsf{Pr}hi(-\mathsf{Pr}hi^{-1}(p)-\lambda)=\mathsf{Pr}hi(\mathsf{Pr}hi^{-1}(p)+\lambda)$ +10218,$L_x^{x+dx}=L_0^{x+dx} - L_0^x$ +10219,$g(S_{X}(x))$ +10220,$\mathsf{Pr}r(B\le t) = 1/2 + 1_{t>1/2}(1/2)$ +10221,$\iota = \dfrac{g(s)-s}{1-g(s)}$ +10222,$Y\circ T=g(X\circ T)$ +10223,$\mathsf{TVaR}_1$ +10224,$R_\lambda$ +10225,$q^+(p)=\sup\ \{ x\mid \mathsf{Pr}r(X < x) \le p \}$ +10226,$s(j) = (1+M-j)/M$ +10227,$x_1\leftrightarrow y_1$ +10228,$\mathsf{E}[X_i/X|X>a]$ +10229,$\bar Q_{act} = \bar Q - F_0$ +10230,$F(x)=1-s$ +10231,$\phi(p)=g'(1-p)\ge 0$ +10232,$\mathsf{Pr}r(Z>\mathsf{E}(Z))$ +10233,$\mathsf{E} X +\lambda_1 {(X-\lambda_2 \mathsf{E} X)^+}_1$ +10234,"$(fun1a.south -| fun5a.east)+(\smlspc,-\smlspc)$" +10235,$\mathsf{P}(\omega)$ +10236,$X\wedge a \le X$ +10237,$X=X(x_i)=\sum_i X_i(x_i)$ +10238,$\rho(X+c)=\rho(X) + c$ +10239,$h\in\mathscr P$ +10240,$a'$ +10241,$c(1)$ +10242,"$135,000 or more, no less than $" +10243,"$\int Zd\mathsf P = \int d\mathsf Q/d\mathsf P\, d\mathsf P = \int d\mathsf Q =1$" +10244,$q=S(a)$ +10245,$\displaystyle\int_\Omega X(\omega)\mathsf{Pr}r^*(d\omega)$ +10246,$A\in\mathscr{G}_Y$ +10247,$\rho(0)=0$ +10248,$Q_\epsilon$ +10249,$\text{AEP}(y\text{ year aggregate PML})=1/y$ +10250,$\theta=-\dfrac{1}{\mu}$ +10251,"$\bar P^a(1,0)<\bar P^a(0,1)$" +10252,$\mathsf{Pr}r(M=m)=\frac{r}{1+r}\frac{1}{(1+r)^m}$ +10253,$r=4$ +10254,$p\delta(p)$ +10255,$Q(a)=1-g(S(a))$ +10256,$10^6$ +10257,$f<1$ +10258,$\mathsf{Pr}r(\cup_i E_i)=\sum_i \mathsf{Pr}r(E_i)$ +10259,"$Z_1,\dots, Z_t$" +10260,$\bar M = \bar P_i - \bar S_i$ +10261,$\mathcal N_X(X_i)$ +10262,$R:=R_0 + R_1 + \cdots R_{T-1}$ +10263,"$=(\text{Corr}(y, \hat y))^2$" +10264,$(1-\alpha)\phi_m^e + \alpha\phi_m^o$ +10265,"$\rho(Y)\le b_{X,q}(Y)$" +10266,$tx_1/x_t$ +10267,"$B\mapsto p(t, B)$" +10268,$2\left( y\log\frac{y}{m} + (1-y)\log\frac{1+m}{1+y} \right)$ +10269,$\alpha f$ +10270,"$a_i=\mathsf{E}[X_i] + k\mathsf{cov}(X_i, X)$" +10271,"$((t_1,t_2),(t_3,t_4))$" +10272,$1 excess attachment $ +10273,$T(p)=q(p)$ +10274,$v = \omega r$ +10275,"$x=0.1, M=1.5,\sigma=0.75, K=6$" +10276,$1 - T$ +10277,$t>t_2$ +10278,"$g(s) = \min(1, s + d(1-s) + (\delta^{\star} - d) \sqrt {s(1-s)})$" +10279,$\frac{\partial f}{\partial \mu}=\frac{\partial f}{\partial \mu}\frac{f}{f} = \frac{\partial l}{\partial \mu}f$ +10280,$(3+2)/2=5/2$ +10281,$0!=1$ +10282,$g(uv)$ +10283,$\displaystyle\int_0^\infty xg'(1-F(x))f(x)dx = -xg(S(x))\vert_0^\infty + \displaystyle\int_0^\infty g(S(x))dx=\displaystyle\int_0^\infty g(S(x))dx$ +10284,$kN$ +10285,$k=1$ +10286,$\mathit{EGL}_{ro}(a)=P(X_{-1}\wedge a) - P(X_{-1}\wedge a_{ro}) \ge 0$ +10287,$F(x)$ +10288,$(l-X)^+$ +10289,$s=1/4$ +10290,$e^{\mu_L}-1$ +10291,$\mu^*(M\cap E)=\mu(E)$ +10292,"$h^i = \lim_{\epsilon\downarrow 0}(h_{i,\epsilon}-h_0)/\epsilon$" +10293,$F_t(x) = \mathsf{P}(X \le x \mid \mathscr F_t)$ +10294,$t^*-\epsilon$ +10295,$\sum_j x_j L_{ij} - \zeta - \eta_i\le 1$ +10296,$\iota(s)$ +10297,$\lambda_t \Omega$ +10298,$\int c(y)dy\neq 1$ +10299,"$\rho\in\mathcal R^h_{X,r}$" +10300,$q_A \le q_B$ +10301,$\rho(X)=0$ +10302,$\mathsf{CTE}_p <= \mathsf{CTE}^+_p <= \mathsf{TVaR}_p$ +10303,$R_0(t)>(1-t)P(0)$ +10304,"$\Omega:=\tau(\mathrm{int}\,\Theta)$" +10305,$g_j$ +10306,$B-p(\nu(p) + il(p))$ +10307,$F_1$ +10308,$\lambda X + (1-\lambda)Y$ +10309,$\mathsf Q \in \mathcal Q$ +10310,$0\le p_00.99]$ +10320,"$(\Omega, \mathcal{F}, \mathbb{P})$" +10321,$ of $ +10322,"$\mathsf{Pr}^x f(x, Tx) = \mu^tP^x_t\,f(x,t)$" +10323,$g(st) = 1= g(s)g(t)$ +10324,"$\{h_j = h(s_j): j=L,\dots,m\}$" +10325,$p=F(a)=1-q$ +10326,$R_M$ +10327,$s^{1/2}$ +10328,$P_0=\Delta X_0$ +10329,$\mathsf{var}phi(X+m) = \mathsf{var}phi(X) + m$ +10330,$AVaR_\lambda(X) = \max_{\mathcal{B}bb Q\in {\cal Q}_\lambda} E_{\mathcal{B}bb Q}(-X)$ +10331,$(\alpha+1)/\alpha=1/(2-p)$ +10332,$\rho_{t+1}(X) = \rho_{t+1}(Y) \implies \rho_{t}(X) = \rho_{t}(Y)$ +10333,$P=a - \nu(a-L)$ +10334,$d(y;\mu) = 2\left( \dfrac{(y^+)^{2-p}}{(1-p)(2-p)} - \right.$ +10335,"$U:[0,1]\to[0,1]$" +10336,$\mathsf{E}_{\mathsf Q}$ +10337,$ since $ +10338,$(1 - \nu F(a)) da$ +10339,$0 \le X_i(a) \le X_i$ +10340,$r=0.086$ +10341,$p(x)$ +10342,$H(x)=y$ +10343,$h(0.05) = 1-g(1-0.05) = 0.0203$ +10344,$d(y;\mu) = y\log\left(\dfrac{y}{\mu}\right) - (y-\mu)$ +10345,$\lambda\to\infty$ +10346,$\lambda=\mathsf{E}[X]<1$ +10347,$\tilde p_i = 1-\sqrt{1-\bar p_i}$ +10348,$D_2$ +10349,$\displaystyle\int$ +10350,$l(y;\mu)=y\tau^{-1}(\mu)-\kappa(\tau^{-1}(\mu))$ +10351,$E=\Omega$ +10352,$e'=0.24$ +10353,$\Delta X_j$ +10354,$R_i(t)$ +10355,"$\mathsf{TI,MON,CX}$" +10356,$I=P_{act}-\mathsf{E}[U]$ +10357,$\mathsf{Pr}r(X=x)=0$ +10358,$\le 1/(1-\alpha)$ +10359,$< 10^{-22}$ +10360,$A0$ +10363,"$g, p, A=g^a, m$" +10364,$\bar{\mathbf M}$ +10365,$\delta$ +10366,$p=10^{-6}$ +10367,"$w_e,w_o$" +10368,$C_1(t)=C_2(t)$ +10369,$\rho(\cdot\mid\mathscr F_1)$ +10370,$\sigma(W)$ +10371,$X_{T_x}=x$ +10372,$\delta A$ +10373,$E[X_{n+1} \mid \mathcal{F}_n] \leq X_n$ +10374,$Z_\epsilon$ +10375,$\nabla \times (\mathbf{v} \times \mathbf{B})$ +10376,$g^o_h$ +10377,${}^{[<4]}$ +10378,$\{ Z\mid \rho(X)=\mathsf{E}[XZ] \}$ +10379,"$v_A,v_E$" +10380,$\rho(X)=\rho(\mathsf{E}[X]+X-\mathsf{E}[X])=\mathsf{E}[X] + \rho(X-\mathsf{E}[X])$ +10381,$g(S(x))\to d$ +10382,$0\le \omega\le 1$ +10383,$F_n(x)\to F(x)$ +10384,$l$ +10385,$s'(t)$ +10386,$fT$ +10387,$\zeta\in\mathscr{P}$ +10388,$s=s_m=1$ +10389,$\rho$ +10390,$k_1>1 \implies k_1^2>k_1$ +10391,$\alpha_{2}$ +10392,$c(y; \lambda)$ +10393,$s=S(a)$ +10394,$X=20$ +10395,"$\tau(a-\rho_{a,\tau}(X))$" +10396,$b_1 = r s_y/s_x$ +10397,$\theta=\lambda\nu$ +10398,$0 \le p_i \le p^\star \le p_j \le 1$ +10399,$\mathsf{E}_{\mathsf{Q}}[\tilde X-X] \le \rho(\tilde X-X)$ +10400,$-A(-X)$ +10401,$D_i-N_i$ +10402,$\alpha_i(x) S(x)$ +10403,$e^{\theta y}>0$ +10404,$y=x$ +10405,$\mathsf{VaR}_{0.75}(X)=90$ +10406,$X_n$ +10407,$d^\ast = 1-(1-g^\ast)/(1-s^\ast)$ +10408,$=14\times 16+2=226$ +10409,$s_j < 1$ +10410,"$EL_i = E[X_i] = \int x_i f(x)\,dx$" +10411,"$r(y,x_j\mid \text{others})=t(b_j) / \sqrt{t(b_j)^2+n-(k+1)}$" +10412,$tR_1(t)>P(1)$ +10413,$\mathsf{E}(X\wedge a)=\int_0^a S(x)dx$ +10414,"$\int f\,d\mu$" +10415,$\bar\delta=\bar\iota\bar\nu$ +10416,$\tilde Q$ +10417,"$1,461.71 \$" +10418,"$a_{0,0}:=a(Y_{0,0})$" +10419,$a_2>a_1$ +10420,$r(s)$ +10421,$L_{250}^{\infty}$ +10422,"$E[Y\,dG/dF]$" +10423,$\hat\rho(A)\ge \rho(A)$ +10424,$t=b$ +10425,"$x=4, M=1.5,\sigma=0.75, K=6$" +10426,$\nu(n)$ +10427,$Y'$ +10428,$\mathsf{Pr}r(Agg > x) \approx \text{frequency}\times \mathsf{Pr}r(Occ > x)$ +10429,$AW$ +10430,$m^2(m-1)$ +10431,$\hat p_s(\theta)$ +10432,$\mathcal{B}\subset\mathscr{F}$ +10433,$S_j:=S(X_j)$ +10434,$\exp(\sigma \sqrt tZ)=\exp(\sigma^2t/2)$ +10435,"$[s_0, s_1]$" +10436,$\phi_t = \Delta_t+\phi_s$ +10437,$B=\{p_\alpha\}$ +10438,$\{ X=x \}$ +10439,$C_i(t^*)=R_i(t^*)$ +10440,$T_n$ +10441,$M^i=\beta^i g(S)-\alpha^i S$ +10442,$XSSE = \infty$ +10443,$\mathsf{E}(L) = F^{-1}(p)dp$ +10444,$1_A(x)=1$ +10445,$F^-1$ +10446,$\sigma^2/n_i$ +10447,$Y=X_1+\cdots +X_N$ +10448,"$\mathbf{1} := (1,1,\dots ,1)$" +10449,$e^{\theta x}j(x)$ +10450,$S(x)=1$ +10451,$\mathsf{TVaR}_\alpha$ +10452,$P+Q=1$ +10453,$\mathsf{E}[(X\wedge a)\_1(a)]$ +10454,"$4,429.00 | $" +10455,$\rho_m(Y)$ +10456,$\mathbf{b}=(X'X)^{-1}X'y$ +10457,$\mathsf{Pr}r(L\ge l)$ +10458,$\delta_p/\nu_p = \iota_p$ +10459,$t-1$ +10460,$ct$ +10461,$P=vL + da$ +10462,$\delta=\iota/(1+\iota)$ +10463,$\mathsf{Pr}(A\mid\mathscr{G})$ +10464,$t_1=s_L$ +10465,"$X,Y$" +10466,$\Delta Q_{gc}(a)$ +10467,$s_R \le s_{R+1} \le s^{\star}$ +10468,$\displaystyle\int_0^\infty xg'(S(x))f(x)dx$ +10469,"$r=0.05, 0.15, 0.25$" +10470,$t-s$ +10471,$\frac{1}{2}mv^2$ +10472,$\forall A\in\mathcal{A}$ +10473,$X:=Y$ +10474,$=\mathsf{E}_{\mathsf Q}[X]$ +10475,"$0,1,1,1,2,3, 4,8, 12, 25$" +10476,$X_0 < X_1 < \dots < X_{n'}$ +10477,$D f(x_0)$ +10478,$X^-(p)$ +10479,$\rho(X)=c$ +10480,$\rho(X_1)=\rho(X_2)$ +10481,$\phi_{R+1}$ +10482,$g(s) = sv + d$ +10483,"$^{\,4,7}$" +10484,$0\le\beta\le \gamma\le 1$ +10485,$\tilde S(x)=g(S(x))$ +10486,$T(U)$ +10487,"$(s(t),m(t))$" +10488,$\wedge$ +10489,$a_1=a(W_1)$ +10490,"$(\mu,\sigma^2)$" +10491,"$\mathsf{biTVaR}_{0,1}^{0.0476}$" +10492,$\mathsf{Pr}hi'(z)=\phi(z)$ +10493,$R_j$ +10494,$\tilde Z = \mathsf{E}[Z\mid X]$ +10495,$\bar P_{2}$ +10496,$S(x) > 1-p$ +10497,"$\Omega=[0,1]\times [0,1]$" +10498,$1-p=0.9$ +10499,$\bar P_x$ +10500,"$m_0, s_1, m_1, s_2, m_2$" +10501,"$\mathsf{E}[X_i(a)\,g'(S_{X\wedge a}(X\wedge a))]$" +10502,$s = S(x)$ +10503,$P_{i}(a) =\beta_i(a) g(S(a))$ +10504,"$X \in \{0, 22, 28, 36, 40, 55, 65, 100\}$" +10505,$2.38 \times 10^{24}$ +10506,$\mathsf{E}[X_\theta]=\kappa'(\theta)=\tau(\theta)=\mu$ +10507,$\mu\in \mathscr{P}$ +10508,$p_i < p_j \le p^\star$ +10509,$P_y$ +10510,$p=0. $ +10511,$W(\cdot)$ +10512,$X_0=x_0$ +10513,$X = 1$ +10514,$g(1-t)^2=(1-kt)^2=1-2kt+k^2t^2$ +10515,"$[0,1]\to [0,1]\times [0,1]$" +10516,"$H:(\Omega,\mathscr{G})\to(\mathbb{R},\mathcal{B}B(\mathbb{R}))$" +10517,$j(x)$ +10518,"$\bar Q_{0,t}:=a_{0,t}-\bar P_{0,t}$" +10519,$v<1$ +10520,"$n=1,\dots,6$" +10521,"$g(s)=w+(1-w)s, s>0$" +10522,$\lambda_{obj}$ +10523,$ at $ +10524,$\mathsf{TVaR}_p = 20(0.55x_{67}+x_{68}+x_{69}+x_{70})/71$ +10525,"$\mathsf{E}[X_{i,2}(a)]$" +10526,$w_t=\zeta_1z_1+\dots +\zeta_tz_t$ +10527,$t \geq T(\omega)$ +10528,$1/(1+r_f)$ +10529,$\mathsf{TVaR}_1(X)$ +10530,$2K + U = 0$ +10531,$\le a$ +10532,$\mathsf{E}(X)=$ +10533,$6.3\times 10^{11}$ +10534,$V_X$ +10535,$x=q(1-g^{-1}(1-\tilde p))$ +10536,$1 - \alpha$ +10537,$m =$ +10538,$=\mathsf{E}[X_i(a)]$ +10539,"$k, b$" +10540,$\rho(-1_{B_l}) \le \rho(-1_{B_r})$ +10541,"$\sigma(L^\infty, L^1)$" +10542,"$[l_i, r_i)$" +10543,$v(A)=\lambda(\pi_1(A))$ +10544,$Pr(X<1.5)={p:.3f}$ +10545,$M_{X+Y}(t)=\mathsf{E}[e^{t(X+Y)}]=\mathsf{E}[e^{tX}e^{tY}]=\mathsf{E}[e^{tX}]\mathsf{E}[e^{tY}]=M_X(t)M_Y(t)$ +10546,"$, if $" +10547,$\beta_2>\alpha_2$ +10548,$\mathsf{XTVaR}_p(X)$ +10549,"$L=L(e,t)$" +10550,$\rho^E$ +10551,$a_0+a_1$ +10552,$10X$ +10553,$q_B(p)=\sup B$ +10554,"$30,000 per accident up to $" +10555,$C > cx/a$ +10556,$\gamma_i(x)$ +10557,$\phi_W(a)=\mathsf{E}(W/Y \mid Y>a)$ +10558,$X^i(\epsilon)=(1+\epsilon) X^i(0)$ +10559,$|X_\alpha|$ +10560,$m_1$ +10561,$ar_0$ +10562,$\mathsf{j}(a) = \max\{j:X_j < a \}$ +10563,$\mathsf{E}[X]+k\mathsf{Var}(X)=a(X)$ +10564,$Nk$ +10565,$\hat q(p)=x$ +10566,$10^{-5}$ +10567,$q=10$ +10568,$1/8.6$ +10569,$\mathsf{Pr}r(V\ge v)$ +10570,$0\le \lambda \le 1$ +10571,"$50.00) of the amount allowed on each property, casualty or fidelity claim in the classes under Subsections B through F of this section, shall be deducted from the claim and included in the class under Subsection I of this section. Claims may not be cumulated by assignment to avoid application of the fifty dollar ($" +10572,$X_1(u_1)$ +10573,$s(t)$ +10574,"$X_1(x_1), \dots, X_n(x_n)$" +10575,"$700,000 (\$" +10576,$\mathsf Q(X>a)/P_X(X>a)=g(S(a))/S(a)$ +10577,$R_a$ +10578,$a\to 2$ +10579,$ into aggregate premiums $ +10580,"$u_0,u_1,\dots,u_k$" +10581,$-\alpha$ +10582,"$S(1-t,t;x)$" +10583,$\alpha_i(x) = \mathsf{E}[X_i /X \mid X> x]\not=\mathsf{E}[X_i\mid X> x]/\mathsf{E}[X\mid X>x]$ +10584,$n-2$ +10585,$\mathsf{E}[X_i(1) \mid X(\mathbf{v}) = q_{\mathbf{v}}(p) ]$ +10586,$\mathsf{E}[W\tilde X] \le \rho(\tilde X)$ +10587,"$\partial d/\partial\mu=-2\,\partial l/\partial \mu$" +10588,$\mathsf{E}[X\mid \mathcal F']$ +10589,$\beta^3 g(S)$ +10590,$R = g^k \pmod{p}$ +10591,$\bar P_t = \rho(Y_{t})$ +10592,$(\sum_i \lambda_i)(\kappa(t+\theta) - \kappa(\theta))$ +10593,$X_j^2$ +10594,$p_iq}$ +10597,$Q(a)=h(F(a))$ +10598,"$\mathbf {D^f\rho_{X\wedge 30,X}(X_1)}$" +10599,$g'(1)=\phi(0)$ +10600,$\sigma_2=\sin(\pi\theta/2)$ +10601,$z_i < \zeta$ +10602,$\rho(X)=\mathsf{E}_\mathsf{Q}(X)$ +10603,$1_D$ +10604,$\displaystyle\int X(\omega)d\omega$ +10605,"$T_{700,100}$" +10606,$2.576\times 6.258$ +10607,$t=q-s$ +10608,$\phi'(p)=f(p)/(1-p)\ge 0$ +10609,$X \le_{cx} Y$ +10610,$A_x=\bigcap \{A\in\mathscr{F}\mid x\in A\}\in\mathscr{F}$ +10611,$\mathcal R'\subset \mathcal R^c$ +10612,$1/\sqrt{x}$ +10613,$\bar Q=a-\bar P$ +10614,$\rho(X)\le\liminf \rho(X_n)$ +10615,$(X−x^∗)I_{B_i}$ +10616,"$(s^*, g(s^*))$" +10617,$S(a)=d\bar S(a)/da$ +10618,$\mathsf{E}[X_m\mid X_{m+n}=x]=mx/(m+n)$ +10619,$g_j<1$ +10620,$g(1-p)=1$ +10621,"$278,755,325 | 33.2% | $" +10622,$2^{-(i+j)}$ +10623,$\mathbf {X_2}$ +10624,$\mathsf{TVaR}_{p^\ast}(X)=\bar P$ +10625,$X>0$ +10626,$A\in\mathscr{G}$ +10627,"$b \in_{R} \{2,\dots,p-2\}$" +10628,$\mathcal R^c\subset \mathcal R^h$ +10629,$\rho_r$ +10630,"$(Bob)+(0,-2.5)$" +10631,$X\ge x_p$ +10632,$\Theta\not=\{0\}$ +10633,$\mathsf{E}[X\mid \mathcal F_t](\omega)$ +10634,"$c_1+c_2=(c(1) + c(1,2) - c(2) + c(2) + c(1,2) -c(1))/2=c(1,2)$" +10635,$\mathsf{E}[X] \le \bar P \le \sup X$ +10636,$s=0.4$ +10637,"$i=1,2,\dots$" +10638,$\mathsf E[X]\rho(N) \le \rho(A)$ +10639,$S;g(S)$ +10640,"$\alpha = 1, \kappa = 0.2$" +10641,$a(t)$ +10642,$E'=X\setminus E$ +10643,$C_2$ +10644,$\beta = \displaystyle\frac{\mu^{1-p}}{(p-1)\sigma^2} = \mu /\lambda \alpha$ +10645,$\phi(1)\le 1$ +10646,"$(1-W_{i,j})\Omega^X_j + W_{i,j}\Omega^X_i = c-c_r$" +10647,$q_V(p)=0$ +10648,$\mathbf{T}^+\mathbf{r}$ +10649,$\beta = m/\alpha$ +10650,$\kappa'(\theta)=\mu$ +10651,$-1\le X_n\le 0$ +10652,$\mathsf{Pr}(A\mid\mathscr{G}) = 1_A$ +10653,$r_f/(1+r_f)$ +10654,$P_i(a)=\phi_i(a)P(a)$ +10655,$\rho(Y)=\rho_F(X)-\alpha(F)$ +10656,$m^p$ +10657,$\sin(x)$ +10658,"$(Alice) + (0,-1)$" +10659,$B\subset \Omega$ +10660,$\rho(X) = \sup_{Q\in\mathcal{Q}} \mathrm{E}_Q(X)$ +10661,$\mathcal D(X)+\mathsf{E}[X]$ +10662,$\bar\zeta_t^2$ +10663,$\mathsf{E}_\mathbb{Q}(Z \mid X)=\mathsf{E}(Z \mid X)$ +10664,$\bar P_i(a)$ +10665,$(\alpha S)'(x)=-\kappa_i(x)f(x)/x$ +10666,"$x=A,L,S$" +10667,$g(s)=x^{1/2}$ +10668,$\delta(p) F(x)=dF(x) + (\delta^*-d)\sqrt{FS}$ +10669,$c_0(y)=c(y)e^{l(y;y)}$ +10670,$k<\sup X$ +10671,$X_n\uparrow 1$ +10672,"$\int |X_n(\omega) - X(\omega)|^p\, \mathsf{P}(d\omega)\to 0$" +10673,$N=(1-\alpha)M$ +10674,$\mathsf{Pr}r(T=n)$ +10675,$X_t = \exp(B_t - t^2/2)$ +10676,$Z_2$ +10677,"$\{4,5\}$" +10678,"$a_{0,1}$" +10679,$0\lt p \lt 1$ +10680,"$\{\mathbf{X}^{}: j=1,\dots,M\}$" +10681,$\rho(X \wedge a) + \delta Q$ +10682,$A \in \mathcal{A}$ +10683,$0\le \alpha \le 1$ +10684,$\Omega=N\cup (N+1)$ +10685,"$\rho^E(t_1,t_2,\phi_n)$" +10686,$Y+Z$ +10687,$\iota = \delta/\nu$ +10688,$1=g(s)+h(p)$ +10689,$q^+(p) := \sup\ \{x \mid F(x) \le p \} = \inf\ \{ x \mid F(x) > p \}$ +10690,$\alpha\le 1$ +10691,$-Y\ge 0$ +10692,$\mathbf{B}(1)=\mathbf{P_3}$ +10693,$F^{-1}(U)$ +10694,$Cq(\alpha)$ +10752,$H(X)>-H(-Y)$ +10753,$\rho=0$ +10754,$1-\tilde p=g(1-p)=g(S(x))$ +10755,"$884,531,173 | 6.9% | $" +10756,$\mathscr G$ +10757,$\mathsf{Pr}(A)$ +10758,$f(x)\le f(y)$ +10759,$a\le (P(1+\iota)-S)/\iota$ +10760,$X(T(U))$ +10761,$X_s$ +10762,$S(x-)=1$ +10763,$P=P(a)$ +10764,"$C_{2,\cdot}$" +10765,$k>1$ +10766,$nb$ +10767,$E\setminus E\in R$ +10768,$m(1)=0$ +10769,$a_2$ +10770,$n \ll p$ +10771,$N'/\alpha a\le w$ +10772,"$X=[x;\ p] = [x_1,\dots, x_n;\ p_1,\dots, p_n]$" +10773,"$490,000 or \$" +10774,$a_{d} = \mathsf{E}[Y_{d}]+4\sigma(Y_{d})$ +10775,$(\mathsf{E}(X_i)-\mathsf{E}(X\wedge a))/\mathsf{E}(X_i)$ +10776,$dv$ +10777,$y=kx$ +10778,"$\{\mathsf{E}[X_i\,Z] \mid \rho(X)=\mathsf{E}[XZ] \}$" +10779,$\epsilon$ +10780,$Y_n$ +10781,$g(s)=s^0.3$ +10782,$X_u=X=u_1X_1 + u_2X_2$ +10783,$\alpha/\beta$ +10784,$\{X > \mathsf{VaR}_p(X)\}$ +10785,$\mu h<\infty$ +10786,$q_Y(p)=e^{q_X(p)}$ +10787,"$[\mathsf{E}[X_1], P(1)]$" +10788,$k ρ(-X)$ +10789,$f(x)<\infty$ +10790,$s_3$ +10791,$t\mapsto \tau$ +10792,$g'(S(X))$ +10793,$\bar P = \bar S + \bar R$ +10794,$\alpha_i'(x) \to 0$ +10795,$v_X = \text{Var}_{0.99}(X)$ +10796,"$n=1,2,\dots, m-1$" +10797,"$S(\mathbf{v}, x)$" +10798,$p \leq q$ +10799,$1-\hat p=g^{-1}(1-p)$ +10800,$\mathsf{E}_q(X_2)$ +10801,$P(a)da$ +10802,$Z=1$ +10803,$c(\alpha)e^{\alpha x}g(x)$ +10804,$1-S(x)=F(x)$ +10805,$=kA$ +10806,$X=\mathsf E[Y\mid X]$ +10807,$\bar L_i$ +10808,$\mathbf {X_{g}}$ +10809,"$[x, x+dx)$" +10810,$\triangleright$ +10811,$g=\mathsf{E}(G^3)=\nu^3 \mathsf{skew}(G')+3c+1$ +10812,$\mu=-\sigma^2/2$ +10813,$O(mn\log(n))$ +10814,$S_{i}(a)=\alpha_i(a) S(a)$ +10815,$\lambda_t=\lambda \mu_t$ +10816,$F(a)/\nu F(a)=1/\nu=1+\rho$ +10817,$\tilde p=g(p)$ +10818,$d<1$ +10819,"$\min(X,a)=X \wedge a$" +10820,$E(X)=\int xf(x)dx = \int xdF(x) = \int (1-F(x))dx$ +10821,$\nabla Q$ +10822,$\ge 5$ +10823,$\mathsf P(A)$ +10824,$ as $ +10825,$\lim_{s\to 1} (g(s)-s)/(1-s) = \lim_{s\to 1} 1-g'(s)$ +10826,$r_D=1-D/L$ +10827,$C_t=D_1 + \cdots + D_t$ +10828,$g(\mu)=\eta$ +10829,"$(0,0),\ (1,0),\ (1,1)$" +10830,$X ∈ L^p$ +10831,"$j=0,\dots,m=8$" +10832,$\bar S'(x)=S(x)$ +10833,$1-$ +10834,$\alpha=1.9$ +10835,$g(s)=1-\sqrt{1-s}$ +10836,$200.06 \$ +10837,"$a(X,p)$" +10838,$\beta>0$ +10839,$\mathsf{E}[X_1\mid X > a^*]$ +10840,$4l + 2$ +10841,"$1, 2, 10$" +10842,$x\in A$ +10843,"$(A.south east) + (-0.07mm,0)$" +10844,$\bar A$ +10845,$y\ge x$ +10846,$a^\star$ +10847,$u=0.1$ +10848,$V(\mu)=e^\mu$ +10849,"$38,287,000 | 34.2% | $" +10850,$\prec_n$ +10851,$\rho(X_n(t))+t\pi$ +10852,$g\leftrightarrow \rho$ +10853,$\kappa_i(X)$ +10854,$1-m\le 1$ +10855,$B\in\mathsf{E}E$ +10856,$\mathbf {\mathsf{P}(X_2)}$ +10857,$100G$ +10858,$=v_f \mathsf{E}_Q\left[\dfrac{X_i}{X}(X\wedge A)\right]$ +10859,"$\rho\in \mathcal R^h_{X,r}$" +10860,$\Omega\times\Omega$ +10861,$Z=d\mathsf{Q}/d\mathsf{P}$ +10862,"$\{p_0, p_1, \dots, p_n\}$" +10863,$\hat F(t)=\phi(-2\pi t)$ +10864,$n=3$ +10865,"$X(\epsilon^1,\dots)=\sum_i X^i(\epsilon^i)$" +10866,$\mathsf E[\phi(\bar s)]^2$ +10867,"$\ge 50,000$" +10868,$m^2$ +10869,${}^2S(t)=\mathsf{E}[(X-t)_+]$ +10870,$L_\infty\subset L_p \subset L_\sigma\subset L_1$ +10871,$Y=f(X)$ +10872,$D=1$ +10873,$Y_k$ +10874,$x_l$ +10875,$\mathsf{TVaR}_1=E[X]$ +10876,$g(\mu)=\log\mu$ +10877,"$args, dots$" +10878,"$\delta(s,t)\ge 0$" +10879,$\mathsf{E}_q(X_1)/q$ +10880,$ is constant. This NEF is regular because $ +10881,$Q(a)=d\bar Q/da = 1-P(a)$ +10882,$20+8t$ +10883,$X=(X\wedge a) + (X-a)^+$ +10884,$\mathsf{E}[X_2\mid X_1]$ +10885,$X_g$ +10886,$\pm \mathsf{Pr}hi^{-1}(22/23)= \pm1.71$ +10887,$D\rho_X(X_i)=\mathsf{E}_{\mathsf{Q}_X}[X_i]$ +10888,$\mathbf{T}^+$ +10889,$\kappa_i(q(1-g^{-1}(1-\tilde p)))$ +10890,$\mathsf{E}_\mathsf{Q}[Y \mid X] = \mathsf{E}[Y \mid X]$ +10891,$p=\text{Pr}[L^* > A]$ +10892,$s\ge t$ +10893,$1/N$ +10894,$(\delta^*-d)\int_0^a \sqrt{F(x)S(x)}dx$ +10895,$\mathit{NPV}_1 = \bar Q - \bar Q_{act} = F_0$ +10896,$P = \mathsf{E}[X] + \pi \mathsf{SD}(X)$ +10897,$v_x = (1+r_x)^{-1}$ +10898,$X = X_1 + X_2$ +10899,$c(y)=\dfrac{1}{\pi}\dfrac{1}{1+y^2}$ +10900,$v^{n-1}$ +10901,$\phi_i(a)=\mathsf{E}(X_i/Y \mid Y>a)$ +10902,$F(q^-(p))\ge p$ +10903,"$D^f\rho_{X\wedge a,X}(\cdot)$" +10904,$\rho(X)=\max_{Q\in\mathsf{Q}} \mathsf{E}_Q(X)$ +10905,$\bar P(a)=\displaystyle\int_0^a g(S(x))dx$ +10906,$s_2$ +10907,$\tan{}$ +10908,"$\mathsf{TVaR}_0=\mathrm{ess\,sup}$" +10909,$n-a_{n\!\urcorner}$ +10910,$\alpha^1 S$ +10911,$\alpha_i(k) = \mathsf E_p[\kappa_i(X)/X\mid X > X_k]$ +10912,"$1,000,000) Table MD for policies around \$" +10913,$R_1(t)\to \mathsf{E}[X_1]$ +10914,$k(\theta)=\left(\int c(y)e^{\theta y}dy\right)^{-1}$ +10915,$\{Z\circ T\mid T:\Omega\to\Omega\text{\ PPT}\}$ +10916,$\mathcal{F}_n$ +10917,$=E(X_i \mid X=a)$ +10918,$V(\mu)=\mathsf{Var}(\mathsf{CP}_1)=\mu x_2$ +10919,$r_1=0.1$ +10920,$\leftrightarrow$ +10921,$\Delta p$ +10922,$A=X+Y$ +10923,$(\delta^*-d)\sqrt{pq}=$ +10924,$V_X(m)=m$ +10925,$V=P_P-P_R$ +10926,$\rho(X)<\infty$ +10927,$\dfrac{\iota}{1+\iota} p$ +10928,$X_i(a)=X_i$ +10929,$-1/\theta$ +10930,$\mathrm{mv.mv} \not=\rho(A)+\rho(B)$ +10931,$1-\omega_I$ +10932,"$T_{m_2}(Y) \le b_{X,r+v(m_2)}$" +10933,$n\mathsf F^{-1} =n\bar{\mathsf F}$ +10934,$Z=d\mathsf Q / d\mathsf P\ge 0$ +10935,$f'(x_0)$ +10936,$\mathsf{E}_\mathsf{Q}(Y \mid X) = \mathsf{E}(Y \mid X)$ +10937,$B$ +10938,$\mathsf{TVaR}_p$ +10939,$\beta_i(k) = E[\kappa_i(X)Z(X)/X\mid X > k]$ +10940,$\displaystyle\int_\Omega X(\omega)p(\omega)\mathsf{Pr}r(d\omega)$ +10941,$\check g$ +10942,$f_{\hat i}$ +10943,$\mu_x = -d\log(\tpx)/dt = \lim_{t\downarrow 0} {}_tq_x/t$ +10944,$Z(t\mathbf{X})=tZ(\mathbf{X})$ +10945,$-9$ +10946,$EL$ +10947,$\alpha_Y$ +10948,$\mathsf{TVaR}_{0.95}(X)=1000$ +10949,$p=1-\exp(-t)$ +10950,$\beta g(S)-\alpha S$ +10951,$\mathscr{G}amma(n)=(n-1)!$ +10952,$\mathsf{Pr}hi$ +10953,$\mathsf{E}[g]\ge 1$ +10954,"$[t-dt, t]$" +10955,$\mathsf{E}[X_i(a)]$ +10956,$M=\sup(X)\le\infty$ +10957,$\mathsf{Var}(G)=c$ +10958,$50 of the amount allowed on each claim in the classes under subsections 2 to 6 shall be deducted from the claim and included in the class under subsection 8. Claims shall not be cumulated by assignment to avoid application on the $ +10959,$\phi(X)$ +10960,$s_{R+1}=s$ +10961,$v=1/(1+r)=1-d$ +10962,$d\nu=d\mu/\alpha$ +10963,$ρ(kX)$ +10964,"$v:\mathcal M\to [0,\infty]$" +10965,$K_\delta$ +10966,$V(X)>0$ +10967,$\mathsf{Pr}(\mathsf{CP}(\lambda)=0)=e^{-\lambda}$ +10968,$= \rho(B(s_l)) (1 -g(s)) + \rho(B(s_u)) g(s)$ +10969,$\delta(t)$ +10970,$Z_k \succeq_2 (Z_k\mid N)$ +10971,$\mathsf E[(a-X)^+]$ +10972,$\delta_i=\delta$ +10973,$r_0=1$ +10974,$X_{2c}$ +10975,$e^z/z$ +10976,"$\mathbf{x}=(x_1,x_2)$" +10977,$Q(x) = \nu(F(x))F(x)$ +10978,"$66,000, an additional amount, say \$" +10979,$\mathsf{TVaR}_{p_i}$ +10980,"$(lee.east |- lee.north)+(0.25,0.25)$" +10981,$Y=g(X)$ +10982,"$\Omega_i^X := (1-T_{R-1})\rho_e^L(X,s_i)$" +10983,$\mu_1 = \mu_2$ +10984,$M(a)=\mathsf{E}(X\wedge a)+d_iN(a)+(v-\nu^*)\displaystyle\int_0^a \sqrt{F(x)S(x)}dx$ +10985,$\bar M_t = \bar P_t - \mathsf{E}[Y_{t}]$ +10986,"$45,399.20 + \$" +10987,$P_t f$ +10988,$X=mC + nH$ +10989,$\left( g(S(x_{(j)}))-g(S(x_{(j-1)})) \right) / ( x_{(j)}-x_{(j-1)} )$ +10990,$a=Q+P$ +10991,$\rho_m(X)$ +10992,$\mathsf{E}[X]+k\mathsf{var}(X)$ +10993,$+\epsilon$ +10994,$X-b\le 0$ +10995,$y_{n+1}$ +10996,$\mathsf{E} X + c{ X-MX }$ +10997,$(S_t-a)^+$ +10998,$\phi_R$ +10999,$\mathsf{E}[X_i\tilde Z]=\rho_g(X)/2$ +11000,$\mathbf{B}$ +11001,$P=vEL + da$ +11002,$8+11.1667=19.167$ +11003,"$5/8, 1/4, 1/8$" +11004,$2\left( e^{-2y} + e^{-m}(y-m - 1) \right)$ +11005,$P=\rho(X\wedge A)$ +11006,$g(S)\not=q\phi$ +11007,$\mu=\log(\theta)$ +11008,$\mathcal D(X)=\rho(X)-\mathsf{E}[X]$ +11009,$X\le a=a_1+a_2$ +11010,"$\lambda([a,b]) = b-a$" +11011,$\beta_i(X)$ +11012,$s = f/n$ +11013,$\mathsf{E}[(A-L)^+]$ +11014,$0\le p < 1$ +11015,$q_A(p) = \sup A$ +11016,$p=F(x)$ +11017,$E(r_{FS}(p))$ +11018,$0 < \alpha\le 1$ +11019,"$1 million auto accident, a $" +11020,$X=X(v)$ +11021,$1\times m$ +11022,$\rho_{\min{}}(L_i)=\rho_i(L_i)$ +11023,$\kappa_T(y)= -\theta=\log(w(y))$ +11024,$E_i$ +11025,$\mathsf{E}[u(w-X)] = u(w-c)$ +11026,$\omega_3$ +11027,$0.8 \ge p < 0.9$ +11028,"$(a,b)$" +11029,"$\mathrm{ED}(\mu, \sigma^2 / n_i)$" +11030,$m(x)=S(x)+\delta(p)F(x)=S(x)+dF(x)+(\delta^*-d)\sqrt{F(x)S(x)}$ +11031,"$i_L \in \{R,\dots,i^\star \}$" +11032,"$\int_{[0,p]} \dfrac{\mu(dt)}{1-t}$" +11033,$\rho(c) = c$ +11034,$\rho(X) = \mathsf{E}[X] + c\mathsf{E}[X-\mathsf{E}[X]]^+$ +11035,"$\{\omega\mid p(\omega, A(\omega))=1 \}$" +11036,"$[0.2, 0.85]$" +11037,"$\kappa_i(\mathbf{x}, x)$" +11038,$\rho(X\wedge a) = \sum\rho(X_i(a))$ +11039,$r \leq p$ +11040,$P+Q<1$ +11041,$\mathsf E[XZ]$ +11042,$\mathsf{Pr}(B=1)=\mathsf{Pr}(X>x)=p$ +11043,$m\le 4$ +11044,$μ = w_1 δ_{α_1} + w_2 δ_{α_2}$ +11045,$sp}$ +11052,$a(X(\mathbf{v}))$ +11053,$\iota(s)=w/(1-w)$ +11054,${\cal Q}_\lambda =\{ \mathcal{B}bb Q\ll \mathcal{B}bb P \ | \ d\mathcal{B}bb Q/d\mathcal{B}bb P\le1/\lambda \}$ +11055,$D^n\rho_X(X_i)$ +11056,$\hat p=1-g^{-1}(1-p) > p$ +11057,$1^+$ +11058,$\mathbf {\vert S\vert}$ +11059,$X_2=1000$ +11060,$z=1$ +11061,$V_{R_1}(m+1)$ +11062,$\rho(1)=1$ +11063,$\beta_{1}$ +11064,$X= (X_1 + x) + (X - X_1 - x)$ +11065,$q(p')$ +11066,$q(U_X) < m$ +11067,$x^{n-1}e^{-x^2/2}$ +11068,"$\langle X_i, \zeta \rangle$" +11069,$-\kappa_1(-\kappa(\theta))=\theta$ +11070,$f_0\in\mathcal K$ +11071,$Z(x)$ +11072,$\Lambda = \dfrac{E( r_{U} ) - r_{f}}{\sigma_{r_{U}}}$ +11073,$1_{X < q(1-s)}$ +11074,$\epsilon\to 0$ +11075,$\bar p$ +11076,$f^{-1}(\mathcal{B}B(\mathbb{R})) = \sigma(A_n)$ +11077,"$\mathsf{Tw}_2(\mu,\sigma^2)$" +11078,"$\mathsf{MON,NORM,TI}$" +11079,$\mathsf{E}[X\wedge a] + d(a - \mathsf{E}[X\wedge a])$ +11080,"$350,000,000), or if at any time the authority's available capital is insufficient to pay benefits and continue operations, the authority shall have the power to assess participating insurance companies subject to the maximum limits as set forth in this section and Section 10089.30. The assessment shall be limited to the amount necessary to pay the outstanding or expected claims of the authority and to return the authority's available capital to three hundred fifty million dollars ($" +11081,$\mathsf{Pr}r(\emptyset) =0$ +11082,$g(S)\Delta X'$ +11083,"$\mathcal{W}(g, W)\subset \mathcal{W}$" +11084,$\omega\mapsto q(\omega)=F^{-1}(\omega)$ +11085,"$500mm, enough to materially impair their franchise, is judged to be 0.4%. This has a corresponding risk-neutral value of 2.5%. However, they believe that a loss over $" +11086,$\alpha\phi_{m}^o$ +11087,$\bar y$ +11088,$g - s$ +11089,$\sum w_i=1$ +11090,"$0.06 \times (64,861 - 7,500)=3,442$" +11091,$T_2 := (\sum_{j=0}^{n} p_j - \pi)X_n$ +11092,$\mathcal{M}=\{\delta_1\}$ +11093,$0\in\Omega_p$ +11094,"$(1, 2, 3, \dots)$" +11095,$0.5+U/4$ +11096,$c_k-G=\gamma_k$ +11097,"$X=(0,1,0,0,\dots,0)$" +11098,"$\max\,\{ \mathsf{E}[\zeta X] \mid \zeta\in\mathcal{M} \}$" +11099,${}_tq_x=1-\tpx$ +11100,$F^-(p)$ +11101,$q^-(p)=\sup\ \{ x\mid \mathsf{Pr}r(X < x) < p \}$ +11102,$n:=\nabla_yG/\|\nabla_y G\|$ +11103,$V(m)=m^3$ +11104,$g(s)=g(1-p)$ +11105,$\sigma(\mathcal{A}\otimes\mathcal{B})$ +11106,$\sum_0^0=0$ +11107,$I_i$ +11108,$\mathsf{E}_q[X]$ +11109,$X\not\equiv 0$ +11110,$x_2\leftrightarrow y_1$ +11111,"$P,Q,R$" +11112,$a(\mathbf{x}) =\mathsf{VaR}_p(X(\mathbf{x}))= q_p(\mathbf{x})$ +11113,$189K/\$ +11114,$\iota\nu=\delta$ +11115,$u''(z+t)$ +11116,$X(\mathbf{x})=\sum_i x_iX_i$ +11117,$a_i':=\sum \alpha_i(1-S)\Delta (X\wedge a)$ +11118,$(y-t)/V(t)$ +11119,$\max(X)=1$ +11120,$L / (1-e-\pi)$ +11121,$s=1-p=0.2$ +11122,"$h:[a,b]\to[0,1]$" +11123,$=P=\mathsf{MV}(X\wedge a)$ +11124,$\Vert X-Y\Vert := \sup_{\omega\in\Omega} |X(\omega) - Y(\omega)|$ +11125,$\mathsf{E}[v_2(X)]-\mathsf{E}[v_1(x)]^2$ +11126,$a=\mathsf{VaR}_{1-g^{-1}(\tau)}(X)$ +11127,$e^{-rT}S_T$ +11128,$\sigma_Z$ +11129,$P = A - Q$ +11130,$m(1+2m+\frac{a^2+1}{a^2}m^2)$ +11131,$dz_Adz_L=\rho\sigma_A\sigma_L$ +11132,$\mathsf{Pr}(|X_n(\omega)-X(\omega)|>\epsilon)\to 0$ +11133,$\pm 1$ +11134,$\mathsf{E}[YZ_\epsilon]\to\mathsf{E}[YZ]$ +11135,$\{p \ge p_-\}$ +11136,$\mathsf{E}[X\mid T] = \phi(T)$ +11137,$X_i=x_i$ +11138,$T_1 = 20°C = 293.15$ +11139,"$X:(\Omega,\mathscr{F})\to(\mathbb{R},\mathcal{B}B(\mathbb{R}))$" +11140,$r_P < r$ +11141,$y\mapsto Z/\lambda$ +11142,"$X\in \mathcal A_{t,t+1} + \mathcal A_{t+1}\iff -\rho_{t+1}(X)\in\mathcal A_{t+1}$" +11143,$\beta_Q=(a/Q)\beta_A + P/Q\beta_L$ +11144,$f(x)dx=dp$ +11145,$\rho(A_k) \le \rho(A_0) + k \rho(N) \le \hat\rho(A_0) + k\rho(N)=\hat\rho(A_k)$ +11146,"$\rho(P,R,a)$" +11147,$x\mapsto \sin(1/x)$ +11148,${}^{[>89]}$ +11149,$\mathsf{E}[X_i\mid X](x)$ +11150,$U_1=u_1$ +11151,$\delta\to 0$ +11152,$dt=g'(1-s)ds=\phi(s)ds$ +11153,$\sigma(G)$ +11154,$\tau(\theta):=\kappa'(\theta)=\mu$ +11155,$X_2=100$ +11156,$g(S(\infty))=0$ +11157,"$I=[0,1]$" +11158,$T(p)=\mathsf{TVaR}_p(X)$ +11159,$18.7-18.3=0.0.4$ +11160,$\theta_s$ +11161,"$X_{i}=\mathsf{CP}(j_n({x_i})\delta, {x_i})-j_n({x_i})\delta {x_i}$" +11162,$g_n(x)=f_j(x)$ +11163,$g(t)=1$ +11164,$g(s) = t_{df}(\mathsf{Pr}hi^{-1}(s)+\lambda)$ +11165,$\mathsf{EPD}_s(X)$ +11166,$d\bar S_i/da$ +11167,$t>x$ +11168,$=P + r(P+S)$ +11169,"$j=1, \dots, n$" +11170,$\mathcal S(X)=\mathsf{VaR}_p(X)$ +11171,$0=p_0 < p_1 < \cdots < p_n < 1$ +11172,$\{X_t > a_t\}$ +11173,"$=\displaystyle\int_{T^{-1}(B)}^{\phantom{X}} \mathsf{var}phi(T) \,d\mathsf{Pr}\quad$" +11174,$J(x) - J(x+dx) \approx j(x)dx$ +11175,$\mathbf {F(x)=\mathsf{Pr}r(X\le x)}$ +11176,$X(ω)$ +11177,$\mathsf{Pr}(B\mid \mathscr{G})$ +11178,$\mathsf{E}[(X-x)^+]$ +11179,$\mu=\log M-\sigma^2/2$ +11180,$\mathsf{E}[X\mid X>x]/\mathsf{Pr}r(X>x)$ +11181,$T^2 \propto r^3$ +11182,$\mathsf{var}(\sum C'_i)=v_{res}^2 \sum c_i^2$ +11183,$\mathcal F_1$ +11184,$\mu_U = 15$ +11185,$m=6$ +11186,$\mathsf{CX}$ +11187,$m_j=0$ +11188,$R = P - U$ +11189,"$\{\lambda_t\}, with $" +11190,$\log(g) = a + b\log(s)$ +11191,$P+Q=a$ +11192,$=\mathsf{E}(X-c)_-=\int_0^c (c-x)f(x)dx$ +11193,$X_n(2/3)$ +11194,"$s\in[k,1]$" +11195,$0.417 < p < 0.791$ +11196,"$Y_{1,0}$" +11197,$j(x)=x^{-\alpha-1}$ +11198,$\mathbf {\alpha_2S\Delta X}$ +11199,$M_{X+Y}=\mathsf{E}(e^{it(X+Y)})=\mathsf{E}(e^{itX}e^{itY})=\mathsf{E}(e^{itX})\mathsf{E}(e^{itY})=M_X(t)M_Y(t)$ +11200,$a((1+\epsilon)X)$ +11201,$A_i=\{X=x_i\}$ +11202,$g(s)=cs$ +11203,$\pi-\lambda\mathsf{E}[X]$ +11204,$h'(j) = 100$ +11205,$\nu\ge 0$ +11206,$K_\theta(t)=\kappa(t+\theta)-\kappa(\theta)$ +11207,$E_{\mathsf{Q_X}}$ +11208,$B(s)$ +11209,$X\preceq Y$ +11210,$P_k+Q_k=\Delta X_k$ +11211,$\theta\mapsto -\kappa_1(\theta)$ +11212,"$Q_0, Q_{i,\epsilon}$" +11213,$v(A)$ +11214,"$1.2 billion annually**. On this basis, the CEA's operating budget could be up to $" +11215,$R_0(t)=0$ +11216,"$(1-s, 1-g(s))$" +11217,$P_i=\mathsf{E}_Q[X_i]$ +11218,$\bar S(a)=\mathsf{E}(X\wedge a)$ +11219,$F_1 \prec_1 F_0$ +11220,$-v$ +11221,$\alpha_i(a)S(a)$ +11222,$\bar Q_{d}=a_{d}-\bar P_{d}$ +11223,$da = p(a)da + (1-p(a))da$ +11224,$\mu_L=r_L + \pi$ +11225,$\mathsf{E}[1_{U < s}]=s$ +11226,$g(w s_1 + (1-w)s_2) \le w g(s_1) + (1-w) g(s_2)$ +11227,"$700,000,000), any policy issued or renewed on or after that date shall provide, less any applicable deductible, not less than three thousand dollars ($" +11228,$X^{\oplus 1}$ +11229,$F\subset M$ +11230,$\bar M(a):=\bar P(a)-\bar S(a)$ +11231,$(a_i)_i$ +11232,$F(x)\ge p\iff q^-(p)\le x$ +11233,"$(0,g_0)$" +11234,$\mathsf{E}[X\mid X\ge \mathsf{VaR}_p(X)]$ +11235,$Q_0$ +11236,$S_{\mathbf{v}}$ +11237,"$\rho(Y) \ge a_{X, r-\alpha(F)}(Y) + \alpha(F)$" +11238,$U=X+Y$ +11239,$\rho(X_1\wedge a_1)$ +11240,"$. It falls into the compensated IACP, case 3 group, discussed in [Part III](./2020-10-20-Probability-Models-for-Insurance-Losses/), and takes any real value, positive or negative, despite only having negative jumps. It has a thick left tail and thin right tail. Its mean is zero, but the variance does not exist. A tilt with $" +11241,$p^*=0.7501$ +11242,$\hat p=1-g^{-1}(1-p)$ +11243,"$1000 \, \text{kg/m}^3$" +11244,$n^{-1}\mathbf M'\mathbf M$ +11245,"$t=1,2,\dots$" +11246,$R_1(t)=P_1(t)/t$ +11247,$\omega_4$ +11248,$\mathsf E(G)= f + \mathsf E(G') = 1$ +11249,$2.725$ +11250,$dp=f(x)dx$ +11251,$\iota^*=0$ +11252,$\approx 27\%$ +11253,"$I(q,p)=0$" +11254,$A = -\log(p) = 5.298$ +11255,$(1-\theta)\mathsf{Wang}+\theta\mathsf{Dual}$ +11256,$g(a+X)=a+g(X)$ +11257,$\mathsf{E}\zeta=1$ +11258,$v=1/1.15 = 0.87$ +11259,$\alpha_i(x)=\mathsf{E}[X_i\mid X]$ +11260,$s_0=s_0(s_1)$ +11261,$0.909+0.273=1.182$ +11262,$\rho(X)=\mathsf{E}_\mathbb{Q}(X)$ +11263,$B=\mathsf{E}[X 1_{X > x}]-(1-p)x=\mathsf{E}[(X-x) 1_{X \ge x}]=\mathsf{E}[(X-x)^+]=$ +11264,"$100,000 and \$" +11265,$\mathbf {d=1}$ +11266,"$\rho_e^L(X,s)$" +11267,"$\langle \mu, X+a \rangle = \langle \mu, X \rangle + a$" +11268,"$\forall X,Y,t\ge 0$" +11269,$X_S\ge \mathsf{E}[X_T\mid \mathscr{F}_S]$ +11270,$f_X(a)$ +11271,$\bar F(a) = a-\bar S(a)$ +11272,$N_t - A_t = 1 - T$ +11273,$s_m=1$ +11274,"$P(·, ω)$" +11275,$\mathsf{Pr}r(X < x)$ +11276,$P(t)> (1-t)P(0) + tP(1)$ +11277,$\{S\le T\}$ +11278,$LR_{\mathsf{Wang}}$ +11279,"$B\mapsto p(\omega, B)$" +11280,$\bar \nu$ +11281,$(x-a)^+\wedge b$ +11282,$\bar q(s/2)\le 2\bar q(s)$ +11283,${}^{[>13]}$ +11284,$X\wedge a(X)\le Y\wedge a(Y)$ +11285,$1=\delta(p) + \nu(p)$ +11286,$\mathsf{Pr}r(\{\omega_2\})=2/3$ +11287,$\alpha_i(x)=\mathsf{E}\left[\frac{X_i}{X}\mid X\right]$ +11288,$e^{-\lambda}>0$ +11289,$l_p<0$ +11290,$x=0.5$ +11291,"$(\omega_I,1]$" +11292,$P(t)=\infty$ +11293,$\mathsf{E}[X_2(a)\mid X_1(a)=x] \le a-x$ +11294,$S(x)\approx 1$ +11295,$\lambda(1+m^2/\lambda^2)$ +11296,$E[X_\tau]$ +11297,$R(n)$ +11298,$\gamma=\mathsf{Pr}r(X>\mathsf{E}[X])$ +11299,"$\int_B \mathsf{Pr}(A\mid\mathscr{G})(\omega)\,\mathsf{Pr}(d\omega)$" +11300,$\mathsf{E}(X_i ; X \le a)$ +11301,$p\mapsto g(1-p)$ +11302,$A_n \downarrow A$ +11303,$I_k$ +11304,${}_tp_x\mu_{x+t}$ +11305,$k\in\mathbb Z$ +11306,$P(x)=S(x)$ +11307,$\mathsf{E}[X_0]=80$ +11308,$a(\mathbf{v})$ +11309,$\tilde p=p_a$ +11310,$S=1$ +11311,"$\langle \mu,X \rangle$" +11312,$\mu_{t}$ +11313,$0.4$ +11314,$\rho_g(X)=\mu+\lambda\sigma$ +11315,"$(s_0,g_0)=(0,0)$" +11316,$\mu(F)=0$ +11317,$\alpha_j'(x)<0$ +11318,$\mathsf{Pr}r(N>32)=7.37\times 10^{-9}$ +11319,$g(S)=1$ +11320,$se=0.067$ +11321,"$(fun3a.south -| fun6a.south east)+(\smlspc,-\smlspc)$" +11322,$x=60$ +11323,$B(s_0/2^{n+1})2^n\bar q(s_0)=s_0\bar q(s_0)/2$ +11346,$s=k^{-1}(m + ra)$ +11347,$\rho_{(g)}(X)=$ +11348,"$1,353.02 | \$" +11349,$\bar q_{X_1+X_2}(s) \le 2\bar q(s)$ +11350,$A(1_{X>x})$ +11351,${}^{[<190]}$ +11352,$l(t)<\infty$ +11353,$K(t)=\log M(t)$ +11354,$r_K = \exp (\lambda) - 1$ +11355,$X=X\wedge a + (X-a)^+=\sum_i X_i(a) + (X-a)^+$ +11356,$\mu+R\mathbf{AU}$ +11357,$M(s)=\mathsf{E}[X^s]=\mathsf{E}[e^{s\log(X)}]$ +11358,$\mathsf{TVaR}_p\leftrightarrow\delta_p$ +11359,$\sigma=3$ +11360,"$a_1,a_2\in A$" +11361,$\mu=T\lambda$ +11362,$B(p)=0$ +11363,$m + ra$ +11364,$\mathsf{E}[X] + \pi\mathsf{E}[((X-\mathsf{E}[X])^+)^2]^{1/2}$ +11365,$(X(\omega_1)-X(\omega_2))(Y(\omega_1)-Y(\omega_2))\ge 0$ +11366,$P=L+r(a - P)=vL + da$ +11367,$\mathsf{Pr}(\cdot\mid\mathscr{G})$ +11368,$l(p)$ +11369,$\rho_L^E(t_1)=(1-w)\rho_L^1 + w\rho_L^2$ +11370,"$5,000), provided that if the underlying policy of residential property insurance does not cover structural loss, the amount of contents coverage after deductible shall be not less than five thousand dollars ($" +11371,$g(s)=1-(1-s)^\beta$ +11372,$p_j$ +11373,$\omega_0$ +11374,$V=(a - X)^+$ +11375,$-\rho(-X_i)$ +11376,$p < 1$ +11377,$\mathsf P(\{x\})=0$ +11378,"$\zeta_1=0.15, 0.5, 0.85$" +11379,$\mu(a+b\mu)$ +11380,"$(Alice)+(0,-1)$" +11381,$X_{t+1}$ +11382,$\mathsf{E}[tX_1] \le P_1(t) \le \rho(tX_1)$ +11383,$\tilde \mathsf{Pr}i = v^T\mathsf P_s X_s + RA_0$ +11384,$10^{1+6+12}=10^{19}$ +11385,$t_1-\epsilon$ +11386,$h(0.9) = 1-\sqrt{0.1} = 0.684$ +11387,$\mathsf{Pr}r(X_n=0)=1-1/n$ +11388,$R=L+1$ +11389,$\theta < 0$ +11390,$d=0.114$ +11391,$\alpha=\mathsf{TVaR}$ +11392,"$U_i = \rho_i + U_{1, i}^Z$" +11393,$1/\sqrt\alpha$ +11394,"$0, 1, \dots, n - 1$" +11395,$p\mapsto \mathsf{TVaR}_p(Y)$ +11396,$p_{j-}\le p\le p_j$ +11397,$A(X)-B(X)$ +11398,$10^{-15}$ +11399,"$\sigma=0.1, 0.2, 0.3$" +11400,$g(t)=t^α$ +11401,$\rho_t(-\rho_{t+1}(X))\le \rho_t(\rho_{t+1}(Y))$ +11402,$\mathsf E[X] < P < \max(X)$ +11403,$\mathcal E(X)=\mathsf{E}[(p X^+ + (1-p)X^-)/(1-p)]$ +11404,$(\delta^*-d)\sqrt{S(x)F(x)}$ +11405,"$p_0,\dots, p_m$" +11406,$\mathsf{E}[(X\wedge a)(a)]$ +11407,$|b|$ +11408,$n = 2$ +11409,"$524,936,856 | 9.0% | $" +11410,$X\succ Y$ +11411,$(-1.2186) \cdot 0.0090 = -0.0110$ +11412,$\rho(X)\ge X$ +11413,$\kappa_1'(-\kappa(\theta))=1/m$ +11414,"$Z_{t+1},\dots,Z_T$" +11415,$t \to \infty$ +11416,$(T\lambda)g = \mu^t g(t)l(t) \le \mu h < \infty$ +11417,${}^{[<111]}$ +11418,$1-G(x)\sim x^{-\alpha}(L_1(x) + L_2(x))$ +11419,"$0,1,\dots, T$" +11420,"$(p,q(\hat p))$" +11421,$\mathsf{Q}$ +11422,$s < p$ +11423,$> 0.9$ +11424,$\hat F / |\hat F|$ +11425,$f_Y$ +11426,$m=et$ +11427,$\delta_{p_1}$ +11428,$\tilde p=\tilde F(F^{-1}(p))$ +11429,$J/K$ +11430,$x=2$ +11431,$f(\mu)= \mu$ +11432,$\mathcal A$ +11433,$0 0$ +11452,"$\mathsf{biTVaR}_{0,0.9}^{0.3138}$" +11453,$\mathsf{E}[X_\lambda]=\lambda\mathsf{E}[X_1]$ +11454,$\kappa^i(x) := \mathsf E[X^i|X=x]$ +11455,$Z_A=(1-p)^{-1}1_A$ +11456,$\lim_{s \downarrow 0}1/g'(s)$ +11457,"$M = \{M_t, t \geq 0\}$" +11458,$\mathsf E[\exp(r_1Z_1 + r_2Z_2)]$ +11459,$\mathrm{conv}(S)$ +11460,$\mathsf{E}[X\mid\sigma(T)]$ +11461,$g(u) = m'u / (r(u) - m'u)$ +11462,$U = A$ +11463,$\sigma=1.667$ +11464,$\rho(X)=E_Q[X]=\int XfdP$ +11465,$c_{min} := c_r + \Delta_m \Theta_m^X$ +11466,$\bar P(x)$ +11467,$\mathbf {1_{X>x}}$ +11468,"$Y_i\sim \mathrm{ED}(\mu, \sigma^2)$" +11469,$R_3$ +11470,$0.20$ +11471,$t < T$ +11472,$\bar X^a \pm \sigma_a$ +11473,"$135,545,515 | 69.8% | $" +11474,$\mathbf T$ +11475,$1+r=\sum_\omega q(\omega)P(\omega)$ +11476,"$\lambda,\mu$" +11477,"$500 million. In­stead, it found it could syndicate $" +11478,$\sum (X\wedge a)p$ +11479,$10^{-4} \times$ +11480,$\text{VIF}>10$ +11481,$v(A\cup B) + v(A\cap B)\ge v(A) + v(B)$ +11482,$a \in \mathbb{A}$ +11483,$\tilde Z\in\mathscr{P}$ +11484,$e^\theta/(1-e^\theta)$ +11485,$8.5$ +11486,$\tau^{-1}$ +11487,$f \circ g$ +11488,$R_Q$ +11489,$\pi(X)=\mathsf{E}_g(X\wedge \alpha(X))=\int_0^{\alpha(X)} g(S(t))dt$ +11490,$\tau(\omega_I) < 0$ +11491,$\theta=\tau^{-1}(\mu)$ +11492,$F_u^{-1}$ +11493,$dgS$ +11494,"$d(x,\omega)$" +11495,$c(S\cup\{i\})=c(S)+c(i)$ +11496,$e^\theta$ +11497,${}^{[>60]}$ +11498,$\mathsf{LOSS}=\mathsf{LR}\ \mathsf{PREM}$ +11499,$1-0.271 = 0.729$ +11500,$\mathbf {x_1}$ +11501,$\mathcal{A}\otimes\mathcal{B}$ +11502,$\bar P=\bar S+\bar M$ +11503,$\rho_m(X)\le \rho_{m_0}(X)$ +11504,$\phi_{R+1}^u$ +11505,$\beta_2g-\alpha_2S$ +11506,$x^{-3/2}$ +11507,$\tau < t+d$ +11508,$\mathsf{E}[h(X_i)L(X)]$ +11509,$\approx 10^{-1}$ +11510,$P(X_{0}(a))$ +11511,"$[0, \epsilon_2]$" +11512,$1-\nu p$ +11513,$\forall B\in\mathscr{G}$ +11514,$P(P(\omega))$ +11515,"$\eta,\zeta$" +11516,$Y = \operatorname{frac}(2\omega)=\{2x\}=x -\lfloor x\rfloor$ +11517,$B\in\mathcal{B}$ +11518,$L_0^{a+y}=L_0^a+L_a^{a+y}$ +11519,$\log-sd \sigma$ +11520,$\rho(X) + c = \rho(X+c)\ge \rho(X) + \mathsf{E}[cg]$ +11521,"$(fun5a.south west)+(-0.5*\wspcer,-0.5*\medspc)$" +11522,$dp=\exp(-t)dt$ +11523,$\mathsf{E}[X_{t+1} \mid \mathscr{F}_t] = X_t$ +11524,$\sup(X)$ +11525,$\mathsf Q(A)=\mathsf{E}[Z1_A]$ +11526,$p\to 1$ +11527,$M^-=\sum_i M_i^-$ +11528,$(P-S)/(a-P)\ge \iota$ +11529,"$h_{i,\epsilon}-h_0\to 0$" +11530,$X_2=t$ +11531,$\rho_i(F_i)$ +11532,$\beta_k^i$ +11533,$\mu(dp)=f(p)dp$ +11534,$\alpha_i(a)$ +11535,$P=L + \iota Q$ +11536,$\mathsf{Var}(X_\theta)=V(\mu)=\kappa''(\theta)=\tau'(\tau^{-1}(\mu)=1/(\tau^{-1})'(\mu)$ +11537,$\mathsf{P}(1_{U < s}=1)=\mathsf{P}(U < s)=s$ +11538,$A_t = T$ +11539,$q(p)\phi(p)$ +11540,$(\alpha+1)/\alpha\to 1$ +11541,$F:\mathbb{R}^n\to\mathcal{Z}$ +11542,$\bar{X^a} = \mathsf E[X^a|X]$ +11543,$\alpha>-2$ +11544,$p=0.001$ +11545,$10X + 20$ +11546,$eG$ +11547,"$18,062.50 - \$" +11548,"$i=1,\dots, M$" +11549,$\mathsf{Pr}r(X>a)=0$ +11550,$kd} \sigma_s^2$ +11583,$xS(x)|_0^\infty$ +11584,"$968,000/4, or \$" +11585,$Q=0$ +11586,$\alpha=-1$ +11587,$E_Q(X_i(a)) = E_Q(E_Q(X_i(a)\mid X))$ +11588,"$I(q,p) \ne I(p,q)$" +11589,$p\ge p_0$ +11590,$\mathsf{E}[X_i]=0$ +11591,$\chi^2$ +11592,$M(t)=\mathsf E[\exp(tx)]$ +11593,$S \ge s_0$ +11594,$x-\log(x)\ge 1$ +11595,$\tilde Z=\mathsf{E}[Z\mid X]$ +11596,$\zeta=(1-p)^{-1}1_{W}$ +11597,$E'=\Omega\setminus E\in\mathcal F$ +11598,$r_i+a_i$ +11599,$v^TX$ +11600,$s_1(k_1 - k_0) + k_0 uv$ +11601,$F$ +11602,$\omega < p^-$ +11603,$x=q(1-g^{-1}(1-p))$ +11604,$t1-p$ +11608,$Q_k+P_k>\Delta X_k$ +11609,$g_\tau(1)=1$ +11610,$X_i(a$ +11611,$g(x)=e^{2\pi i x\theta}$ +11612,$\epsilon\mu(A\cap B)\le \nu_0(A\cap B)$ +11613,"$M(a)=M(X,a)$" +11614,$S(x)=\mathsf{Pr}hi((-x+\mu)/\sigma)$ +11615,$S_u(t)=\text{Pr}(X_u>t)$ +11616,$A=A_1\cup\cdots\cup A_n$ +11617,$s \ne s^\ast$ +11618,$U=4$ +11619,$X=f(Z)$ +11620,$T=2$ +11621,$0.1 < s < 0.2$ +11622,$H(n + \text{prev hash} + \text{value})x_2}$ +11633,$D(F)$ +11634,$x_0 \in \{ x \mid F(x) \ge p \}$ +11635,$X = \sum_i \Delta_iX$ +11636,$a\wedge b$ +11637,$\mathrm{Q}$ +11638,$SiO_4^{4-}$ +11639,$\mathsf{E}[X\mid T]=\mathsf{var}phi(T)$ +11640,$Z(\omega)=\dfrac{1}{1+r}\dfrac{\mathsf Q(\omega)}{\mathsf{P}(\omega)}$ +11641,$r\not=c$ +11642,$\sigma(Z)=\sqrt{\mathsf{var}(Z)}$ +11643,$\mathsf{E}(X_i/X \mid X \le a)$ +11644,$=p$ +11645,$\rho_X(X_i) \ge \mathsf{E}[X_i]$ +11646,$\mathsf{E}_\mathsf{Q}[X]=\mathsf{E}[XZ]$ +11647,"$P(\omega, \cdot)$" +11648,"$X_{t,1}$" +11649,"$E, F$" +11650,$\bar q(s)=q(1-s)$ +11651,$L=\sum_s l_s B_s$ +11652,$P(E_1\cup E_2)=P(E_1)+P(E_2)$ +11653,$ because it has a cusp). In the case $ +11654,$a=M(a)+Q(a)$ +11655,$\mathsf{Pr}(A\cap G)=\mathsf{Pr}(A)\mathsf{Pr}(G)$ +11656,$\rho(\tilde X)=\rho(X) + \rho(\tilde X-X)$ +11657,$\zeta^2$ +11658,$(1-z)^{-a}$ +11659,$g(s) = 1 - (1 - s)/(1 + r_f + Ck(s))$ +11660,$y=(\log(x)-\mu)/\sigma$ +11661,$F(x_{\max{}})-F(x_{\min{}})$ +11662,$\hat\rho(A) \le \rho(N)\rho(X)$ +11663,$\mathcal{F}$ +11664,"$1,000,000,000) plus costs of issuance and sale of those revenue bonds or other debt and amounts paid or payable to bond issuers and providers of credit support and letters of credit for, and interest on, those revenue bonds or other debt, regardless of the frequency or severity of earthquake losses incurred after the creation of the authority. Once the authority has levied policy surcharges in a total amount of one billion dollars ($" +11665,"$823,500, or 82.35% *x* \$" +11666,$L = \text{E}[L^*\wedge A]$ +11667,$\mathsf{E}[XZ]=\mathsf{E}[\mathsf{E}[XZ\mid X]]=\mathsf{E}[X\mathsf{E}[Z\mid X]]=\mathsf{E}[X\tilde Z]$ +11668,$P_2\ge (\rho(X_1)-P_1) + \rho(\mathsf{E}[X_2\mid X_1])\ge \rho(\mathsf{E}[X_2\mid X_1])$ +11669,$M_{2}\Delta X$ +11670,$v(A)\le v(B)$ +11671,$sqrt{st}$ +11672,$\mathbf{H}\mathbf{y}= \mathbf{Xb} = \hat y$ +11673,$\mathsf{Pr}r(A)>0$ +11674,$r_2=0.5$ +11675,$(A-L)^+$ +11676,$R^S=g^mA^R$ +11677,"$I_{k,n}$" +11678,$\mathsf{E}[X_0g'(S(X))]$ +11679,$\dfrac{\mathsf{E}[X_i | X]}{X}$ +11680,$j(x)=x^{\alpha-1}$ +11681,$Y_j$ +11682,$d\omega$ +11683,$\epsilon v_1$ +11684,$\mathsf{E}_\mathsf{Q}[X\wedge a] + \tau a$ +11685,$\alpha^3 S$ +11686,"$g'(s)=(1-p)^{-1}1_{[0,1-p]}$" +11687,$b_h$ +11688,$\mathsf{Var}(U)>\mathsf{Var}(X)$ +11689,$\rho(X)=\mathsf{E}(X\theta)$ +11690,$\iota_p$ +11691,$\log(g')$ +11692,$\neg b/p$ +11693,$a = \omega^2 r = v^2 / r$ +11694,$10^{-18}$ +11695,$(1-p)(1-\alpha)=(p-1)(\alpha-1)=-1$ +11696,$n=16$ +11697,$\rho(X)=\mathsf{E}[X\mid A]$ +11698,"$i=0,1$" +11699,$AR$ +11700,"$1,013,863 | \$" +11701,$u$ +11702,$\rho()$ +11703,$V=1_{X\le x^\ast}$ +11704,$|S|$ +11705,$X=x_l$ +11706,$F(x-) = \lim_{t\uparrow x} F(t)$ +11707,$(g(s_0)-g_0)/s_0 \ge g'(s_0)$ +11708,$\mathsf{TVaR}_{p_1}(X) = r$ +11709,$\{\mathcal{F}_t\}$ +11710,$p_Y<0.5$ +11711,$T = 20°C + 273.15 = 293.15 K$ +11712,$\sum_j Y_j = 0$ +11713,$da>0$ +11714,$I_i=(X-a_i)^+\wedge y_i$ +11715,$a=\bar P(a) + \bar Q(a)=\bar S(a) +\bar M(a) +\bar Q(a)$ +11716,$H(X)0.5$ +11727,$\mathsf{WCE}_p(X) = \mathsf{TVaR}_p(X)$ +11728,$1-g$ +11729,"$g_2(s)=\min(2.5s, 1)$" +11730,$\zeta_1=1/\sqrt{2}$ +11731,$\mathsf{E}_\mathsf{Q}[X_i\mid X]$ +11732,$T(X) = \sum_{i=1}^{n} X_i$ +11733,$\delta=0.13043$ +11734,$(\alpha S)'(x)=-\mathsf{E}[X_i\mid X=x]f(x)/x$ +11735,$\mathsf P(X\ge x_p)=1-p$ +11736,$h_1\le w$ +11737,$F^{(-2)}=[F^{(2)}]^*$ +11738,$\tau \ge t+d$ +11739,$=1-M_i/P_i$ +11740,$. Definition of normal cone to $ +11741,"$10,280$" +11742,$t\mapsto s(t)$ +11743,$366.4$ +11744,$m(m+1)^2$ +11745,$t-2$ +11746,$k<0$ +11747,"$\{(s_j,g_j): j=0,\dots,L-1, R+1,\dots,m\}$" +11748,"$k=1,\dots,m$" +11749,$1\not\in S$ +11750,"$(0,b)$" +11751,$sd/\sqrt{n}$ +11752,$u_3=1-e'-\mathsf{E}_Q[L]/P$ +11753,$Q = M/\iota$ +11754,$\mathsf{E}_\mathsf{Q}(X_i \mid X)=\mathsf{E}(X_i \mid X)$ +11755,$\mathsf{E}_{\mathsf Q}[\cdot]$ +11756,$d(y;\mu)>0$ +11757,$S(M-) > 0$ +11758,$10$ +11762,$g_k(x) = 1-(1-x)^k$ +11763,$ϕ_s(X)$ +11764,"$100,000, and a minimum premium of \$" +11765,$\Omega=\tau(\mathring\Theta)$ +11766,$0.01 \text{ kWh}$ +11767,"$w_{p_1,p_2}\delta_{p_1} + (1-w_{p_1,p_2})\delta_{p_2}$" +11768,$x_1+y_1 \le x_1+y_2\le x_2+y_2$ +11769,$Z_\epsilon\to Z$ +11770,$U/2$ +11771,$s = \mathsf{Pr}r(U \ge U_s)$ +11772,"$-\tpx\,\mu_{x+t}$" +11773,$v=\omega_I$ +11774,$a-\bar S(a)=\bar R(a)+\bar Q(a)$ +11775,$g'(s)$ +11776,${}^{[>12]}$ +11777,"$\langle X(\epsilon), \zeta_{x+\epsilon} \rangle$" +11778,$\sigma=\mathsf{TVaR}_s$ +11779,$q=-100$ +11780,$q=S$ +11781,$\partial\rho(X)=\{Q_0\}$ +11782,$\mathsf{Pr}(N) = 1/2$ +11783,$\mathsf{E}[X_1\mid X=x]$ +11784,$\mathsf{Var}$ +11785,$W=0$ +11786,$f(s)\le s$ +11787,$m/s^2$ +11788,$\mathsf{E}[X\mid A]$ +11789,$. The distribution is very thick tailed and does not have a mean ($ +11790,$t>t_*$ +11791,$p\gg n$ +11792,$\mathsf{P}[\cdot]$ +11793,$m\in\mathbb R$ +11794,$\rho^x_h(s)(X)$ +11795,"$(\x*.75, -2)$" +11796,$\rho(X) = \int_0^\infty g(S(x))dx$ +11797,$\mathsf{Pr}r(\mathsf{var}nothing) =0$ +11798,"$\rho(Y)\ge \rho_{m_X}(Y) \ge a_{X,r}(Y)$" +11799,$\mathsf Q(A)=\int_A f(\omega)\mathsf P(d\omega)$ +11800,$p(\nu(p)-l(p))$ +11801,$\mathsf E[|X_1|]<\infty$ +11802,$k=4$ +11803,"$\mathsf{E}[\psi(S,T)] = \displaystyle\int_M \mathsf{Pr}_T(dt)\int_L\psi(s,t)P^T_S(ds\mid t)$" +11804,$u(0)=0$ +11805,$\mathbf{H} = \mathbf{X}(\mathbf{X}' \mathbf{X})^{-1} \mathbf{X}'=(h_{ij})$ +11806,$\mathsf{VaR}_p(X)=x$ +11807,"$1,000 to \$" +11808,$k=st$ +11809,"$(fun6.north west)+(-\smlspc,\smlspc)$" +11810,$\tpx$ +11811,$X_1-X_2$ +11812,$X(\omega)=1/\omega$ +11813,$P = 3.1035$ +11814,$\rho_g(V)= g(F(x^*)) \ge F(x^*)=\mathsf{E}[V]$ +11815,$U_R\rho_R^E(t_2)$ +11816,$a < b_h$ +11817,$t^2/2$ +11818,"$(-\mu(U)/2, \mu(U)/2)$" +11819,$q(p)\phi(p)dp$ +11820,"$\mathbb{Q} = \left \{ q:I(q,p) \le I^* \right \}$" +11821,$X_k^1$ +11822,$\mathsf{E}[X_i/X \mid X \le a]$ +11823,$lsc(\rho)$ +11824,$M=P-L$ +11825,"$X\,\Delta S$" +11826,$0 \ge \rho(X_n) \ge -\rho(-X_n) \uparrow 0$ +11827,$N:\mathbb{R}^n\to\mathcal{X}$ +11828,$1\le p \le \infty$ +11829,"$(x_{1,i}, x_{2,k(i)})$" +11830,$Y_\nu=X_\nu/\nu$ +11831,$\displaystyle\int_0^\infty xg'(S_X(x))dF_X(x)$ +11832,"$[0,p)$" +11833,$1-\mathsf{P}(X=0)$ +11834,$u=0$ +11835,$g(A)/p=59.142$ +11836,$\mathsf E[X] = \mathsf E[\mathsf E[X\mid I]]$ +11837,$\hat g(s)-g(s)$ +11838,$\mathsf{E}[X_0]$ +11839,$\times$ +11840,$\mathsf{E}(X)$ +11841,$\rho^*$ +11842,$\mathsf{E}_\mathsf{Q}(X_i\mid X)$ +11843,$A\leftrightarrow 1_A$ +11844,$\upsilon$ +11845,$i=0.04$ +11846,$dP/dt\not=0$ +11847,$\mathsf{E}[Yg'(S(X))]$ +11848,$\mathsf{Var}(Y)=npq=\mu(1-p)=\mu(1-\mu/n)$ +11849,$A_t=\mathsf E[q(W_T) \mid B_t]$ +11850,$1/(1-p)>1$ +11851,$K_s = 1-g(s)$ +11852,$q(s)=q(p)$ +11853,"$(S,d)$" +11854,$X\wedge a = \displaystyle\sum_i X_i(a)$ +11855,$\mapsto m'^2 - bm' + b^2/4 + bm'- b^2/2 + c = m'^2 - (b^2/4 - c)$ +11856,"$\rho_{g_k}(X) = \mathsf{E}[\max(X_1,\dots,X_k)]$" +11857,"$\zeta_1,\dots, \zeta_T$" +11858,"$X = (X_1, X_2, \ldots, X_n)$" +11859,$ for integer $ +11860,$\bar P_1=\mathsf{E}[X_1g'(S_X(X))]$ +11861,$\int_-^\infty dG(z) / (z+\tau)^n$ +11862,$d\bar S(a)/da=S(a)$ +11863,$10^{-8} - 10^{-11}$ +11864,$\lambda=0.73$ +11865,$p=1/2$ +11866,$s\to\infty$ +11867,$k=3$ +11868,$D_3$ +11869,$R=a-X$ +11870,$p\left(1 + \dfrac{\mu^2}{p^2}\right)$ +11871,$\sigma=0.50$ +11872,$\rho_c(X)=\mathsf{TVaR}_{0.8}(X)=8.5$ +11873,$T=T_Z\circ T_X$ +11874,$\text{Poisson}(\lambda'/\phi)$ +11875,$\rho(X) = \rho(X\wedge a) + \rho((X-a)^+)$ +11876,$X_\infty=\mathsf{E}[Y\mid\mathscr{F}_{\infty^-}]$ +11877,$-(1-p)g''(1-p)\ge 0$ +11878,$X-X_1$ +11879,"$(Bob) + (0,-3)$" +11880,"$\gamma_i(\mathbf{x}, x)$" +11881,$x_j$ +11882,$p=0.50$ +11883,$O(h^5)$ +11884,$\mathsf{E}[Z(X)]=1$ +11885,$c_{ii}$ +11886,$a=\alpha(X)$ +11887,$\mathsf{E}[X \mid X > x] = \mathsf{E}[X 1_{X > x}] / \mathsf{Pr}r(X > x)$ +11888,$\mathsf PX$ +11889,$\mathsf{TVaR}_{p_1}(X)\le r$ +11890,$R_1(t)\approx R_1(0)$ +11891,$\rho_{m_i}(Y)\to\rho_m(Y)$ +11892,$\iota^{**}$ +11893,$a/X$ +11894,$V=V(t)$ +11895,$\log(1-\mathsf{Pr}hi(x))$ +11896,"$\mathbf {\Delta\,g(S)}$" +11897,$\kappa_i(X) = X_i$ +11898,$\Delta_0s_0=g(0+)$ +11899,$\mathsf{TVaR}_p(X)=x+\mathsf{E}[X \mid X > x]=x+\mathsf{E}[X \mid X\ge x]$ +11900,$O(n\log(n))$ +11901,$\mathsf{Pr}hi(z_i)$ +11902,$\mathsf Q\in\mathcal Q$ +11903,"$j=0,\dots, m-1$" +11904,$0 \leq s_1 \leq s_0 \leq 1$ +11905,$Y=Y_l+Y_s$ +11906,$g(p)=p$ +11907,$g(p)=\displaystyle\int_0^p\phi(1-t)dt=\displaystyle\int_{1-p}^1 \phi(t)dt$ +11908,$AR\succ BR$ +11909,$N_e$ +11910,$U_{i}^{< j >} = U_{i}^{< k >} = A_{i}$ +11911,$J(x)=x^{-\alpha}$ +11912,$\sup_{\omega\in\Omega} (f(\omega)+g(\omega)) \le \sup_{\omega\in\Omega} f(\omega) + \sup_{\omega\in\Omega} g(\omega)$ +11913,$y_i=b_0+b_1 x_i + e_i$ +11914,$\nu N(a)$ +11915,$\hat\rho(A) = \rho(N)\rho(X) \le \rho(A)$ +11916,$M^2$ +11917,$-\alpha >0)$ +11918,$a_{ro}:=\mathit{VaR}_{p}(X_{-1})=10743.5$ +11919,$S(x) = 1-F(x)$ +11920,$d\tilde p=g'(S(x))f(x)dx$ +11921,$S(M-)$ +11922,$Z_{xn}$ +11923,$F_Y^{-1}(V)=q_Y(V)$ +11924,$\lambda=0$ +11925,$k > 0$ +11926,"$Z\sim\mathrm{ED}^*(\theta,\lambda)$" +11927,$\iota>0$ +11928,$p<0.9$ +11929,$s_{j-1} < s < s_j$ +11930,$\alpha/\beta=\mu\alpha/(\alpha+1)$ +11931,$X_0 = X -\mathsf EX$ +11932,$q^-(p)$ +11933,"$\Delta m_{21}^2 \approx 7.53 \times 10^{-5} \, \text{eV}^2$" +11934,$k\le k_0$ +11935,$P=\rho(X)$ +11936,$m(a) = 1 - \nu F(a)$ +11937,$\sigma_1=0$ +11938,$y=mx + m-m\log(m)$ +11939,$\approx 10^{-3}$ +11940,"$\{1,2,\dots,10000\}$" +11941,$r = R/(A - EL - R)$ +11942,$X\circ f$ +11943,$g^{ak} = (g^k)^a$ +11944,"${}_tp_x\,\mu_{x+t}$" +11945,$\xi$ +11946,$D^n\rho_{X\wedge a}(X_i)$ +11947,$\mu_L=0.03$ +11948,$\zeta\in Y$ +11949,$g(S(a))/S(a)$ +11950,$1/CV^2$ +11951,$f(\mu) = 2 / \mu$ +11952,$t\in\Omega$ +11953,$BY \succ AR$ +11954,"$14 billion, or some \$" +11955,$l(y;\mu)=-y/\mu - \log\mu$ +11956,$0\le f(x)-f(y)\le x-y\ \forall 0\le y < x$ +11957,$(1-p)/(p\nu(p)^2)$ +11958,$R_0(t) < (1-t)P(0)$ +11959,"$104 million pretax writeoff, resulting in a $" +11960,${}_0V = 1$ +11961,$\lambda = m(\alpha-1)$ +11962,$\pi(p)$ +11963,$r(u) - m'u$ +11964,$\mathcal M_\rho=\{ m \}$ +11965,$w = \sum_i w_i$ +11966,$\Delta \mathit{MV}_{gc}(a)$ +11967,$\rho(X)=E_Q(X)$ +11968,$M \implies\ g$ +11969,$S(s)$ +11970,$a<\infty$ +11971,$5 \times 10^{19}$ +11972,$-(\nu-l)-l=-\nu$ +11973,"$P((1+\epsilon)v_1, v_2, a+da)=P^a((1+\epsilon)v_1, v_2)$" +11974,"$[0,1]\to[0,\infty)$" +11975,$ | \$ +11976,$\alpha=(p-2)/(p-1)$ +11977,$ since the contact function $ +11978,$g(s)=s^{0.7}$ +11979,"$X_{i,j} \leftarrow \kappa_{i}(X_j)$" +11980,$\rho(A_k) \ge k\mathsf E[N]$ +11981,"$\mathcal A_t = \mathcal A_{t,t+1} + \mathcal A_{t+1}$" +11982,$g^{-1}(p)=p^2$ +11983,"$X_{t-2,3}$" +11984,$\sum_i \mathsf{VaR}_{p^*}(X_i) = \mathsf{VaR}_{0.996}(X)$ +11985,$\alpha X + (1-\alpha)Y$ +11986,$f_0(x) = \lim_n g_n(x)$ +11987,$ipl(p)$ +11988,$\phi(s)=\mathsf{E}[e^{isX}]$ +11989,$\mathsf{E}[X]=1/\beta$ +11990,$k_i=a_i/x_i$ +11991,$x+b$ +11992,$\alpha_i(x)S(x)=\mathsf{E}[(X_i/X)1_{X>x}]$ +11993,$br+v$ +11994,$\mathsf{E}[\kappa_i(X)g'(S(X))]$ +11995,"$\mathsf{TVaR}_{0.95}(X)=\int_0^{1000}g(S(x))\,dx$" +11996,$Q_k^i$ +11997,"$\rho(X) = \mathsf{E}(\zeta X) = \langle \zeta, X \rangle$" +11998,$\mathscr{M}$ +11999,$\zeta + \dfrac{1}{1-\alpha}\sum p_i\eta_i\le \omega$ +12000,$m(1+2\frac{m}{p} +\frac{1+a^2}{a^2}\frac{m^2}{p^2})$ +12001,$n-3$ +12002,$nu(B)=$ +12003,$\phi(Y)$ +12004,$q(p)=\sum_i \mathsf{E}[X_i\mid X=q(p)]$ +12005,$e^{- \kappa(\theta)}>0$ +12006,$\mathsf{E}[X_i/X \mid X\ge a]$ +12007,"$\{(s_j, g_j)\} \cup \{(0,0), (1,1)\}$" +12008,"$c=1,2,3$" +12009,$P^i = \sum_j q_j \kappa_i(j)\Delta X_j$ +12010,$\beta g(S)$ +12011,$g(S_0)=1$ +12012,$k\ge k_0$ +12013,$R'_1(1) \le 0$ +12014,"$(C.north east)+(1.5, 0)$" +12015,$\beta_i(x)$ +12016,"$X^{\tau_n} = \{X_{t \wedge \tau_n}, t \geq 0\}$" +12017,$y_n$ +12018,$\mathbf {\mu}$ +12019,$g'S_t$ +12020,$P_t$ +12021,$\mathsf{E}(X_i \mid X=a)$ +12022,$X+X^{n}$ +12023,"$0,1,\dots,k$" +12024,$X\wedge 1$ +12025,$\mathsf{COMON}$ +12026,$1-F(x)=1-p$ +12027,$f(x)=(x-d)^+1_{\{x \le m \}}$ +12028,$\{Y=y\}$ +12029,$G:\mathbb{R}^n\to\mathcal{X}$ +12030,$Wf/(1-g)$ +12031,"$\rho^*(\mu)\ge \sup_{a\in\mathbb{R}} \{ \langle \mu,X+a \rangle - \rho(X+a) \} = \sup_{a\in\mathbb{R}} \{ a\mu(\Omega) -a+ \langle \mu,X \rangle - \rho(X) \}$" +12032,$\mathscr{P}=\{ \mathsf{Q} \mid \mathsf{Q} \ll \mathsf{P} \}$ +12033,$\beta_2$ +12034,"$\mathsf{Pr}hi(a, b)=(\phi(f(a)), \phi(b))$" +12035,"$1,2$" +12036,$q_Z(U)\in\mathscr{P}$ +12037,$u(x)=(1-e^{-\pi x})/\pi$ +12038,$0=p_1 0$ +12049,$X_i=U_i$ +12050,"$\mathsf{PML}_{n, \lambda}(X)=\mathsf{PML}_{n, \lambda}$" +12051,$V_1=V_2$ +12052,$\rho(X\mid \mathscr F_1) =\mathsf E[X g'\mathsf{Pr}r(X>x\mid \mathscr F_1) ]$ +12053,$\mathsf{E}(Z\mid X)=Z$ +12054,$35.40 \* | \$ +12055,$s=1-t$ +12056,$P_1(t)\to P(1)$ +12057,"$1,000,000 and the treaty covers losses for the layer \$" +12058,$p_i=0$ +12059,$P_X(A)=0$ +12060,$\epsilon(t-\mathsf{E}_q(X_2))$ +12061,$\mathbb{Q}(\sqrt{2})$ +12062,$cv$ +12063,$T = \min\{ t:U(t)\le 0 \}$ +12064,"$(0,0,0,0,0,0,0,5,0,5)$" +12065,$\mathsf{E}[X]+k\mathsf{Var}(X)$ +12066,$t-\delta t$ +12067,$\mathsf{Pr}r(X\ge p_{j_\pm})$ +12068,$\phi_R-\phi_{m}^e$ +12069,$\beta_i(X_4)$ +12070,"$375,088,155 | -5.5% | $" +12071,$\theta_s>0.5$ +12072,$\dfrac{g(s)}{s}$ +12073,$m\in H_r$ +12074,$g_i=u_i^{1/b} \alpha_Y + 1$ +12092,$V(2)$ +12093,$f_0(x) = \sup_n f_n(x)$ +12094,$b = g/(1-g)$ +12095,$P_1(t)/t<\rho(X_1)$ +12096,$k=c/(e^c-1)$ +12097,$-m_2/(1-s_2)$ +12098,$t_2<0.5$ +12099,$K_\theta(t)=\theta t + \dfrac{t^2}{2}$ +12100,$f=\alpha g + (1-\alpha)h$ +12101,$<\mathsf{E}[X]$ +12102,"$B_3=[-k, \epsilon]$" +12103,$P-\mathsf{E}[U]$ +12104,$U=\sum_i U_iP_i / P$ +12105,$0 \le f(X) \le X$ +12106,$\mathsf{FATOU}$ +12107,"$EL = E\lbrack X_{0}\rbrack = \sum_{j = 1}^{M}{\min\left( X_{0}^{j},A \right)p_{j}}$" +12108,"$=2\displaystyle\int_\mu^y \dfrac{y-t}{V(t)}\,dt \ge 0$" +12109,"$\mathrm{ess\,sup}(X)=\sup\{x\mid \mathsf{Pr}r(X>x)>0 \}$" +12110,$\rho(Y)\ge a_Y-2^{*}$ +12111,$Z=Z_X$ +12112,$\rho(X)\le\liminf\rho(X_n)$ +12113,$\sup X=\inf$ +12114,$L-f(L)$ +12115,$p_0$ +12116,$\mathsf{E}[X]=\lambdax) \approx n\mathsf{Pr}r(Y>x)$ +12120,$dp=dF(x)$ +12121,"$(x, y)$" +12122,$s_I=s/\omega_I$ +12123,$\{n\mid X(n)\not =0\}$ +12124,$m^3/p^3$ +12125,$e^x=\sum_{n\ge 0} x^n/n!$ +12126,$wq_X(p)+(1-w)q_Z(p)$ +12127,$a=1/2$ +12128,$1/16$ +12129,"$(\mu, \sigma^2)$" +12130,$W(0)\neq 0$ +12131,$g'(s)=\beta(1-s)^{\beta-1}$ +12132,$\rho_a(0) = \rho(0 \wedge a(0)) = \rho(0 \wedge 0) = \rho(0) = 0$ +12133,$_0(X) =\mathsf E[X]$ +12134,$\mathcal A=\{X\mid \rho(X)\le 0 \}$ +12135,$\Delta X\wedge a$ +12136,$S(x)=e^{-\mu x}$ +12137,"$[-k, \epsilon] \succeq [0, \epsilon - k]$" +12138,"$26,354. The loss of \$" +12139,$P = \log(\mathsf{E}[e^{\pi X}])/\pi$ +12140,"$u_1, \dots, u_t$" +12141,$\mathsf{E}_\mathsf{Q}[X_i\mid X]=\mathsf{E}[X_i\mid X]$ +12142,$v(a-L)$ +12143,"$\mathscr{G}=\{\emptyset, \Omega\}$" +12144,$S(x)/P(x)$ +12145,$\alpha_Y \le \alpha_X$ +12146,"$x_{1,i}+x_{2,k(i)}$" +12147,$c(x)$ +12148,$\alpha_2(98)=0.9$ +12149,$E(X_i \mid X=x)f_X(x)$ +12150,$\mathsf{var}phi (X) \ge 0$ +12151,$ and so $ +12152,$g(s)=1\wedge(s/0.35)$ +12153,$A+\delta$ +12154,$(1-r_0)\delta_1$ +12155,$v^{(\mathrm{time\ to\ payout})}\rho(\mathrm{risk now})$ +12156,$a < \max(X)$ +12157,$E_Q$ +12158,"$(X_1,\dots,X_k)$" +12159,$\rho(X)=\max_{\mathsf Q\in\mathscr{Q}} \mathsf{E}_{\mathsf Q}[X]$ +12160,$(ng)$ +12161,$\mathsf{E}_{\mathsf{Q}}[X\wedge a] = \rho(X\wedge a)$ +12162,$X_i=X_i\sum_i \partial C/\partial x_i + \partial N/\partial x_i$ +12163,${}^{[<169]}$ +12164,$\tau_i$ +12165,$110K; and \$ +12166,$\theta-\log(1+\theta))=y$ +12167,"$x_j=0, 1$" +12168,"$\rho(X)=\mathsf E[X] + \lambda \mathsf{cov}(X, Z)$" +12169,$x=X(p)$ +12170,$S_a=0$ +12171,"$j=1,\dots, n$" +12172,"$642 billion of gross written premium in 2017, and, after reinsurance, net earned premium was $" +12173,"$cos(0)*sin(90)*(1,1)$" +12174,$\mathsf{E}[r_{FS}(s*)] = +\infty$ +12175,"$i,j$" +12176,$s = \iota_U - \iota^\star(k_r + a_r)/(1-a_c)$ +12177,$\forall A\in\mathscr{F}$ +12178,$\rho(X)\ge \rho(Y)$ +12179,$w=E[w|s=0.1]=0.06405$ +12180,$m=1$ +12181,"$(Bob)+(0,-3.25)$" +12182,$x\mathsf{E}[X_i/X\mid X>x]$ +12183,$\infty-\infty$ +12184,$1_{U>p}$ +12185,"$c(1,3)-c(3)$" +12186,${}^{[<77]}$ +12187,"$\mathsf{E}[(X_i-\mathsf{E} X_i)(X-\mathsf{E} X)]/\mathsf{SD}(X)=\mathsf{cov}(X_i,X)/\mathsf{SD}(X)$" +12188,$\delta_p$ +12189,$t = g(S(x))$ +12190,"$[s_L,s_m=1]$" +12191,"$p(\cdot, y)$" +12192,$r_h=\mu_L=0$ +12193,$\ne$ +12194,$\mathsf{biTVaR}(Y)=\mathsf{TVaR}_{p^\ast}(Y)$ +12195,$R(X)$ +12196,$\mathsf{E}_{\mathsf Q}[.]$ +12197,"$(-\infty,\infty)$" +12198,$\zeta=\zeta_X$ +12199,"$\beta(X) = \int\check g(S_X(x))\,dx$" +12200,$\mathsf{TVaR}_{p*}(X)=a$ +12201,$X_T=X$ +12202,$Z_\mathit{lift}$ +12203,$-br$ +12204,$\kappa_p(\theta)$ +12205,$s =$ +12206,$E[X_i \mid X \ge a]$ +12207,$2.576$ +12208,$Z(S_X(x))=-(x-\mu)/\sigma$ +12209,"$U = X\wedge A := \min(X,A)$" +12210,$\mathsf{SD}$ +12211,$\le 2$ +12212,$\alpha\not\equiv 0$ +12213,$\pi_i$ +12214,$0\le x \le 1000$ +12215,"$X_{0,1},X_{0,2},\dots, X_{0,N}$" +12216,$\mathsf{Pr}r(B)=0$ +12217,$f(x)=(\sqrt{2\pi}x)^{-1}\exp(-(\log(x)-\mu)^2/2\sigma^2)$ +12218,$\mathsf{E}[X_0] + \mathsf{VaR}_p(X_1)$ +12219,"$\mathcal{M}_{X,c}=\mathcal{M}$" +12220,$f_{\max{}}$ +12221,$E_0+A$ +12222,$\mathsf{E} X + c{X-\tau }_p$ +12223,$-\log(1-\mathsf{Pr}hi(x))$ +12224,$\mathbf N$ +12225,$D_1$ +12226,$PQ = P/Q$ +12227,$X=X_n$ +12228,$n \times 1$ +12229,$23.81 / 34.05 = 70$ +12230,$a_l\mu$ +12232,$X\preceq_1 Y$ +12233,$\mathsf{Pr}r(X< q(p))\le p \le \mathsf{Pr}r(X\le q(p))$ +12234,$3^{-n}$ +12235,"$71,487,378 | 22.3% | $" +12236,$\{A_n\}$ +12237,$\mathbf {Q_{2}\Delta X}$ +12238,$st$ +12239,$X=(X_1-x_0) + (X_2-X_1) = Y_1 + Y_2$ +12240,"$\mathsf{E}[\min(X,a)]=\mathsf{E}[X\wedge a]$" +12241,$w/s = g'(s-) - g'(s+)$ +12242,$\kappa(\theta)=-\log(-\theta)$ +12243,$\mu T$ +12244,$\bar q_{X_1+X_2}(s)=q_{X_1+X_2}(1-s)$ +12245,$g^e(s^\star)=g^u(s^\star)$ +12246,$A = fX + Y$ +12247,$\rho(L) = F^{-1}(1-g{-1}(1-p)) dp > \mathsf{E}(L)$ +12248,$I^\star$ +12249,$\delta^2 p + \nu^2 q-(p-\nu)^2=p(1-p)$ +12250,$\lfloor pN\rfloor$ +12251,$\iota(a)$ +12252,$9 = 2^3 + 1 = 2^{(2^1 + 1)} + 1$ +12253,$(1-\lambda)(1+\gamma)$ +12254,$\rho(Y) = \rho(Y-X + X) \le \rho(Y-X) + \rho(X)$ +12255,$X_t \neq \mathbb{E}[0 \mid \mathcal{F}_t]$ +12256,"$[a_Y, b_Y]$" +12257,$a=\mathsf{E}[X\mid X\ge a^*] =\mathsf{E}[X_1\mid X\ge a^*] + \mathsf{E}[X_2\mid X\ge a^*]$ +12258,$1-p\le s\le 1$ +12259,$\mathbf{Tm} = \mathbf{r}$ +12260,$MX$ +12261,$a0$ +12292,"$(\Omega, \mathscr F, \mathsf P)$" +12293,$0\le p^*\le 1$ +12294,$(T\lambda)\{\lambda_t\Omega=0\}=0$ +12295,$-1_{B_l}$ +12296,${}^{[>75]}$ +12297,$A(1_{X_1>x_1}+1_{X_2>x_2}) \le A(1_{X_1>x_1}) + A(1_{X_2>x_2})$ +12298,$a>b_h$ +12299,$S_{\mathbf{v}}(a)$ +12300,$\mathsf{Amb}(X)$ +12301,$f_{\mathbf{v}}$ +12302,$u^{(4)}<0$ +12303,$\preceq_k$ +12304,$\dfrac{\partial l}{\partial \theta}=y-\kappa'(\theta)$ +12305,$E(X^k)\le E(Y^k)$ +12306,$g(0.x_1x_2x_3...) = 0.x_2x_4\dots$ +12307,$x_1=q(p)$ +12308,$x_0 \gtreqqless a_1$ +12309,$-10$ +12310,$\mathsf{VaR}_{0.99}(X)$ +12311,$B_n$ +12312,$a(X)\le a(Y)$ +12313,$\rho\ge 0$ +12314,$a = P + Q$ +12315,$u(x)=-v(-x)$ +12316,"$(s,m)$" +12317,$\mathsf{E}_{\mathsf Q}[X_i(a)]$ +12318,$(x-d)^+ \wedge l$ +12319,$Y\in L^0$ +12320,$S(\mathbf{x}; a)$ +12321,$G'$ +12322,$\bar P_\tau(a)=\bar P(a) + \tau(a-\bar P_\tau(a))$ +12323,$a=q_X(0.99)$ +12324,$p\approx 0.01$ +12325,$\mathsf{P}[ \cdot ]$ +12326,$\rho^E_L(t)$ +12327,$\mathbf {M_2\Delta X}$ +12328,$M(x)=P(x)-S(x)$ +12329,$\phi(\mathsf E[\bar s])^2$ +12330,$f(A)=B\cap f(L)$ +12331,$\mathsf{T}$ +12332,$\mu_t$ +12333,$x_{i+1}=x_i+1$ +12334,$\mathsf{TVaR}_{0.8}(X+tX_1)$ +12335,$2^1\rightarrow 3^1-1=2 \rightarrow 1 \rightarrow 0$ +12336,$\lambda_t=\lambda$ +12337,"$i=R,\dots,i^\star$" +12338,$dP/dx$ +12339,$\lambda=0.34273$ +12340,"$D_i(X_1,\dots,X_n; a)$" +12341,"$A\mapsto p(t, A)$" +12342,$\bar p_i = \bar p_{i-1}+p_i$ +12343,$T_{x/n}$ +12344,"$144,000 (=\$" +12345,$\mathsf{TVaR}_1=\sup$ +12346,$D\rho_X(X_i)$ +12347,$-1_{A^c}$ +12348,$p=(\alpha-2)/(\alpha-1)$ +12349,$w_l$ +12350,$\mathsf E[N\mid I]=0$ +12351,$\mathsf{E}(X_i \mid X\le a)$ +12352,$S_m=\mathsf{P}(X>X_m)=0$ +12353,"$\bar P_{t,0} = D\rho_{W_t}(Y_{t,0})$" +12354,$V(a)$ +12355,$S_X(x)=\mathsf{Pr}hi(-(x-\mu)/\sigma)$ +12356,$c(y)=\dfrac{e^{-1/(2y)}}{\sqrt{2\pi y^3}}$ +12357,$x=q^-(p)$ +12358,$M(x)/Q(x)$ +12359,$\mu=7.8044$ +12360,$q_{\mathbf{x}}=F_{\mathbf{x}}^{-1}$ +12361,$\Theta_p$ +12362,$\mathsf{Pr}r(N=n)=p_n$ +12363,$\delta+\nu$ +12364,$c=-\mathscr{G}amma(-a)(c_1 + c_2)\cos (a\pi/2)$ +12365,$c > 1/2$ +12366,$\kappa'(\theta(\mu))=\mu$ +12367,$\rho_g(Y)$ +12368,$\alpha(Q)$ +12369,$\nabla \alpha\propto \pi$ +12370,$\delta_0$ +12371,$e_i$ +12372,$+/-$ +12373,$g \circ S$ +12374,$\kappa^i(X_k)$ +12375,$E \subseteq \mathbb{R}$ +12376,$\iota(p)=\delta(p)/\nu(p)$ +12377,$s=1-p$ +12378,$X(\omega) = \omega$ +12379,$> r$ +12380,$x^*=\mathsf{VaR}_p(X)$ +12381,$B(b)<0$ +12382,$1+2c(1-\mathsf{Pr}r(Z>\mathsf{E} Z))$ +12383,$d(y;\mu)$ +12384,$(a-X)^+=0$ +12385,$\beta_0$ +12386,$\tilde p > p$ +12387,$X(t)=X$ +12388,$\Delta_0 s_0 = g(0+) = T_L$ +12389,$\mathsf{E}_Q(Y)$ +12390,$X_2'$ +12391,"$\mathbf {D^f\rho_{X\wedge 30,X}(X_2)}$" +12392,$\lambda=\mu^{2-p}(\alpha+1)/\alpha\to \mu(\alpha+1)/\alpha$ +12393,$\mathsf{E}[(X-\mathsf{E} X)^+]$ +12394,$\mathsf{Pr}r(X>\mathsf{VaR}_p(X))=1-p$ +12395,$c_Y$ +12396,$t\in M$ +12397,$L_0^{l_1} + L_{l_1}^{l_1+l_2} = L_0^{l_1+l_2}$ +12398,$\alpha \uparrow 2$ +12399,${}^{[>37]}$ +12400,$(P-\mathsf{E}[U])/(A-P)\ge \rho$ +12401,$n=5$ +12402,$X \mapsto bX$ +12403,$a=\sum_i a_i$ +12404,$1/\mu$ +12405,"$\mathcal R_{X,r}=\{ \rho\in\mathcal R\mid \rho(X)=r \}$" +12406,$a={a}$ +12407,$\beta_1(x)\le \alpha_1(x)$ +12408,$m>1$ +12409,$\mathsf{VaR}_{0.95}(X)$ +12410,$0 \le s_1 \le s_L$ +12411,"$\mathsf{biTVaR}_{p_0,p_1}^{w}$" +12412,$-\rho(-H)=\rho(H)$ +12413,$-g''(s)=\alpha(1-\alpha)s^{\alpha-2}$ +12414,$S(x)\gg 0$ +12415,$a=0.2$ +12416,$wx + (1-w)y\in C$ +12417,"$\int e^{\theta y}c(y)\,dy< \infty$" +12418,"$1,000). The 99^th^ percentile loss amount is \$" +12419,$X_1+({n}-X_2)$ +12420,$V_s$ +12421,$q(1-g^{-1}(1-p))/q(p)$ +12422,$\sum_t \sigma_t^2 = 1$ +12423,$x=y^4$ +12424,$E_Q(X \mid \mathcal{G})$ +12425,$p=(2+\alpha)/(1+\alpha)$ +12426,$=V=P+r(P+S)-rS-L=eL+\rho S$ +12427,$\mathsf{Pr}r(X_1=1)=p$ +12428,$r_a+r_l$ +12429,$\theta=-\dfrac{1}{2\mu^2}$ +12430,$\int_0^1 g(s)ds - 0.5$ +12431,$\hat p > p$ +12432,$\text{E}(G)=1$ +12433,$r_O=0.02041$ +12434,$Z=g'(S_X(x))$ +12435,$\mathit{EGL}_{gc}(a)$ +12436,"$\Omega=(0,1)$" +12437,"$\langle \nabla\zeta, N \rangle + \langle \zeta, \nabla N \rangle$" +12438,$aX+Y$ +12439,"$\mathcal{A}=\{\mu\in \mathscr{P} \mid \langle \mu,X \rangle \le \rho(X) \ \forall X\in\mathcal{X}\}$" +12440,"$\displaystyle\int_B \mathsf{E}[X\mid\mathscr{G}]\,d\mathsf{Pr}$" +12441,$0=s_0 \le s_L$ +12442,"$g = 9.81 \, \text{m/s}^2$" +12443,$S\Delta X$ +12444,$A=G\triangle P$ +12445,$\mu=1/c$ +12446,$s+1 - (s - T +1))=T$ +12447,$ah_1$ +12448,$a(\mathbf{v}) =\mathsf{VaR}_p(X(\mathbf{v}))= q_{\mathbf{v}}(p)$ +12449,$x=(y-\mu)/\sigma$ +12450,$r\times$ +12451,${}^nS_X(t)\le {}^nS_Y(t)$ +12452,$\tilde X=X\wedge a$ +12453,$\alpha=-2$ +12454,$G_t$ +12455,$Y_i$ +12456,$T_2$ +12457,$\rho(X) = \sup_{\mathcal{B}bb Q} \{ E_{\mathcal{B}bb Q}(-X) - \alpha(X) \}$ +12458,"$(1-g(s), 1-s)$" +12459,$g(s)=(1-p)^{-1}s\wedge 1$ +12460,$\check g(st)=vs = \check g(s)\check g(t)$ +12461,"$\mu, p, \sigma$" +12462,$\mathsf{E}[XY\mid\mathscr{G}]=X\mathsf{E}[Y\mid \mathscr{G}]$ +12463,$\partial \tau^{-1}/\partial \mu = 1/(1+\mu^2)$ +12464,$e_x=\sum_t {}_tp_x$ +12465,$K'(0)=\kappa'(\theta)=\mu$ +12466,$\mathbf {a_{2}}'$ +12467,$p=F(x)=\mathsf{Pr}r(X\le x)$ +12468,$\frac{d}{dp}(1-p)^{-1}=(1-p)^{-2}=q^{-2}$ +12469,$Y=-X$ +12470,$0.7168 \times 591.25 = 423.8$ +12471,$L(t)=S(T^{-1}(t))=\{ S(\omega)\mid T(\omega)=t \}$ +12472,$a = v^2 / r$ +12473,$a\le \rho(X)\le b$ +12474,"$Y_{0,t}:=\sum_{d>t} X_{0,d}$" +12475,$a_0=x_0$ +12476,$L_\sigma=F_L^{-1}(\tau_\sigma^{-1}(U))$ +12477,$0.5 < t < 1$ +12478,$\nu (\Delta X_k-L_k)$ +12479,$\mathsf{TVaR}_\alpha(X)\le \omega$ +12480,$(-2N\log(1-p))^{1/2}=22.49$ +12481,$s_3=1$ +12482,$(1+r)Z=\mathsf Q/\mathsf{P}$ +12483,"$100K, \$" +12484,$\rho_{g}$ +12485,$\mathcal E$ +12486,$\bar P_{act} = \bar P + F_0 > \bar P$ +12487,$\phi(p) = g'(1-p)$ +12488,"$\pi_h^{L,R}$" +12489,$k_i$ +12490,$-\rho(-X)$ +12491,$\mathsf{MON}'$ +12492,$q_\zeta$ +12493,$V(\mu)=\mu^3$ +12494,$M_i(t)=C_i(t)$ +12495,$c(y;\nu)$ +12496,$\beta_i(a)g(S(a))=\mathsf{E}_{\mathsf{Q}}[(X_i/X) 1_{X>a}]$ +12497,$m(1-\frac{m}{n})$ +12498,$\max_\mathsf{Q} \mathsf{E}_\mathsf{Q}[X] - \alpha(\mathsf Q)$ +12499,$x=S^{-1}(g^{-1}(u))$ +12500,$B(p)$ +12501,$s_R=1$ +12502,$t_* \le t^*$ +12503,$q(0)$ +12504,$\bar P(\infty)=\mathsf{E}[q(U)\phi(U)]$ +12505,$\mathbf S=\mathbf C\mathbf C^t$ +12506,$=\mathsf{E}(X_i(a))$ +12507,$v_k(x)=x^k$ +12508,$0.2= 0.25/1.25$ +12509,"$\bar P_{0,0}$" +12510,$E[M_\tau] \ge 1$ +12511,"$(\s, 4-\s)$" +12512,$x\mapsto \mathsf{E}(X_i\wedge x)$ +12513,$\kappa_{T_\nu}(y)$ +12514,$\mathsf{Pr}(X\in A)=0$ +12515,$Y_1=X_1 + \dots + X_n$ +12516,$a(f)=\dfrac{gs_g}{1-f-fgs_g}$ +12517,$\alpha > 2$ +12518,"$H(A, L, t)$" +12519,$\rho(-1_{A^c}) = \rho(-1_{B_l} - 1_{B_r}) = \rho(-1_{B_l}) + \rho(-1_{B_r})$ +12520,$g(uv)\le g(u)g(v)$ +12521,"$p_1\in(0,1)$" +12522,$p<0$ +12523,$\beta=\dfrac{c_1-c_2}{c_1+c_2}$ +12524,$N':=kNT$ +12525,"$F_{df_m,df_e}=df_e/(df_2-2)$" +12526,$\rho_c(X)$ +12527,$\nu^{\star}$ +12528,$p-1=28$ +12529,"$f_x(x_i, \hat x_i)$" +12530,$Z(u)=sum_i u_iX_i$ +12531,"$(rep.east) + (1.5, 1.5)$" +12532,"$\Omega=\{ 1,2,3,4,5,6 \}$" +12533,$\mu_0=\mu_1$ +12534,$X_t=\mathsf E[q(W_T) \mid B_t]$ +12535,$r_s$ +12536,$R_1(t)= \bar P^a_1(t)/(1-t)$ +12537,$s_R < s_e < s^\star < s_u < 1$ +12538,$(1 - \nu F(a))$ +12539,$m_3 := m_2$ +12540,"${1+1}*(1,.5)$" +12541,$X(\mathbf{v})=\sum_i v_iX_i$ +12542,$(r_1+r_2)/2$ +12543,$\rho_1(X)(U_1)$ +12544,$\mathsf{E}(X) = \displaystyle\int_0^\infty xf(x)dx = -xS(x)\vert_0^\infty + \displaystyle\int_0^\infty S(x)dx = \displaystyle\int_0^\infty S(x)dx$ +12545,"$\mathbf{T_0}=(\mathsf{TVaR}_{p_j}(X_i))_{i,j}$" +12546,$D=\sum_{i\in I} D_i$ +12547,$\mathsf{E}_\mathsf{Q}[Y\mid X]\mathsf{E}[Z\mid X] = \mathsf{E}[YZ \mid X]$ +12548,$R^S=g^{kS}$ +12549,$ is the MGF of a gamma with shape $ +12550,$\mathsf{E}[X_i (X\wedge a)/X \mid X=x] = \mathsf{E}[X_i\mid X=x] (x\wedge a)/x$ +12551,$GF$ +12552,$0 1/3$ +12590,$B \in\mathcal B_p$ +12591,"$(y,x)$" +12592,$0 < s < 1$ +12593,$1/\beta$ +12594,$\alpha_\epsilon=\alpha$ +12595,$X'\Delta g(S)$ +12596,$\mathsf{E}[X\mid T=t]$ +12597,"$(A.north east)+(0.1, -0.05)$" +12598,"$(0,1,2,3,4,8,8,8,8,9)$" +12599,"$C(S_0, a, t)$" +12600,$W_0=Y_{0} + W_1$ +12601,$\{s_j\}$ +12602,$A=\bigcup_i A_i$ +12603,$t > 2/3$ +12604,$\sigma=2.58$ +12605,$G>c(x)$ +12606,$=\exp(8.7103 + \mathsf{Pr}hi^{-1}(0.995)\times 1)$ +12607,$A\subset \Omega$ +12608,"$p=1,2$" +12609,$g^1(s_{R+1})=g^2(s_{R+1})=g_{R+1}$ +12610,$xy^4 / (x^2 + y^8)$ +12611,$f_x$ +12612,$X(\omega)=q(T(\omega))$ +12613,$0.33$ +12614,$\mathsf{E}(X_i/X ; X > a)$ +12615,$s_1 > s_0^2$ +12616,$\mathsf{Var}(\sum C'_i)=v_{res}^2 \sum c_i^2$ +12617,$\mathsf{Q}'(\Omega_a) =\mathsf{Q}(\Omega_a)$ +12618,$A = \alpha(X)$ +12619,$t=0.06405$ +12620,"$ also gets more precise. For example, with $" +12621,$F(x)=\mathsf{P}(\{X\le x\})$ +12622,"$5,000 and \$" +12623,$X ^ a = \displaystyle\sum X_i(a)$ +12624,$Q_1=0.125$ +12625,"$[s_0=0,s_R<1]$" +12626,"$(m,p)$" +12627,$\bar P=\bar P_1+\bar P_2$ +12628,$\| Z \|^*= \sup \{ \mathsf{E}[YZ] \mid \| Y \| \le 1 \}$ +12629,$M_i=\beta_ig(S)-\alpha_iS$ +12630,$S_g = g\circ S$ +12631,$s=s(Z_i)$ +12632,$\sigma_1=0.5$ +12633,"$(-1/2,0)$" +12634,$g''(p)=-\phi'(1-s)\le 0$ +12635,"$\mathsf{biTVaR}_{0,1}$" +12636,$\rho_{(g)}(X)=\int xg'(S(x))f(x)dx$ +12637,$\rho(X)=\rho_{m_0}(X)$ +12638,$\tpx=e^{-1}$ +12639,"$(3,4)$" +12640,$\approx\sqrt{2Np}$ +12641,$\mathbf R_{\ge 0}$ +12642,$g_0$ +12643,$P(\hat s)=\mathsf{E}[\hat s]=s$ +12644,$\rho(X)=-\tilde\rho(-X)$ +12645,"$F_X,q_X$" +12646,$f(y;\mu)=tbd$ +12647,$X_i\sim X$ +12648,$D$ +12649,$\mathsf{EE}=\mathsf{F}'\mathsf{F}$ +12650,$=\mathrm{MV}(X\wedge a)$ +12651,$\mu_=0.03$ +12652,$=P=\mathrm{MV}(X\wedge a)$ +12653,$\Delta \tilde p\times T$ +12654,$a=a(\mathbf{v})$ +12655,$L_0$ +12656,"$P_i = E[X_i g'S(X)] = \int g(S(x))\,dx = \int x_ig'S(x)f(x)\,dx$" +12657,$\rho\mapsto a^\rho(\ \cdot\ ;\ \cdot\ )$ +12658,$0\le\beta<1$ +12659,$Q_P=a-P_P$ +12660,$\downarrow$ +12661,$\mathsf{E}[\log(X)]$ +12662,"$X=X(U_1, U_2, U_3)$" +12663,$\nabla_x f= \nabla_xq_\alpha -\nabla_x G$ +12664,$v-\nu^*=\delta^*-d$ +12665,$\mathsf{E}[X_i /X \mid X]$ +12666,$\sup_n \| X_n \|< \infty$ +12667,$\mathsf{E}[X|X>x]=x+\mathsf{E}[X]$ +12668,"$u,v$" +12669,$X=C(\bar x)+N(\bar x)=$ +12670,$S(x)=(k/(k+x))^\beta$ +12671,$\lim_{t\downarrow 0} F(x+t)=F(x)$ +12672,$g(x)\ge x$ +12673,$\epsilon_-$ +12674,"$0,0,0,1,2,5,8,12,23,40$" +12675,"$\zeta=(5,4,3,2,1)$" +12676,$\mathsf{E}\_\mathsf{Q}[(X\wedge a)\_2(a)]$ +12677,$\beta$ +12678,"$(A.north east)+(0.2, -0.05)$" +12679,$\mathsf{E}[X_1\wedge 2272] = 732.3$ +12680,$\lfloor x \rfloor$ +12681,$\hat\rho_{\omega_I}(X)=\hat\rho(X)$ +12682,"$(fun2a.south west)+(0,-2*\ )$" +12683,"$2 Billion. The CEA must obtain reinsurance contracts in an amount of at least 200 percent of the total initial cash contributions from participating insurers. To meet this requirement, the CEA has obtained a two-year contract for $" +12684,$a\ge c$ +12685,$=1-\nu F(a)$ +12686,"$\forall E\in\mathcal{A},\ \exists N\in\mathcal{B}:\ QN=0$" +12687,$\mathsf{TVaR}_{0.95}(Y)=0.8\mathsf{E}[X]=2000$ +12688,"$j=7,8,9$" +12689,$\hat\mu=y$ +12690,$X_i(x_i)$ +12691,$P + \rho_i(F_i) < \rho_i(X_i) \iff P < \rho_i(X_i) - \rho_i(F_i)$ +12692,${}^{[>58]}$ +12693,$46.156+5.5=51.656$ +12694,$r \approx 1.496 \times 10^{11} \ \text{m}$ +12695,$\kappa_X(\theta)=\lambda(e^\theta-1)$ +12696,$\bar P$ +12697,$F_Y$ +12698,$X\mid \mathscr F$ +12699,$\psi=1_{A\times B}$ +12700,$\sigma(1-t)=g'(t)$ +12701,$1_{Ua)=\mathsf{Pr}hi(-d)$ +12711,$\sup$ +12712,$w_i^y := f(X_i^y)/g(X_i^y)$ +12713,$g(s)q$ +12714,$-q_{-Y}^-(1-p)$ +12715,$\mathsf{E}[X_i\mid X = x_p]$ +12716,$\omega=\sqrt{s}$ +12717,$\rho_c(Y)=\mathsf{E}[Y]$ +12718,$\int_0^\infty g(S(x))dx$ +12719,$\mathsf{E}[aX]=a\mathsf{E}[X]$ +12720,$\mathsf{CTE}_p(X)=(12+25)/2=18.5$ +12721,$\mathsf{Pr}(B\cap A)$ +12722,$e^\theta/(1+e^\theta)$ +12723,$Y\preceq_2 X$ +12724,$r_f=0$ +12725,$g''(t)=-\phi'(1-t)\le 0$ +12726,$\lambda^Q$ +12727,$P = \mathsf{E}[X] + \pi \mathsf{E}[|X-\mathsf{E}[X]|^p]^{1/p}$ +12728,"$1,493.70 \$" +12729,$\mathsf{TVaR}_{p_j}$ +12730,$ and the average thickness of the difference in support sets must be zero because the two support sets have the same measure $ +12731,$g(0+)=\lim_{t\downarrow 0} g(t)\ge 0$ +12732,$O(mn\times n^2)$ +12733,"$k_1, k_2$" +12734,$n_i$ +12735,$\mathsf{VaR}_p(X)$ +12736,$(X_n)_{n \geq 1}$ +12737,$\mathsf{E}[Z_j\mid X]$ +12738,$p_i + \mathrm{Sh}(\eta_i)$ +12739,$3000$ +12740,$950\times 1.045^{-4}+100=896.63$ +12741,$f_0\in L^1(\mathbb R)$ +12742,$q \cdot X$ +12743,$\rho^e(s)$ +12744,$\mathsf{S}=\mathsf{C}'\mathsf{C}$ +12745,$b=1-a$ +12746,"$(I, \mathcal B, \mathsf P)$" +12747,"$L(e,t)\sim D(et, \phi)$" +12748,$\sigma_i$ +12749,$-(1-v^n)$ +12750,$R_1(t) < \rho(X_1)$ +12751,"$\mu=10, \sigma=2$" +12752,"$1,9,4,4,2,$" +12753,"$\int |X_n(\omega) - X(\omega)| \,\mathsf{P}(d\omega)\to 0$" +12754,"$(\nu,\nu,\dots,\nu,\nu+10\delta)$" +12755,$1 \times 10^{24}$ +12756,$\mathsf{TVaR}_p(X)=\mathsf{E}[X\mid X >\mathsf{VaR}_p(X)]=\sum_i\mathsf{E}[X_i\mid X>\mathsf{VaR}_p(X)]$ +12757,$e^\mu$ +12758,$\approx$ +12759,$\mathsf{E}(X_i\mid X)$ +12760,"$\eta\gg \zeta:[0,1]\to\mathbb{R}$" +12761,$1+x^2$ +12762,$c^{p-2}\to\infty$ +12763,$1+t$ +12764,"$1,000,000/\$" +12765,$X_i=F_i^{-1}(u_i)$ +12766,$E[X_1(a)]$ +12767,$a(h_1+h_2)\le 2aw$ +12768,$-\mathsf{E}_Q[-X]$ +12769,$\preceq_2$ +12770,$f(x)=\sin(x)$ +12771,$(1+r)\mu$ +12772,"$\rho(X,\mathsf Q)$" +12773,$\theta\in\Theta$ +12774,$R_1(t)>tP(1)$ +12775,$q < p$ +12776,$\int_0^\infty xg'(S(x))dF(x)=\int_0^\infty g(S(x))dx$ +12777,$(1-p)\gamma(dp)$ +12778,$i=r$ +12779,$A)$ +12780,$\bar P_{d}=\rho(Y_{d})$ +12781,$\mathsf{E}[X\mid Y=y]$ +12782,"$(\omega'=1, \omega'')\in B_k$" +12783,$q_p(\mathbf{x})=\mathsf{VaR}_p(X(\mathbf{x}))$ +12784,$R_2(t)$ +12785,$\theta < 1$ +12786,"$(Bob) + (0,-2)$" +12787,$\mathcal{A}_{\rho} = \{ X \mid \rho(X) \le 0 \}$ +12788,$P_j=\sum_{i=0}^j p_i$ +12789,$P(X_{-1}\wedge a)$ +12790,$\mathbf{P_i}$ +12791,"$[s_R,1]$" +12792,$\phi'(p)\ge 0$ +12793,$\rho^E_R(t)$ +12794,$X=q(\omega)$ +12795,$t=\pm 3$ +12796,$(0.5)(20)+(0.5)(30)=25$ +12797,$R^2=0.82$ +12798,"$\mathrm{Tw}_{3-p}(\mu, \sigma^2)$" +12799,$X \times Z$ +12800,$0.45 \times 8.6 = 3.9$ +12801,$1500 \times (0.1075 - 0.017) = 135.75$ +12802,"$X), log(d$" +12803,$k =(A-\mu_U)/\sigma_U$ +12804,$\mathsf{E}_Q[X]=-E_Q[-X]$ +12805,$P(t)>(1-t)P(0)+tP(1)$ +12806,$1/\nu=1+\rho$ +12807,$\rho(X)=\mathsf{E}_\mathsf{Q}[X]=\mathsf{E}[XZ]$ +12808,$L+1$ +12809,$M=\sum_i M_i$ +12810,$R_2(t)\approx \mathsf{E}[X_2]$ +12811,$t\mapsto e^{-2\pi it}$ +12812,$\mathcal M_\rho\subset\mathcal M$ +12813,$h(s)$ +12814,$W'\subset\mathcal W$ +12815,${}^{[>109]}$ +12816,$\mathcal F_1 = \sigma(N)$ +12817,$\mathsf{TVaR}_1(X)=\max(X)$ +12818,$m=m_1/a$ +12819,$s_R \le t < s^\star < s \le s_m=1$ +12820,$B=\{ \omega\in\Omega \mid \zeta(\omega)>0 \}$ +12821,"$\Vert X \Vert = \mathrm{ess\,sup}(|X|)$" +12822,$U = X\wedge A$ +12823,$p=0.8$ +12824,$1\le x \le 2$ +12825,"$(s_{R+1},g_{R+1})$" +12826,"$a, b$" +12827,$g(S_{\mathsf{j}(a)})(a-X_{\mathsf{j}(a)})=(0.5)(80-11)=34.5$ +12828,$\xtext$ +12829,$y_2$ +12830,$j=7$ +12831,$\mathbf {X_1/X}$ +12832,$1=S(a) + \delta F(a) + \nu F(a)$ +12833,$\mathsf E[XY]\not=\mathsf E[X]\mathsf E[Y]$ +12834,$\rho(X+tY)\ge \mathsf{E}_{\mathsf Q_X}[X+tY]$ +12835,$kS=m+Ra$ +12836,$\sum \alpha_i=1$ +12837,$\mathbb{R}_+=[0\infty)$ +12838,$\mathsf{E}_\mathsf{Q}[0]=0$ +12839,$X(\omega)\ge a'$ +12840,$\delta_p+\nu_p=1$ +12841,$a_t$ +12842,$Z=g'(S(X))$ +12843,$\rho(0)=\rho(0+0) = \rho(0) + \rho(0)$ +12844,$S=T=$ +12845,"$(s_j,g_j=h_j)$" +12846,$N_t = 1$ +12847,$a=\mathsf{TVaR}$ +12848,"$(X', Y')$" +12849,$e^{\mu+\sigma^2/2}$ +12850,$0.03$ +12851,$\rho(B(s_u)) - \rho(B(s_l))$ +12852,$\rho(X)= (1+r_f)^{-1}\mathsf{E}_Q(X)$ +12853,$a\theta^2=c$ +12854,$\omega_I > s$ +12855,$(1-\nu_p-il_p)/(\nu_p-l_p)=\iota_{1/2}$ +12856,"$\mathsf{z}=(z_1,\dots,z_k)$" +12857,$g(S)\Delta X$ +12858,$\phi(p)=g'(1-p)=b(1-p)^{b-1}$ +12859,"$(2, 3)$" +12860,$LR_{PH}$ +12861,$\sum_i \mathsf{Var}(X_i)<\infty$ +12862,$\kappa'(\theta)=-\dfrac{1}{\theta}$ +12863,$\rho(-k_i 1_{A_i}) \le c < 0$ +12864,$ | \$ +12865,$1/s$ +12866,"$\mathsf P,\mathsf Q_2,\dots,\mathsf Q_r$" +12867,$\mathsf{Pr}i^b_{m}(X)$ +12868,$s>{s_equity}$ +12869,"$\subset [\essinf X ,\mathrm{ess\,sup} X]$" +12870,"$(fun2.north west)+(-\ , \ )$" +12871,$\pi_X(t)\le \pi_Y(t)$ +12872,$\rho_\mu$ +12873,$Y=\sum_i X_iY_i$ +12874,$m(1-m)$ +12875,$u_3$ +12876,$(dt)^{3/2}$ +12877,$R_1(t_1-\epsilon)a_1$ +12889,$\kappa=0$ +12890,$LR_{CCoC}$ +12891,$X=30$ +12892,$p_i \le p^* \le p_j$ +12893,$\rho(X+Y)=\rho(X) + \rho(Y)$ +12894,"${}^{[<37,38]}$" +12895,"$\Omega\subset (0,b)$" +12896,"$6,000,000,000) for the last 180 days of any calendar year, the board shall relieve all participating insurers of their obligation to pay additional earthquake loss assessments under this chapter, by an amount equal to the amount of available capital in excess of six billion dollars ($" +12897,$10^2 - 10^4$ +12898,"$19,760,000 | 15.4% | $" +12899,$\rho(X)=50=:r$ +12900,$L'=L'(s)$ +12901,$f\in L^1(\mathbb R)$ +12902,"${}^{[<67,49]}$" +12903,$\mathit{RDS}_k$ +12904,$\rho_{m_i}(Y)\ge \rho(Y)-2^{-i}$ +12905,$=\mathsf{E}[X_i/X \mid X > a]$ +12906,$X_1 = fI + N$ +12907,$l(t)$ +12908,$N(1-p)$ +12909,$\rho_\theta(X) \equiv \theta\mathsf{TVaR}_{0}+(1-\theta)\mathsf{TVaR}_{1}$ +12910,"$\iota_1, \dots, \iota_m$" +12911,$10^{-6} - 10^{-3}$ +12912,$P =\{ Q \mid dQ/dP \le k \}$ +12913,$\mathsf{E}[X]=\int_0^1 q(p)dp$ +12914,$\partial d/\partial\mu=-2(y-\mu)$ +12915,$\rho(X)=P$ +12916,$=\mathrm{MV}(a-X)^+$ +12917,$1/n$ +12918,"$d(g(S(x)))/dx=-g'(S(x))\,dF/dx$" +12919,$8.617 \times 10^{27}$ +12920,"$4,429. This gives a risk load of \$" +12921,$X(\omega) \le Y(\omega)$ +12922,$\rho(X_0) = \mathsf{E}[X_0Z]$ +12923,$\mathsf{TVaR}_p(X) = \mathsf{VaR}_p(X)$ +12924,"$\langle \zeta_{\bar x}, N(\bar x) \rangle$" +12925,$(u_2-u_1)/(u_3-u_2)$ +12926,"$100,000, \$" +12927,$\rho(\mathsf E[X\mid \mathscr F])\le \rho(X)$ +12928,$E\cap U_0=E$ +12929,$l(kX)=k\rho(X)$ +12930,$\int_0^q = \int_0^{\mathsf{E}_q(X_2)} + \int_{\mathsf{E}_q(X_2)}^q$ +12931,$L_d^{d+l}(x)=(x-d)^+ \wedge l$ +12932,$q(\epsilon)=q+\epsilon\mathsf{E}_q(X_1)$ +12933,$0 \rho(X^{\oplus n-1})$ +12948,$\mathsf{P}(\omega_H)=pQ$ +12949,$[0; -k]$ +12950,$X \mapsto X + a$ +12951,$X(a)=a=90$ +12952,$\sum_{i}U_{i} = U$ +12953,$g(X_n)=1$ +12954,"$p=0,1$" +12955,$\rho(X) =\mathsf{TVaR}_p(X)$ +12956,"$a\,\mathsf{E}\left[\dfrac{X_1}{X}\mid X\ge a \right]$" +12957,$p_{j-}< p_j < p_{j+}$ +12958,$1 = m(x) + \nu F(x) = S(x)+\delta F(x) + \nu F(x)$ +12959,$L_c$ +12960,$\mathit{MV}_{ro}(a) = a-\rho(X_{-1}\wedge a)$ +12961,$X>x$ +12962,$S_0 X_0 = S_n X_n = 0$ +12963,$y\in\Omega$ +12964,$P = \mathsf{E}[X] + \pi\mathsf{E}[X]$ +12965,$f_{X+Y}$ +12966,$1=\sum_i \alpha_i(x)=\sum_i \mathsf{E}\left[\frac{X_i}{X}\mid X=x\right]$ +12967,$\mathscr{F}\otimes\mathcal{B}$ +12968,$R^2=0.892$ +12969,"$\sigma=0.5, 1.0$" +12970,$10^{32}$ +12971,"$a(x_1,x_2)=\sqrt{3x_1^2 + 4x_2^2}$" +12972,$2/3$ +12973,$\bar Q'(x)=Q(x)$ +12974,$Z'(g(s))g'(s)=Z'(s)$ +12975,$1/4\le s\le 1$ +12976,$G(\bar x)$ +12977,$m(1+m/\lambda)^2$ +12978,$\int Zd\mathsf P=1$ +12979,$\delta > 0$ +12980,$R_1(t)<\mathsf{E}[X_1]$ +12981,$f^T$ +12982,$L_2$ +12983,$\mathsf{var}(X)=\mathsf{E}(X^2)-(\mathsf{E}(X))^2=\mathsf{E}((X-\mathsf{E}(X))^2)$ +12984,$t > s*$ +12985,$l + sX$ +12986,$d-1$ +12987,"$g, g^2, \dots,g^{q-1}, g^q\equiv 1$" +12988,"$\alpha,\beta,\kappa$" +12989,$Z^*=\mathsf{E}[Z\mid X]$ +12990,$Y_s$ +12991,$\alpha < 0$ +12992,$g(a)=a$ +12993,$\mathsf{VaR}_\pi(X)$ +12994,$T'(p) = (T(p) - q(p)) / (1 - p) > 0$ +12995,$R_1(1)=\bar P^a_1(1)$ +12996,"$\Omega=\{1,2,3 \}$" +12997,"$I=[0,P)$" +12998,$X_1=s$ +12999,$0 | \$ +13000,$Var(G)=c^2$ +13001,"$(X_i,Y_i)$" +13002,$\rho(X)\not=\sum_i\rho(X_i)$ +13003,$r=d/v$ +13004,$\mathsf{Pr}r(p(\omega)=0)=0$ +13005,$S(x_{i-1})-S(x_{i})=S(x_i-(x_i-x_{i-1}))-S(x_i)=-S'(x_i)(x_i-x_{i-1})=f(x_i)(x_{i}-x_{i-1})$ +13006,$\alpha \le 1$ +13007,$\mathsf{TVaR}_0=\max(X)$ +13008,$N(a):=\int_0^a F(x)dx=a-\mathsf{E}[X\wedge a]$ +13009,$X(\cdot)$ +13010,$T_i\circ T$ +13011,$\alpha\le -1$ +13012,$a<100$ +13013,$S_{X\wedge a}(x)=0$ +13014,$P(t)t}]$ +13018,$G=X_1+X_2$ +13019,$q_Z(U)$ +13020,"$[0, \infty)$" +13021,$Q = 5.0449$ +13022,$1-p \ge g^{-1}(1-p) \implies 1-g^{-1}(1-p) \ge p \implies q(1-g^{-1}(1-p))>q(p)$ +13023,$f:\mathbb{R}\to\mathbb{R}$ +13024,$\rho_2$ +13025,"$\zeta, \zeta_t\ge 0$" +13026,"$p=0.98,0.99$" +13027,"$(3,6-4.724)$" +13028,$s=0.9$ +13029,$Q_{1}\Delta X$ +13030,"$(\Omega, \mathscr{F}, \mathsf{Pr}, \{\mathscr{F}_t\}_{t \geq 0})$" +13031,$M_X(t)^n$ +13032,"$X_{t,d+1}$" +13033,$\mathsf{E}[X]+kR(X)$ +13034,$\phi(0)=\mu(\{0\})$ +13035,$\mu = \nu$ +13036,"$\mathsf E[X] \le \mathsf E[\rho(X, P_I)] \le \rho(X)$" +13037,$\mathsf{E}[e^{hX}] = \exp(h\mu+\sigma^2h^2/2)$ +13038,$\frac{1}{2}kT$ +13039,$1 \times 10^{20}$ +13040,$\mathit{NPV}_1 = 0$ +13041,$Q=\nu\mathsf{XTVaR}$ +13042,"$X \sim \mathrm{Tw}_p(\mu, \sigma^2)$" +13043,$G=N+C$ +13044,$RB$ +13045,$\mathsf{E}(X)=\mathsf{E}(X\wedge k) + \mathsf{E}(X-k)_+$ +13046,$=\displaystyle\int_0^\infty x \mathsf{Pr}_X(dx)$ +13047,$g(s)=\mathsf{Pr}hi(\mathsf{Pr}hi^{-1}(s)+λ)$ +13048,$\mu < \frac{1}{2} \sigma^2$ +13049,$\mathsf{TVaR}_{0.75}(X_1)=10$ +13050,$R_2(1) = \bar P^a_2(1)$ +13051,$\exists \mu$ +13052,$X_0=\mathsf E[X]$ +13053,$\bar S(a) = \int_0^a S(x)dx$ +13054,"$\mathsf{biTVaR}_{0,p}^w(X)$" +13055,$\mathbf {g_4(s)=s^{0.9}}$ +13056,$g(S(x)$ +13057,$\rho(X)=\mathsf{TVaR}_p(X)$ +13058,$\zeta_1=0.15$ +13059,$p_1<1$ +13060,$00$ +13066,$F_0 = \bar P_{act}-\bar P = R-\bar M$ +13067,$\rho(X)=x_p$ +13068,$\zeta_2=\sin(\theta\pi/2)$ +13069,$P=\mathsf{E}[U]+R$ +13070,$(\bar\alpha+1)/\bar\alpha=1/(2-p)$ +13071,$x<1$ +13072,$p^*\le p$ +13073,$x=a$ +13074,$-iv$ +13075,"$f(\cdot, \omega)$" +13076,$g(1)-g(0)=1$ +13077,$g(x) = \mathsf{Pr}hi(\mathsf{Pr}hi^{-1}(x) + h)$ +13078,$X\mapsto\int X(\omega)Z(\omega)\mathsf(d\omega)$ +13079,$\mathsf{SRM}$ +13080,$R^2={gres.rsquared:.2f}$ +13081,$a_{n\!\urcorner}$ +13082,$\mathsf{E}[X\mid\mathscr{F}]\preceq_2 X$ +13083,$V$ +13084,"$\iota,\nu,\delta$" +13085,$X(\omega') = \sum_\omega X(\omega)1_\omega(\omega')$ +13086,"$p\in(0,1)$" +13087,$f_1$ +13088,$\mathsf{E}S(X)=q(p)$ +13089,$>1-p$ +13090,$\mathit{EER}$ +13091,${}^{[<244]}$ +13092,$g'(s)=1$ +13093,$q_X(p)\le q_Y(p)$ +13094,$X=X_0+X_1$ +13095,${}^{[>42]}$ +13096,"$i=1,\dots, n$" +13097,"$p\in (1,2)$" +13098,"$1,9,4,4,2,4,$" +13099,$W_0$ +13100,$\lim_n \mathsf{E}_{\mathsf{Q}_n}[X]=\rho(X)$ +13101,$\mathsf{E}[h_\epsilon Y]\to\mathsf{E}[h Y]$ +13102,$A:=g^a \pmod{p}$ +13103,$X+\epsilon X^i$ +13104,"$\Omega=\{\omega_1, \omega_2 \}$" +13105,"$(x_{1,1}, x_{1,2})$" +13106,$a()$ +13107,$\mathcal{B}$ +13108,$p<\infty$ +13109,$\alpha_i(X_u)= \text{E}[u_iX_i \mid X_u > F_u^{-1}(p)] = u_i \partial T/\partial u_i$ +13110,$\mathsf x\mathsf{TVaR}$ +13111,"$q\in[1, \infty]$" +13112,$\mathsf{E}_{\mathsf Q}(X\wedge a)$ +13113,$\lambda=0.3427$ +13114,$\omega\in\Omega$ +13115,$\phi(p)=1$ +13116,$t^*-\epsilon/2$ +13117,$T^{-1}(A)$ +13118,$P=\bar\nu(a)(\bar S + \bar\iota(a) a)$ +13119,$P(a) = \nu S(a) + \delta$ +13120,"$s=0,1$" +13121,"$\mathsf E[X] =\displaystyle\int_0^\infty xf(x)\,dx = \displaystyle\int_0^\infty S(x)\,dx$" +13122,$\mathbf {X_1pK}$ +13123,$X\ge 0$ +13124,$S\ge (1-\epsilon)\mathsf{E}[X]$ +13125,$L(a)$ +13126,$T:\Omega\to\Omega$ +13127,$\iota = M/Q$ +13128,$P=\sum_i P_i$ +13129,$z_1$ +13130,$_1F_1$ +13131,$R_1(t^*-\epsilon)1$ +13143,$p=0.999999$ +13144,$2^{16}$ +13145,$x_i+y_j$ +13146,$\mathsf{E}[X_i \mid X=q(1-g^{-1}(1-p))]$ +13147,$\mathsf{TVaR}_{0.8}(X)=25$ +13148,$t\ge 1$ +13149,$\tilde p=1-(1-p)^{1/\rho}$ +13150,$v^b{}_bq_x(1-{}_b\bar V)$ +13151,$\sum_i P^i$ +13152,$t=t_*$ +13153,$2^{-t}$ +13154,$S_k=Pr(X > X_k)$ +13155,$a\mathsf{E}_{\mathsf{Q}}[...]$ +13156,"$w_{p_i,p_j}\mathsf{TVaR}_{p_i}(Y) + (i-w_{p_i,p_j})\mathsf{TVaR}_{p_j}(Y)$" +13157,$-α(α-1)t^{α-1}$ +13158,$(j)$ +13159,$\mathsf{E}[g]\le 1$ +13160,$p > 1$ +13161,$\mathit{ROE}(s) = fs/(1-f-s)$ +13162,$-norm less than $ +13163,$s^{0.642}$ +13164,$\bar M_i(a)$ +13165,$S(a)=\mathsf{E}[1_{X>a}]$ +13166,"$\mathsf{PH,SA,CX}$" +13167,$R(t)$ +13168,$\rho(aX)=a\rho(X)$ +13169,$w_t = \zeta_1z_1 + \cdots +\zeta_tz_t$ +13170,$\rho_g(X) = \infty$ +13171,$\rho_m(X)=\rho_m(X\wedge k) + \rho_m((X-k)_+)$ +13172,$\bar P(x) = \bar S(x) + \bar R(x)$ +13173,$h'(j) = h(s(j)- h(s(j+1))$ +13174,$\le x$ +13175,$\sqrt{g d}$ +13176,"$w_o,w_u$" +13177,$\rho(X_{-1}\wedge a_{ro})={{mvp_ro}}$ +13178,$B=M$ +13179,$V(m)=m(m-a)$ +13180,$(1-g(s))q$ +13181,$\rho(X)=1.169$ +13182,$\iota=\dfrac{M}{Q}$ +13183,$\mathsf{E}[(-Y)Z]\ge 0$ +13184,"$r_1 = 1643984129.762957 \approx 1,643,984,129.8$" +13185,$q=11$ +13186,$\rho(X_a)= \mathsf{E}_{\mathsf{Q}'}[X_a]$ +13187,"$n=1,2,\dots$" +13188,$\phi(s)=g'(1-s)=(1-s)^{1/b}/(b(1-s))$ +13189,$(Mg+\alpha a)w$ +13190,$x\mapsto (x-a)^+$ +13191,$D/C$ +13192,$j=9$ +13193,$255$ +13194,$\mathsf{E}[X\cdot Z\circ T]=\mathsf{E}[X\cdot Z\circ T_B\circ T_A ]=\mathsf{E}[X \cdot Z\circ T_A]=\mathsf{E}[X\circ T_A^{-1}]=\mathsf{E}[X Z]$ +13195,"$I=[0, 32]$" +13196,"$(X,Y)$" +13197,$a_x=1/\lambda$ +13198,$\nu(dy)=(e^{2y}-1)dy$ +13199,"$\alpha_1,\alpha_2$" +13200,$g^a\equiv n\pmod{p}$ +13201,$\mathsf{Q}_n\in\mathscr{P}$ +13202,$\mathsf{E}_Q[X+Y]=\mathsf{E}_Q[X] + \mathsf{E}_Q[Y]$ +13203,$\mathsf{TVaR}_1( X )$ +13204,"$9 billion. Frequency is 1.74, the long term average annual number of landfalling hurricanes 1851-2017. Losses are modeled with per occurrence limits ranging from $" +13205,$g^-1(p)$ +13206,$WX$ +13207,"$990,000 (= \$" +13208,${}^nS^{-1}_X(t)\le {}^nS^{-1}_Y(t)$ +13209,$\mathsf{E}_Q[X_i]$ +13210,$x=\mathsf{VaR}_{0.99}(X)$ +13211,$1\le i\le j\le n$ +13212,$f^*_i$ +13213,"$1,2,\dots,A$" +13214,${}_tp_x=\mathsf{Pr}r(T_x > t) =\mathsf{Pr}r(T_0 > x+t \mid T_0 > x)$ +13215,"$q_X(U), q_Y(1-U)$" +13216,$\rho^*=\rho(0.5)$ +13217,$f>0$ +13218,$\nu\ll\mathsf{Pr}$ +13219,$(a-X(\omega))^+$ +13220,$B\cup B_t = (B\cap B_t) \cup C_t$ +13221,$(p-\nu-il)/(v-l)$ +13222,$a_t=(1-t)x_0 + ta_1$ +13223,$(k+1)\times 1$ +13224,$106.20 /3.00= \$ +13225,$X_n\le 1$ +13226,"$\langle NF(x), Th_i \rangle+\langle \partial NF/\partial x_i, \zeta_{GF(x)} \rangle$" +13227,$\bar P_i(a)=\mathsf{E}_{\mathsf{Q}}[X_i(a)]=\mathsf{E}[X_i(a)g'(S(X))]$ +13228,$\kappa_i(x)\approx t-\sum_{j\not=i} \mathsf{E}[X_j]$ +13229,$\delta N(a)$ +13230,$\mathcal M(\mathsf{P})$ +13231,"$Binomial(s,N)$" +13232,$Z_1 = Z\circ T_X$ +13233,$g(S_k) + g(1-S_k) \ge 1$ +13234,$T_x=\inf\{ t > 0 \mid X_t\ge x\}$ +13235,$P_Q = \mathsf{P}[(X-a)V(a)]$ +13236,"$M_{i,j}$" +13237,$r'=0$ +13238,"$w_{p_1,p_2}\mathsf{TVaR}_{p_1}(Y)+ (1-w_{p_1,p_2})\mathsf{TVaR}_{p_2}(Y)$" +13239,$S\Delta X'$ +13240,$\sum_{i}X_{i}^{< j >} = X^{< j >}$ +13241,$\mu\left(1+\dfrac{\mu}{p}\right)$ +13242,$S(x)=1=F(x)$ +13243,"$j=0,\dots, N-1$" +13244,"$A_0,\dots,A_r\in\mathscr{F}$" +13245,$x\leftrightarrow u(x)$ +13246,"$(Bob)+(0,-2)$" +13247,$\mathsf{E}[\mathsf{VaR}_{1-(1-p)/nG}(Y)]$ +13248,"$C=[0,\infty)$" +13249,"$(a-X)^+:=\max(a-X, 0)$" +13250,"$\Omega=\{\omega_1,\dots,\omega_n\}=\{\text{Ada}, \text{Bernhard}, \dots, \text{Zeno} \}$" +13251,$. The rate of increase is relatively consistent for small $ +13252,$100K and an aggregate limit on the deductible of \$ +13253,$10^{17}$ +13254,$\lambda t$ +13255,$M_0$ +13256,$P(\mathsf{var}nothing)=0$ +13257,$\alpha_i(k) = E[\kappa_i(X)/X\mid X > k]$ +13258,$\mathsf{E}[X \mid \mathcal F_0]$ +13259,$\nu < 1$ +13260,$\forall X\exists P\forall z[z\subset X\rightarrow z\in P]$ +13261,"$X_1, X_2, \ldots, X_n$" +13262,$(1-g(s))/(1-s)$ +13263,"$y_1,\dots, y_n$" +13264,$a = 0.6565$ +13265,$Ca_3Al_2(SiO_4)_3$ +13266,$(S(x) + \delta(F(x))F(x)) dx$ +13267,$y^2 - 2\sigma y=(y -\sigma)^2 -\sigma^2$ +13268,$s\leftrightarrow 1-s$ +13269,$\mathsf{Pr}(\Omega\mid\mathscr{G})_{\omega_0}=1$ +13270,$d\to\infty$ +13271,$N(\lambda)$ +13272,$Q$ +13273,$(\omega^l)^n = (\omega^n)^{l} = 1$ +13274,$T\mathsf{Pr}$ +13275,$X(\omega)=0$ +13276,$0.75+U/4$ +13277,$D∪ C^c$ +13278,"$(1-g(S(x)),x)$" +13279,$\mathbf {K}$ +13280,"$\rho(P,R,a)=\sqrt{(0.4P)^2+(0.25R)^2+(0.1a)^2}$" +13281,"$J=\{\omega\mid P(\mathscr{G}amma\cap Z, \omega) = \frac{1}{2}1_\mathscr{G}amma(\omega) \ \forall\mathscr{G}amma\in\mathscr{G}\}$" +13282,$X_1=99$ +13283,"$[\omega_I,1]$" +13284,"$3,000 limit and \$" +13285,$t_{n+1}$ +13286,$\mathcal F_1=\sigma(N)$ +13287,$p=0.638$ +13288,"$x=1.5, M=1.5,\sigma=0.75, K=6$" +13289,$n\ge 0$ +13290,$c^*(y)=c(y)e^{l(y;y)}$ +13291,$1\le p\le \infty$ +13292,$M(a)=\mathsf{E}(X\wedge a) + \delta N(a)$ +13293,$g(s)q=0.1839$ +13294,$S(x) + \delta F(x)$ +13295,$1.5\times 10^{37}$ +13296,$t_0$ +13297,$iv^{n-t}$ +13298,"$\rho^\star(X)=\rho(\rho(X, \mathsf P_I), \mu)$" +13299,$\{ Z\not=0 \}$ +13300,$\dfrac{g(s)-s}{1-g(s)}$ +13301,$x_i-x_j$ +13302,$\zeta>0$ +13303,$\mathsf{E}(X_i \mid X \le a)$ +13304,$\mathit{EGL}_{ro}(a)=P(X_{-1}\wedge a) - \rho(X_{-1}\wedge a_{ro}) \ge 0$ +13305,$p_0\not= p_1$ +13306,$\mathsf{TVaR}_\alpha(X)$ +13307,$B_p$ +13308,$\rho(X)\le\liminf_{n\to\infty} \rho(X_n)$ +13309,$\mathsf{Pr}(B)$ +13310,$X+c$ +13311,$D(a)=1_{X>a}$ +13312,"$j(x)/J(1)1_{[1,\infty)}$" +13313,$\displaystyle\int_0^\infty u(x)dF_X(x)$ +13314,"$(-2,-1)$" +13315,$R_0(t)= \bar P^a_0(t)/(1-t)$ +13316,"$x=a,b$" +13317,"$t_2 \in [s_R, 1]$" +13318,$\iota'$ +13319,$\rho=\rho(p)$ +13320,$z_p$ +13321,$\rho=0.12$ +13322,$a\beta_1g(S)$ +13323,"$(0,0,0,0,0,0,5,0,0,5)$" +13324,$\mathsf Q(A)\le g(\mathsf P(A))$ +13325,$\omega<0.4$ +13326,"$\int_0^{100} S(x)\,dx$" +13327,$\delta=\log(1+i)$ +13328,$\text{VaR}_{\alpha}(A)$ +13329,$\alpha > -1$ +13330,$\mathsf{E}[(a-X)^+]=\int_0^a F(x)dx$ +13331,$f(x)=x^2$ +13332,$\mathsf{Q}(\{\omega_i\})=0$ +13333,$20 = 1/0.05$ +13334,$\epsilon < 10^{-5}$ +13335,"$D^f\rho_{X\wedge a,X}(X_i)$" +13336,$\mathsf{E}_{\mathsf{Q}_1}=(3\times 6 + 2\times 2)/ 8 = 11/4$ +13337,$U=F(X)$ +13338,$1/q$ +13339,$\mathsf{E}[X_i/X \mid X > x]$ +13340,$X\circ T=X$ +13341,$ achieves the left-shift and $ +13342,"$(x_1,\dots,x_n)^T$" +13343,$\dfrac{s}{g(s)}$ +13344,$\mathsf{Pr}r(X=\mathsf{E}(X))=0$ +13345,$ is implicit. The dependence on $ +13346,$\sup X_n=1\not=\sup X=0$ +13347,$0\le p_1\le p^* \le p_2\le 1$ +13348,$u_i=\mathsf{Pr}hi(Z_i)$ +13349,$\sigma>0$ +13350,$l(p)= \nu(p)-\sqrt{(1-p)/p}$ +13351,$(1-\alpha)^{-1} \min_c c(1-\alpha) + \mathsf{E}[(X-c)^+]$ +13352,$Y \Leftrightarrow \rho(X)\le \rho(Y)$ +13353,"${}^{[<49,43]}$" +13354,$\beta_i(t)<\alpha_i(t)$ +13355,$x_m$ +13356,$\delta>0$ +13357,$<0.01$ +13358,$\mathsf{TVaR}_{1-s}$ +13359,$\mathsf{E}[XZ_\epsilon]\to \mathsf{E}[XZ]$ +13360,$S(x)=\mathsf{Pr}r(X>x)=1-F(x)$ +13361,$w(x)=e^{kx}$ +13362,$x=\log(m)$ +13363,$l(y;\mu)=\log(a(y) + y\tau^{-1}(g^{-1}(\mathbf x\beta))-\kappa[\tau^{-1}(g^{-1}(\mathbf x\beta))]$ +13364,$\{\omega\mid X(\omega) > x\}$ +13365,$\Delta g(S)$ +13366,$\mathcal{B}bb Q\ll \mathcal{B}bb P$ +13367,"$s_i, i=0,\dots,m$" +13368,"$\mathsf{E}(X_{i,2}(a))$" +13369,"$Z,Z_i$" +13370,"$(\mathcal{B},Y\setminus N)$" +13371,$10^{-1} - 1$ +13372,$s=f'(x)$ +13373,"$300,000, an excess loss premium of \$" +13374,$\infty$ +13375,$c_x-c_{\text{Nov 1}}$ +13376,$k\mapsto k\rho(X)$ +13377,$\tau_n = Z_1 + Z_2$ +13378,$0/0$ +13379,$B_t(\omega)=\omega_t$ +13380,$\mathsf{E}_\mathsf{Q_k}[X_j]$ +13381,$t_1-\epsilon/1$ +13382,$\mathbf {M}$ +13383,$p_1=p_2=p^*$ +13384,$\tau^{-1}(\mu) = \displaystyle\int_{\mu_0}^\mu \dfrac{dm}{V(m)}$ +13385,$B=g^b\pmod p$ +13386,${}^{[>111]}$ +13387,$\bar x$ +13388,$dP = P_R dR + P_{RR}(dR)^2 = P((\mu+ \sigma^2/2) dt + \sigma dz$ +13389,$a(1-p) + \mu p - \sigma\phi(z_p)$ +13390,$x \le A$ +13391,$\mathscr{O}(f)=\{f \circ T \mid T\in \text{MPT}\}$ +13392,$\mathsf{Q} \leftrightarrow Z\in L^1 \rightarrow F_Z$ +13393,$\lambda Y_1$ +13394,$\rho:L_p\to\bar\mathbb{R}$ +13395,"$\phi(s) = \alpha^{-1}1_{[1-\alpha, 1)}(s)$" +13396,$t=1-p$ +13397,"$\rho\in\mathcal R^c_{X,r}$" +13398,$g(X)$ +13399,"$[-100, 1000)$" +13400,$q=1-p=S(x)$ +13401,$\bar P^a(t)=\bar P^a_1(t)+\bar P^a_2(t)$ +13402,$A(-X)=-A(X)$ +13403,$(k_1!)(k_2!)\dots$ +13404,$\theta(p)=q(1-g^{-1}(1-p))/q(p)$ +13405,$52.2+27.8$ +13406,$s_2 < s_1$ +13407,$\hat \theta>\theta_d$ +13408,"$\phi:(L,\mathcal{A})\to \mathbb{R}$" +13409,$M < \infty$ +13410,$\mathsf{P}(\{\omega_i\})=1/4$ +13411,$c_i$ +13412,$a\to\infty$ +13413,$=\mathsf{E}[X\mid X > a]$ +13414,"$i,v$" +13415,$x=\mathsf{VaR}_p(X)$ +13416,$\mathsf{E}_Q[C]$ +13417,$\mathsf P(X\le q_X(p))=p$ +13418,$\nu$ +13419,$\sum_j S_j \Delta X_j = \sum_j p_j X_j$ +13420,"$[0,1-p)$" +13421,$1/X$ +13422,$x^\ast$ +13423,$dt=1/n$ +13424,${}^{[<10]}$ +13425,$\mathit{NPV}_{\infty}$ +13426,$\mathsf{var}phi(X)\le \mathsf{var}phi(Y)$ +13427,$A = X_1 + \cdots + X_d$ +13428,$\mathsf{E}[Y\mid\mathcal F']=\mathsf{E}[Y]$ +13429,$F_{\mathbf{x}}(t)=s$ +13430,$\rho(X)=\sum_i \mathsf{E}_\mathsf{Q}(X_i)$ +13431,$dR=\mu dt + \sigma dz$ +13432,$e_i / (s\sqrt{1-h_{ii}})$ +13433,"$\mathsf{TI,\ MON,\ SA,\ PH}$" +13434,$\lim_{\mu\to 0} V(\mu)/\mu^2=\infty$ +13435,$a=1.75$ +13436,$g(\mathsf{Pr}(A))$ +13437,$f(L) \ge 0$ +13438,$\mathsf{E}[X_ih(X)]=\mathsf{E}[\mathsf{E}[X_ih(X)\mid X]]=\mathsf{E}[\mathsf{E}[X_i\mid X]h(X)]=\mathsf{E}[\kappa_i(X)h(X)]$ +13439,"$h_{i,\epsilon}$" +13440,"$(X_1,\dots,X_n)$" +13441,$\kappa_t(s) = t \kappa(s)$ +13442,$p=0.2$ +13443,${}^{[<58]}$ +13444,$F_X(x)=\mathsf P(X\le x)=\mathsf P\{\omega\mid X(\omega)\le x\}$ +13445,$Y=\log(X)$ +13446,$g^{-1}$ +13447,$x\ge F^{-1}(1-s^*)=:x^*$ +13448,$\omega''$ +13449,$0\le p^\ast\le 1$ +13450,"$a'_{X,r}(Y)$" +13451,$Y_N$ +13452,$r = \nabla r$ +13453,$\rightarrow$ +13454,"$X_n,X$" +13455,$\bar S_i(\mathbf{x}; a) := \mathsf{E}[X_i(\mathbf{x}; a)]$ +13456,$1\le\lambda$ +13457,$s_R=s_m=1$ +13458,$p\ge r\ge 1$ +13459,"$(s_j=0,g_j>0)$" +13460,$\partial \rho(X)$ +13461,$\log(1+e^\theta)$ +13462,$c^{2-p}=1$ +13463,$s$ +13464,$\rho(\lambda X) = \lambda \rho(X) \le 0$ +13465,$\mathbf {\beta_{1}g(S)\Delta X}$ +13466,$p={pp:.3f}$ +13467,$t < 2/3$ +13468,$a_1\not=a_2$ +13469,"$\omega\in(0,\omega_I]$" +13470,$f'=\partial f/\partial\mu$ +13471,$H_k(X)=H_k(Y)$ +13472,"$A_1=[-k,-k]$" +13473,$M^+_F$ +13474,$\rho\leftrightarrow g$ +13475,$g(1-F(x))=1-\tilde p$ +13476,"$\mu_t:=\lambda_t / \int_0^1\lambda_s \,ds$" +13477,$q^-(U)$ +13478,$S(x)=0$ +13479,$k(i)$ +13480,$p>S(x^*)$ +13481,"$k \in_{R} \{2,\dots,p-2\}$" +13482,$v(A\cap B) + v(A\cup B)\le v(A)+v(B)$ +13483,$m_i\in\mathcal{M}$ +13484,$\int_0^x$ +13485,$F:\mathbb{R}^n\to \mathcal{X}^n$ +13486,$a=\mathsf{TVaR}_p(X)$ +13487,"$428, \$" +13488,$X \succeq Y$ +13489,$S=\mathsf{E}[X\wedge a]$ +13490,$2.6 \times 10^{12}$ +13491,$\| \sigma \|_p \le c$ +13492,$\mathcal{B}bb{Q}$ +13493,$dW_AdW_L=\rho\sigma_A\sigma_L$ +13494,$\int_0^\infty xf(x)dx$ +13495,$p_s=R(\theta)=0$ +13496,$m^*(O \setminus A) < \epsilon$ +13497,$f(p)=\alpha(1-\alpha)(1-p)^{\alpha-1}$ +13498,$=\mathsf{E}(X\mid X > a)$ +13499,$X\times Y$ +13500,"$t\in[t, t+dt]$" +13501,${kl:.3f}$ +13502,$Z\le (1-p)^{-1}$ +13503,$\rho_\phi(X)=\displaystyle\int_0^1 q(p)\phi(p)dp=\displaystyle\int_0^1 q(p)g'(1-p)dp=\displaystyle\int_0^\infty xg'(1-F(x))f(x)dx$ +13504,$ds=g'(1-t)dt$ +13505,$-24$ +13506,$Z\ge 0$ +13507,$t \geq \tau_n$ +13508,$g^mA^r == r^s$ +13509,$A\cap B$ +13510,$\mathbf {X_{1c}}$ +13511,$\iota$ +13512,$\epsilon >0$ +13513,$\hat\rho(X)<\rho(X)$ +13514,$s>1-p$ +13515,"$M(X_1, a_1)+M(X_1, a_2)=M(X_1, a_1+a_2)$" +13516,$NPV = aF_0$ +13517,$x=S^{-1}(g^{-1}(s))$ +13518,$f(x)=x$ +13519,"$i=1,\dots,r$" +13520,$\beta/\alpha$ +13521,$\rho\in\mathcal{R}$ +13522,"$6,000,000,000) for the last 180 days of any calendar year, the board shall relieve all participating insurers of their obligation to pay additional earthquake loss assessments under Section 10089.30, by an aggregate amount equal to the amount of available capital in excess of six billion dollars ($" +13523,$Q^*$ +13524,$h=H(A)$ +13525,$U_2$ +13526,$ of paying and $ +13527,$N'/(Mg+\alpha a) - w\le w$ +13528,$\mathsf{TVaR}(p)=(1-p)^{-1}\int_{p}^1 q(s)ds$ +13529,$(1+c)\mu$ +13530,$\xi:\mathcal{A}\times M\to \mathbb{R}$ +13531,$X^i(0.05)$ +13532,$G>q_\alpha$ +13533,"$(-\pi/2, \pi/2)$" +13534,$\bar s = (r_1+r_2)/2$ +13535,$\rho_\alpha(X):=\rho(X\wedge \alpha(X))$ +13536,$\Delta=\mathsf{Pr}hi(d^*)$ +13537,$\mathscr{O}(f)$ +13538,"$0\le U, V\le 1$" +13539,$\text{E}_{\mathcal{B}bb{Q}}[Y\mid \mathcal{G}] \text{E}[Z \mid\mathcal{G}] = \text{E}[YZ\mid \mathcal{G}]$ +13540,$s_0^2 > s_1$ +13541,$X=\mathsf P_TX_s$ +13542,$\mathsf E[g(X_n)]\to \mathsf E[g(x)]$ +13543,$\kappa(\theta)=\dfrac{\alpha-1}{\alpha}\left(\dfrac{\theta}{\alpha-1}\right)^\alpha$ +13544,$\rho(\mathsf E[X\mid I])$ +13545,$P(A)=0$ +13546,$y\in C\subset\mathbf R$ +13547,"$\bar P_i(\mathbf{x},a):=\mathsf{E}_g[X_i(\mathbf{x}; a)]$" +13548,$ because $ +13549,"$392,457 | \$" +13550,$T q(p)$ +13597,$X_0 < \dots < X_{N-1}$ +13598,$\mathsf E[\log(X_1 / X_0)]$ +13599,$=q(p)$ +13600,$\delta^2 p +\nu^2q-(p-\nu)^2=\delta^2 p -p\nu^2 -p^2+2p\nu =p(\delta^2 -\nu^2) -p^2+2p\nu =p(\delta -\nu) -p^2+2p\nu =p\delta -p^2 + p\nu = p-p^2$ +13601,$X_1/X$ +13602,$\mathsf{E}(L) = q(p)\delta$ +13603,$l=l(p)$ +13604,$D_\infty$ +13605,$f(L)=0$ +13606,$c_1$ +13607,$p^*=0.752$ +13608,"$\mathsf{Y}_{i,k}=\mathsf{X}_{i,r(k)}$" +13609,$S\mathsf{Pr}(A) = \mathsf{Pr}(S\in A)$ +13610,$h(1)=1$ +13611,$\omega\in B_0$ +13612,${}_tV$ +13613,$\mathsf{E}[X_2\mid X \ge a]$ +13614,$X(\mathbf{x}) = \sum_i x_iX_i$ +13615,"$\alpha_p = 1- (\| (X-\eta_{p,\alpha})_+\|_{p-1} / \| (X-\eta_{p,\alpha})_- \|_{p})^{p-1}$" +13616,$\{ X >q(p) \}$ +13617,$g'(0)\le 1$ +13618,"$\mu=8.7, \sigma=2.5$" +13619,"$p_1,p_1$" +13620,$U = X + Y$ +13621,"$a(\epsilon^1,\dots)$" +13622,$\rho(X)=\mathsf{E}[h(X)L(X)]$ +13623,$\mathsf{E}(T)=74.25$ +13624,$P/ Q$ +13625,$C=(\mathsf{VaR}_{0.996} - 2EL)(1-\tau)$ +13626,$\frac{p}{p+k}p_k(a(p+k))\frac{\delta_k}{k!}$ +13627,$g'(1-p) \frac{q\wedge \alpha}{q}$ +13628,$P_Q$ +13629,$\lambda < \infty$ +13630,$\hat F(x)$ +13631,$(1-p)^{-1} \min_x x(1-p) + \mathsf{E}[(X-x)^+]$ +13632,$r_f /(1+ r_f)$ +13633,$Z(\omega):=(d\mathsf{Q}/d\mathsf{P})(\omega)$ +13634,$\bar \zeta_t\not=1$ +13635,$\mathsf{E}[X_1\mid X_t=x]$ +13636,$90$ +13637,"$\Theta_s\sim\text{Normal}(\theta_s, \sigma_s)$" +13638,$\bar a_{40}=17.95$ +13639,$Y= IX$ +13640,$\mathsf{E}[D_t D_s] - \mathsf{E}[D_t]\mathsf{E}[D_s] = 0$ +13641,"$\mu,\sigma$" +13642,$E_k$ +13643,$\int_0^1 \mu(ds) = 1 - \alpha < 1$ +13644,$\mathcal{A}_1$ +13645,"$\inf_Q \, \alpha(Q) \in\mathcal{B}bb R$" +13646,"$(asecret.east) + (0,-0.5)$" +13647,$\rho(X_n)\downarrow 0$ +13648,$\Omega'$ +13649,$\Lambda \mu_{U} = \dfrac{E( r_{U} ) - r_{f}}{\sigma_{r_{U}}} \left(\dfrac{\mu_{U}}{\sigma_{U}}\right)$ +13650,$E[Z]$ +13651,"$300,000, and find that the insurance charge is \$" +13652,$f(x)=0$ +13653,$f_{xx}=-1/S_t^2$ +13654,"$(x_1, x_2)$" +13655,$\rho(X)=\mathsf{E}_{\mathsf Q}[X]$ +13656,$d=iv=i/(1+i)=1-v$ +13657,$r_{pq}:=\sqrt{p(1-p)}$ +13658,$\bar Q(a):=a-\bar P(a)$ +13659,$\mathsf{VaR}_{p_0}(X)=\sup X$ +13660,$H=G_0-F$ +13661,$\{r_i\}$ +13662,$-g''$ +13663,"$(fun1a.south -| fun4a.south east)+(\smlspc,-\smlspc)$" +13664,$\mathsf{E}[x_t x_s] = \mathsf{E}[x_t \mathsf{E}[x_s \mid \mathscr{F}_t]] = \mathsf{E}[x_t^2]$ +13665,$F(\omega_1 + \omega_2)$ +13666,$h(s)=s^m$ +13667,$\beta_0+\beta_2$ +13668,$m=K^{-1}Km$ +13669,$p_0=p_1$ +13670,"$\langle \cdot,\cdot\rangle:\mathcal{X}\times\mathcal{M}\to \mathbb{R}$" +13671,$m(1)=m_3=0$ +13672,$\rho(X)+c$ +13673,$\mathsf{Q}\in\mathcal Q$ +13674,$\mathsf{P}$ +13675,$q_Y$ +13676,$R_1(t) 0$ +13697,$\mathsf{E}(Q/X | X\ge x)$ +13698,$v\mathsf E[X_i]$ +13699,$g(0^+) = r/(1+r)$ +13700,${}^{[>331]}$ +13701,$e^{st}-1\approx st + O(s^2)$ +13702,$\lambda=0.1525$ +13703,$\mathsf E[X_i\mid X=x]$ +13704,$r(a-P)$ +13705,$\int rf =\mathsf{E}[r]=\mu$ +13706,$g(S(x))=s$ +13707,$p(\delta_p-il_p)$ +13708,$\tilde X_1\le X_1$ +13709,$\int_0^a g(S(x))dx=\left.xg(S(x))\right|_0^a + \int_0^a x g'(S(x))f(x) dx =\mathsf E[Xg'(S(X))]$ +13710,$v(\Omega)=1$ +13711,"$(X_1,\dots,X_r)$" +13712,$l = jn$ +13713,$x_0\to\infty$ +13714,$a$ +13715,"$\phantom{P}= v\,\mathrm{EL} + d\,\max(\mathrm{loss})$" +13716,"$\rho(X, p^\star)=a(X)$" +13717,$\gamma(ds)$ +13718,$\lambda m^3$ +13719,"$E_1,\dots,E_N$" +13720,$500g=4900N$ +13721,$\mathsf{E}_Q(\cdot)$ +13722,$X=3$ +13723,$\rho(W)=\mathsf{E}[W]+\lambda\sigma(W)$ +13724,$\omega$ +13725,"$D^n\rho_X(X_{i,\cdot})$" +13726,$x=3$ +13727,$p\delta_p/p\nu_p=\iota_p$ +13728,$c\ge 1/2$ +13729,"$p\in [0,1]$" +13730,$. Thus $ +13731,$p=0.4$ +13732,$\rho(X) = \inf\{ \alpha \mid X+\alpha \in \mathcal{A} \}$ +13733,$m=q(p)$ +13734,$W=Z$ +13735,"$\int_{[0,1]}$" +13736,$\tilde \mathsf{Q}$ +13737,$U = X$ +13738,$A\subset \mathbb{R}$ +13739,$\log(1+\mu t + \sigma dW_t)=\mu t + \sigma dW_t +o(dt)$ +13740,$s < 1$ +13741,$\sum_\omega \mathsf Q(\omega) =\mathsf{E}[Z] / \mathsf{E}[Z]=1$ +13742,$t_0 < 0.5 < t_1$ +13743,"$\mathcal{P}_{X,r_X}$" +13744,$M_i(x)+Q_i(x)$ +13745,"$\rho^E(t_i,s_j)$" +13746,$g(\hat S(x))$ +13747,$m_X(·)$ +13748,$\partial P/\partial x$ +13749,"$n={{n}}, p=1/{{p}}={{pf}}$" +13750,$\mathsf{VaR}_{0.996}$ +13751,$X+\epsilon Y$ +13752,$P(t)=(1-t)R_0(t)+tR_1(t)$ +13753,$E(u(X)) \le E(u(Y))$ +13754,$\mathsf{Pr}r(X_n=1)=1/n$ +13755,"$(x_1,\dots,x_n)$" +13756,$\lim_{x\to \infty} xg(S(x))=0$ +13757,$f'(a)$ +13758,$Y=Z/\lambda$ +13759,$\Omega\times\mathscr{F}$ +13760,$\rho(Y):=\max(Y)$ +13761,$\sigma=\sqrt{\log(\mathsf{cv}^2+1)}$ +13762,$x \le 300$ +13763,"$1,9,10$" +13764,"$(0, 1)$" +13765,$\mathsf{Pr}r(X_n=Y)=\mathsf{Pr}r(X=Y)=0$ +13766,$D_s$ +13767,$\mathsf Q(B) = \mathsf P(A\cap B)/\mathsf P(A)=\mathsf P(A\cap B)/(1-p_0)$ +13768,$\mathsf{Pr}(A\mid\mathscr{F})(\cdot)$ +13769,$r \in G$ +13770,$\bar\iota = 0.12$ +13771,$d=(\log(a)-\mu)/\sigma$ +13772,$\mathbf {X_1}$ +13773,$x=x_{90}=\cdots=x_{98} < x_{99} < x_{100}$ +13774,$g'(x)=0$ +13775,$A_{\min}$ +13776,$m(A) = m^*(A)$ +13777,"$1m, essentially only providing \$" +13778,$g'(s-)=g'(s+)$ +13779,$P(t):=\rho(X_t\wedge a_t)$ +13780,$\gamma_i$ +13781,$\sum_i x_i$ +13782,"$1,000 sales per year, and the claim severity is \$" +13783,$L(\nu)$ +13784,$x = 0$ +13785,$\ge\mathsf{E}[X_i]$ +13786,$s < t$ +13787,$X=1_A$ +13788,$\nu(dy)=c(y)dy$ +13789,$\mathrm{L}$ +13790,$x^{\ast}:=\min(x)$ +13791,$\mathsf{F}^{-1}\mathsf{C}$ +13792,$M_k$ +13793,$X\le k$ +13794,$J(1)<\infty$ +13795,$U(-X)\ge U(-Y)$ +13796,$\mathsf{E}_\mathsf{Q}[X\wedge a]$ +13797,$s\ge s_0$ +13798,"$[0,1] \times \{y\} \subset X$" +13799,$(g(S(x^-)-g(S(x)))/(S(x^-)-S(x))$ +13800,$(X-a)^+$ +13801,$0<\alpha_1<\alpha_2<1$ +13802,$A \in \mathcal{C}$ +13803,$b<1-a$ +13804,$\mathsf{LI}$ +13805,$\theta(1+\log\theta)$ +13806,$f'>0$ +13807,"$\mathsf{E}[X_{t,d}\mid \mathcal F_0]=\mathsf{E}[X_{t_d}]$" +13808,$a_x=2$ +13809,"$X_{t+1,1}$" +13810,"$(\Omega\times M, \mathscr{F}\otimes\mathcal{B})$" +13811,$\cdot$ +13812,$\nabla\rho(X)=\{h\}$ +13813,$g(s)=d + sv$ +13814,"$ is time cheap. Indeed, the condition implies the denominator is $" +13815,$g(\sqrt{st})^2$ +13816,$\omega=\omega'$ +13817,$\rho(Y)\ge \rho_m(Y) \ge a_Y$ +13818,$\sum (\hat y_i-\bar y)^2$ +13819,$1/2+1/4+1/6+\cdots$ +13820,$A\subset C$ +13821,$\rho_t(X) = \displaystyle{1}{\beta} \log \mathsf E[e^{-\beta X}\mid \mathscr F_t]$ +13822,$\bar A_{x+b}$ +13823,$1-(1-s)^m$ +13824,$\mathsf{E}[(X-x_l)^+]$ +13825,$1-g(0^+)$ +13826,"$\sum_j X_{i,j}(a)\Delta g(S_j)$" +13827,$A_\omega$ +13828,$p_{k+1}$ +13829,$=\mathsf{E}[X_i/X \mid X \le a]$ +13830,$\omega=1$ +13831,"$\rho(X)=\int_{[0, 1)} \mathsf{TVaR}_p(X)m(dp)$" +13832,"$g(s) = \max(g_m, g^0(s))$" +13833,$g'(1-s)$ +13834,$X\in \mathcal X$ +13835,$R(\mathsf{E}(\Theta))$ +13836,$L^*$ +13837,$g'(S(x))=dQ/dP$ +13838,$F_X$ +13839,"$\Theta_s\sim \text{Normal}(\hat\theta_s, \hat\sigma_s)$" +13840,$\mathsf{E}[r_{FS}(t)]$ +13841,"$\Omega=\{1,2\}$" +13842,$a_0$ +13843,$g(s)\ge d$ +13844,$Z_4$ +13845,$P_{act}-P$ +13846,$0.75$ +13847,$\mu$ +13848,$gS>S$ +13849,$U_1=\mathsf{Pr}hi(Z_1)$ +13850,$\mathsf{NA}(X_i) = \mathsf{CoTVaR}_p(X_i)$ +13851,$qX_i$ +13852,$-(1-s)g''(1-s)\ge 0$ +13853,$j={c+1:.0f}$ +13854,$X'(\omega) \le Y'(\omega)$ +13855,$s_u = (f+1) / (n+1)$ +13856,$J(x)\to\infty$ +13857,"$(L,\mathcal{A}, \mu)$" +13858,$\rho(A_k)\ge \mathsf E[A_k] = k\mathsf E[N]$ +13859,$y\neq\mu$ +13860,$\beta_L$ +13861,$p_{\mathit{cl}}$ +13862,$\rho_g(X)=452.98$ +13863,$p=1/6$ +13864,$w(x)=x$ +13865,$X < A)$ +13866,$\Delta=\{ \mathsf{E}[Y] \}$ +13867,"$S,T$" +13868,$\int_A m(Y)d\mathsf{Pr}=\int_A m(Y(\omega))\mathsf{Pr}(d\omega)$ +13869,"$n_{\text{water, 20°C}} = \frac{2338 \times 1}{8.314 \times 293.15}$" +13870,"$p\,da$" +13871,"$(x_1,x_2]$" +13872,$\hat\theta=0.53$ +13873,$\bar sm$ +13877,$\lambda\mapsto \rho(X+\lambda Y)$ +13878,$P_{x+b}-P_x$ +13879,"$U_L,U_M,U_R$" +13880,$p_{\mathit{pr}}$ +13881,"$\rho = 1000 \, \text{kg/m}^3$" +13882,$\{X_\alpha\}$ +13883,"${}^{[<13,153]}$" +13884,$\rho(X)=\infty$ +13885,$\nu(p)=1/(1+\iota(p))$ +13886,$\mathsf{M}'\mathsf{M}$ +13887,$ is a measure on $ +13888,$g'(S_X(X))$ +13889,$u_j(x) = 1 - exp(-\lambda_j x)$ +13890,"$a\,\dfrac{\mathsf{E}[X_1\mid X \ge a]}{\mathsf{E}[X\mid X \ge a]}$" +13891,$(a-X)^+$ +13892,$\rho(m) = \rho(0) - m$ +13893,$LR_{Dual}$ +13894,$(1-p)^{-1}$ +13895,$\gamma=r_f$ +13896,"$307,733 (\$" +13897,$X_n=1$ +13898,$\exists !x\phi(x)\leftrightarrow \exists x\phi(x)\wedge \forall x\forall y(\phi(x)\wedge \phi(y)\rightarrow x=y)$ +13899,"$q_X(U),q_Y(U)$" +13900,$\sigma=0.25$ +13901,$(\rho)$ +13902,"$X_j=(x_{1j}, x_{2j},\dots,x_{Mj})^t$" +13903,"$\gamma_{a, i}(x)$" +13904,$ policyholders are paid $ +13905,$x x)$ +13921,$r\times r$ +13922,$1.5 billion XS $ +13923,$\mathsf{E}(YZ\mid X)=Z\mathsf{E}(Y\mid X)$ +13924,$\delta_\mu$ +13925,$\mathsf{E}_q(X_1)$ +13926,$P'_1$ +13927,$r\times n$ +13928,$s_u$ +13929,$\Delta_k^s = (1-\alpha)\Delta_k^e + \alpha\Delta_k^o$ +13930,$A=X_1$ +13931,$\phi'(s)=\mu(ds)/(1-s)\ge 0$ +13932,"$n=0,1,\dots$" +13933,$A \subseteq O$ +13934,$g'(1)$ +13935,$\nu(p) F(x)$ +13936,"$0,10,40$" +13937,$LR_{\mathsf{Dual}}$ +13938,"$\Delta m_{32}^2 \approx 2.44 \times 10^{-3} \, \text{eV}^2$" +13939,$\mathsf{E}[e^{kX}]$ +13940,$\mathsf{Pr}r(S_t > a)=\mathsf{Pr}r(X_t > a/S_0)=1-\mathsf{Pr}hi\left([\log(a/S_0)-(r-\sigma^2/2)t]/\sigma\sqrt{t} \right)=\mathsf{Pr}hi(d^*-\sigma\sqrt{t})$ +13941,"$\mathsf{CONVEX,LI}$" +13942,$\mathcal{A}=\{\Omega\}$ +13943,$c<0$ +13944,$h(0.9)/0.9 = 0.76$ +13945,$0 < \lambda \le 1$ +13946,$X=W+Q$ +13947,$U_1$ +13948,$\alpha_{1} \not= 1$ +13949,$\mathsf{SA}$ +13950,$-\log(\cos\theta)$ +13951,$\gamma = \frac{1}{\sqrt{1 - (v/c)^2}}$ +13952,$C^{D+E}$ +13953,$\mathscr{F}_1 \subset \mathscr{F}$ +13954,$xf(x)$ +13955,"$p(\cdot, A)$" +13956,$2^{256}=115792089237316195423570985008687907853269984665640564039457584007913129639936=1.2\times 10^{77}$ +13957,$A_{k_0}$ +13958,$s\uparrow 1$ +13959,$\mathsf{E}[x_t x_{s-1}] = \mathsf{E}[x_t \mathsf{E}[x_{s-1} \mid \mathscr{F}_t]] = \mathsf{E}[x_t x_{t-1}]$ +13960,$\mathbf {\iota=M/Q}$ +13961,$\aleph_0$ +13962,$q = s$ +13963,$\bar\nu(x)$ +13964,$X_{i}^{< 0.99 M >}$ +13965,$\mathsf E(G')=1-f$ +13966,$\mathsf{E}[X^k] \le \mathsf{E}[Y^k]$ +13967,$\mathbf {M\Delta X}$ +13968,"$a_{0,t}:=a(Y_{0,t})$" +13969,$\zeta\in \mathcal{Z}*$ +13970,$\eta(\theta)$ +13971,"$s,t<1$" +13972,$U = a$ +13973,$aggfft*aggfft = aggfft$ +13974,$K_\theta(t)= tbd$ +13975,$S^*$ +13976,$\bar F(a)=\int_0^a F(x)dx = a-\bar S(a) = \bar Q(a) + \bar M(a) = \mathsf{E}[(a-X)^+]$ +13977,$dQ/dp=\phi(p)$ +13978,$1-g(s) = v - vs = v(1 - s)$ +13979,$L_0^l(X)$ +13980,$\mathsf{TVaR}_{0.5}(X_1)=9$ +13981,$A=\rho_{\mathsf{TVaR}}(X)$ +13982,$v=S$ +13983,$X=X_s + X_c$ +13984,$\mathsf{E}_g[X_i \mid X]$ +13985,$\mathbf {\omega_i}$ +13986,$B_t$ +13987,$\{X\le x\}$ +13988,"$(s_{R+1},g_e)$" +13989,$g(S(x-))=1$ +13990,$p<0.05$ +13991,$p\nu_p$ +13992,"$\px=\sum_i\int_B X_i\,d\mathsf{Pr}$" +13993,"$\int_{\mathbb{R}} K_\delta(y)\,dy=1\ \ \forall \delta>0$" +13994,"$(p,q(p))$" +13995,$a\le \dfrac{P-S}{\iota} + P\approx \dfrac{P-\mathsf{E}[X]}{\iota} + P$ +13996,$(z-\zeta)^+$ +13997,$\mathbf {X_{1}}$ +13998,$\kappa_1(10) = \mathsf{E}[X_1\mid X=10]$ +13999,$A=0.00022$ +14000,$X_i(\mathbf{x}; a)$ +14001,$\rho(X)=\int g(S(x))dx$ +14002,"$(X,a_1)$" +14003,"$\bar P(\mathbf{v}, a(\mathbf{v}))$" +14004,$\bar\iota$ +14005,$S/L\ge A/L-1$ +14006,$0.318 / 260.81 = 0.13\%$ +14007,$\theta<0$ +14008,"$^{\,11}$" +14009,$X\ge Y\implies \rho(X) \ge \rho(Y)$ +14010,"$p_B(y)=p(B, y)$" +14011,$c(y)\ge 0$ +14012,$1/\lambda$ +14013,$s < 0.5$ +14014,$s \leq t$ +14015,$\mathsf{E}[g'(S(X))]=1$ +14016,$x \lt y$ +14017,"$(rep.east) + (1.5, -0.5)$" +14018,$R_L=R_f + \beta_L(R_M-R_f)$ +14019,$R_2(t) > R_2(0)$ +14020,$\sigma_\mu(\alpha) = \displaystyle\int_0^\alpha\dfrac{1}{1-u}\mu(du)$ +14021,$\mathsf{E}[Y]=\mu$ +14022,"$-b, b$" +14023,$L=\sum_i L^i$ +14024,$\sqrt{2Np}=19$ +14025,$= n \times 6.022 \times 10^{23} = 2.38\times 10^{24}$ +14026,$\alpha=1/2$ +14027,$G = C + \sum_i N_i$ +14028,$Q_2\Delta X$ +14029,$n=67$ +14030,$\sum_{j=0}^{n} q_j = 1$ +14031,$h(X)$ +14032,"$b_m\le b_{m_0}<\mathrm{ess\,sup} X<\infty$" +14033,$F = m v^2 / r$ +14034,$0.25$ +14035,$X=A+B$ +14036,$\mathbf {Z_3}$ +14037,$0\le \pi\le 0.5$ +14038,"$j=1,2, \dots, d$" +14039,"$j \in \{5,\dots,8\}$" +14040,"$ (MA.south)+(0, -1) $" +14041,$g_n$ +14042,$\omega_6$ +14043,$\omega = \dfrac{d\theta}{dt}$ +14044,$10^{10}$ +14045,$f_{opt} = 1-s/g$ +14046,$(r-i)\sum_t Q_t$ +14047,$\mathit{PV}_{r_X}(X) + \mathit{PV}_{r_f}(\text{UW profit tax})$ +14048,$0.97$ +14049,$\bar S(a+da)-\bar S(a)\approx \bar S'(a)da = S(a)da$ +14050,$-1.805$ +14051,$g'(t)>0$ +14052,"$\sigma_1=0.15, 0.5, 0.85$" +14053,"$n=2,3$" +14054,$(1-s)-(1-g(s)) = g(s)-s$ +14055,$\tilde X_1+\tilde X_2\succeq^2 \tilde X_1$ +14056,$9.67$ +14057,$\mathsf{E}_\mathsf{Q}[\cdot]$ +14058,"$\mathsf{CP}(\lambda, \mathrm{gamma}(\alpha, \beta))$" +14059,$M=E_0+B$ +14060,$\le 1$ +14061,$q=1-p$ +14062,$\mathsf{E}[(X-c_l)^+]$ +14063,$V(\mu)=\kappa''(\tau^{-1}(\mu))$ +14064,$X\wedge a=30$ +14065,"$1,600** = (1+r)-1(\$" +14066,$g'(s)=1-r_0$ +14067,$\mathsf{MON'}$ +14068,$A\setminus B$ +14069,$\mathsf{E}[X\wedge a]= 2.4982$ +14070,$\bar X$ +14071,$S/\check S$ +14072,$X_1 + X_2 \sim 2^{1/\alpha}X$ +14073,$\theta<1$ +14074,$\mathsf{Pr}r(X_{-1}a)>0$ +14088,$C_p$ +14089,$F((k+1/2)b)-F(k-1/2)b)$ +14090,$\xi=p$ +14091,$\lambda_{x+t}=\lambda\mu_{x+t}$ +14092,$S(x) = \mathsf{Pr}r(X>x)$ +14093,$s=0.01$ +14094,$\backslash$ +14095,$\mathsf{E}[\mathsf{E}[X_iZ\mid X]]\not=\mathsf{E}[\mathsf{E}[X_i\mid X]\mathsf{E}[Z\mid X]]$ +14096,"$500,000 of loss, and a per loss limit of \$" +14097,$s=s_1+s_2$ +14098,$p^*=48.25/71=0.6796$ +14099,$Z(s)=\mathsf{Pr}hi^{-1}(s)$ +14100,"$j, p, S, \kappa_1, \Delta X, \Delta(X\wedge a)$" +14101,$\rho_g(X)<\infty$ +14102,$μ = δ_α$ +14103,$\rho_\phi$ +14104,$\mathsf{CTE}_p(X)=(8+12+25)/3=15$ +14105,$\mathsf{P}(X=0)$ +14106,$P=53.565$ +14107,$P^T(F\mid t)=\mathsf{Pr}(F)$ +14108,$\mathsf{E}[U]/P$ +14109,"$\mathrm{Cov}(A,N)=\mathrm{E}(X)\mathrm{Var}(N)$" +14110,$k_1$ +14111,$p\ge 1$ +14112,$\epsilon(\mathsf{E}_q(X_1)-s)$ +14113,$\sigma\in L_q$ +14114,$\bar Q(a)=a - \bar P(a)$ +14115,$\bar M(x)$ +14116,$\lambda dt$ +14117,$a\ge \psi(X)$ +14118,$\mathbf {q}$ +14119,"$\kappa_{T_x}=x\,\kappa_{T}$" +14120,$f(x_p)$ +14121,$\iota=\iota(1-s)$ +14122,"$(N,m)$" +14123,$X=q(p)$ +14124,$C_t$ +14125,$-\mathsf{E}[X_i\mid X=t]f(t)/t=-\kappa_i(t)f(t)/t$ +14126,$0 | \$ +14127,$F(X(\omega))$ +14128,$\sigma=0.4$ +14129,$\alpha=-1/2$ +14130,$\int c(x)dx = 1$ +14131,$G=f+G'$ +14132,$\mathsf{E}[X\mid \mathcal F'](\omega)$ +14133,$\mathcal X^\perp$ +14134,"$\mathsf{E}_P[h_0]=\mathsf{E}_P[h_{i,\epsilon}]=1$" +14135,$\mathsf{E}_\mathbb{Q}$ +14136,$a_{t} = a_{t-1}$ +14137,$8.617 \times 10^{10}$ +14138,$ is an interior point of $ +14139,${}^1S^{-1}=S^{-1}$ +14140,$m$ +14141,$=\mathsf{E}[X_i \mid X=a]$ +14142,$r(X+Y)\le r(X)+r(Y)$ +14143,$\mathsf{Pr}i$ +14144,$\mathsf{E}_\mathbb{Q}(X_i) = \mathsf{E}_\mathbb{Q}(\mathsf{E}_\mathbb{Q}(X_i \mid X)) = \mathsf{E}_\mathbb{Q}(\mathsf{E}(X_i \mid X))$ +14145,$\mu = w \delta_{\alpha_1} + (1-w) \delta_{\alpha_2}$ +14146,$X_\nu/\nu$ +14147,$X_1 = \mathsf E[X\mid \mathscr F_1]$ +14148,$p(\omega)$ +14149,$X_i(a) = aX_i/X$ +14150,$E[Total(a)]$ +14151,$k/T=\beta$ +14152,$g_2(s)=\sqrt{s}$ +14153,$\dots$ +14154,$\alpha+\beta = \iota^\ast/(1+\iota^\ast)$ +14155,$6.258$ +14156,$w=\dfrac{g(s)-s}{1-s}$ +14157,$1<\lambda=k+f$ +14158,"$t\in[\epsilon, 1]$" +14159,$\beta_1(x)>\alpha_1(x)$ +14160,$\mathsf{E}(X\mid\mathscr{G})$ +14161,"$\mathscr{G}=\{ b(a_i,2^{-j}) \}_{i,j}$" +14162,$\rho_{\mathsf{TVaR}}$ +14163,$d=i/(1+i)$ +14164,$\nu(p)$ +14165,"$\mathscr{G}amma(\alpha,\beta)$" +14166,$100* for the first year and \$ +14167,$p<0.7$ +14168,$\mathsf E[X\mid A]$ +14169,$g(S(x)) = 1 - h(F(x))$ +14170,"$\mathsf{Tw}_p(\mu, \sigma)$" +14171,$\inf_\eta \{ \eta + \phi(X_\eta) \}$ +14172,$\text{Var}(G)=a\theta^2$ +14173,$s^\star < s$ +14174,$m_1 r_1 = m_2 r_2$ +14175,$r=3$ +14176,$a\ge 0$ +14177,$m+1$ +14178,$g(uv) m_2 / r_2$ +14184,$0.1$ +14185,$\Delta \tilde p< \Delta p$ +14186,$\leftrightarrow\mathcal P\rightarrow \rho_t(X)=\max_{P\in \mathcal P}\mathsf E_P[X\mid \mathcal F_t]$ +14187,$g'(1)<1$ +14188,$0\le\lambda\le 1$ +14189,$\mathsf{E}_\mathsf{Q}[X_i \mid X]=\mathsf{E}[X_i \mid X]$ +14190,"$\alpha\in [0.6, 1.5]$" +14191,$\mathsf P(A\mid \mathscr G)$ +14192,$\iota^i=\iota$ +14193,$\phi_R-((1-\alpha)\phi_R + \alpha\phi_m^o)$ +14194,$Q_\epsilon \to Q$ +14195,$a_l < b_l$ +14196,$\rho(0X)=\rho(0)=0\rho(X)=0$ +14197,$X_n=X$ +14198,$t0$ +14210,$W_T$ +14211,$\mathsf{E}E$ +14212,$a\alpha_i(a)=\kappa_i(a)$ +14213,$\mathsf{var}phi(X + Y) = \mathsf{var}phi(X) + \mathsf{var}phi(Y)$ +14214,$\mathcal{A}$ +14215,$20𝐾𝐾 + \$ +14216,$F:\mathbb{R}^n\to\mathcal{X}^n$ +14217,"$[0,1]\to\{\text{Explicit Events}\}$" +14218,$r_2=$ +14219,$\rho(X)=\int_0^1 \mathsf{TVaR}_p(X)\mu(dp)$ +14220,$a'=\mathsf{E}[X|A^c]$ +14221,"$\mathsf{Pr}hi, \phi$" +14222,$\mathsf{E}[X] + \pi\mathsf{E}[(X-\mathsf{E} X)^+]$ +14223,$\mathsf{E}[\mathsf{E}[X\mid \mathscr{G}_1]\mid \mathscr{G}_2]=\mathsf{E}[X\mid \mathscr{G}_1]$ +14224,$\mathsf{Pr}(A\mid\sigma(T))$ +14225,$u < A$ +14226,$\mathsf{Pr}(\mathsf B(s)=1)=s$ +14227,"$20,175.75 + \$" +14228,$X_n\uparrow 0$ +14229,$\log(c(y))$ +14230,$C_1(t)=C_2(t)=\bar P^a(t)$ +14231,$L'$ +14232,"$\rho(X)=\sup\{ \mathsf{E}(XZ) \mid Z\ge 0, \mathsf{E}(Z)=1, \mathsf{E}(Z\log(Z))\le\log(1/(1-\alpha)) \}$" +14233,$S_i(x):=\alpha_i(x)S(x)$ +14234,$\mathsf{E}[a \wedge X] = \displaystyle\int_0^\infty \gamma_a(x) S(x) dx$ +14235,"$j=1,\dots,d$" +14236,$f'(\omega) = 1 - s/\omega^2=0$ +14237,$\mathsf{Pr}(Z=0)$ +14238,$\max(A+B)=\max(A) + \max(B)$ +14239,$M_i = M_i^+ - M_i^-$ +14240,$\partial B$ +14241,$P=\int_0^a g(S(x))dx = \int_0^a d + vS(x)dx = v\mathsf E[X \wedge a] + da$ +14242,$\mu A=\mu 1_A$ +14243,$t^*$ +14244,$X(n)$ +14245,$r=(1+\bar\iota)/(1+\tau)-1$ +14246,$S_X(a)$ +14247,"$4,000 | | | | | $" +14248,$Y_{0}$ +14249,${}^2S^{-1}(t)=q\mathsf{TVaR}_q(X)$ +14250,$p^*=1$ +14251,$\rho(X)=\max_\mathsf{Q} \mathsf{E}_\mathsf{Q}[X]$ +14252,"$\mathsf{E}[X_i] + \pi(X)\mathsf{cov}(X_i, X)/\mathsf{SD}(X)$" +14253,"$X\wedge a =\min(X,a)$" +14254,$\mathbb{R}\times \mathbb{R}$ +14255,$\bar P(a)=\rho_g(X\wedge a)$ +14256,$\rho(W_0\wedge a_0)$ +14257,$a\le 1$ +14258,$P^i = E[Z\dot X^i]$ +14259,$\mu\sqrt{1+2\mu}$ +14260,$1 \times 10^{15}$ +14261,"$500,000, up to a limit of \$" +14262,$\kappa_1(x)=\mathsf{E}[N_1/(N_1+N_2)]x$ +14263,$\zeta_iZ_i$ +14264,$Pr(X ≤ 1.5)={fz.cdf(1.5):.3f}$ +14265,"$\mathit{MV}(X, a) = a - \rho(X\wedge a)$" +14266,$A(\text{Bernoulli})$ +14267,$\alpha_k^i S_k\Delta X_k$ +14268,"$3,200 | \$" +14269,$ be the waiting time until accumulated surplus equals $ +14270,$+$ +14271,$\mathcal E(X)=c\mathsf{E}[X^2]$ +14272,$\eta=(1-\alpha)^{-1}1_A$ +14273,$d^*=(\log(A/L) + (r_h-\mu_L + \sigma^2/2))/\sigma\sqrt{t}$ +14274,$\mathsf{E}_\mathsf{Q_2}[X_j]$ +14275,$K(x)=(\pi(x^2+1))^{-1}$ +14276,$M_0=Q_0=0$ +14277,"$g\in D_n^*=\{ g \mid (-1)^{k+1} g^{(k)} \ge 0, k=1,\dots,n-1, (-1)^n g^{(n-1)}\text{ nonincreasing} \}$" +14278,$S(x_1)-S(x_2)\approx f(x_1)(x_2-x_1)$ +14279,$L_k$ +14280,"$({s_equity2:.3f}, {d + v * s_equity2:.3f})$" +14281,$\omega=\exp(-2\pi i / n)$ +14282,$\mathsf{E}[XZ_1]$ +14283,$p_i:=0.5$ +14284,$1-w$ +14285,$\pm\infty$ +14286,"$723,103,276 | | $" +14287,$a_1'=a_0-X_{1}$ +14288,$g(S_j)$ +14289,$\text{VaR}_\alpha$ +14290,$A(X+c)=A(X)+c$ +14291,$2\square^2 + 11$ +14292,$\mathsf{E}_g[X_i]$ +14293,$g(1-p)=1- \tilde p$ +14294,$+\mathsf{NORIPOFF}$ +14295,$Q\in\mathscr{P}$ +14296,$x \ge x^\ast$ +14297,$g(S_{\mathsf{j}(a)})=0.5$ +14298,$g=1$ +14299,$\mathsf{E}[(X-m)(1_{U_X\ge p}-B)]\ge 0$ +14300,$1/g'(s)$ +14301,$\mathsf E[e^{tX}]$ +14302,$\rho(\tilde X\wedge a)\le a$ +14303,$\rho_X(X_i)$ +14304,$\mathsf{Pr}r(E')+\mathsf{Pr}r(E)=\mathsf{Pr}r(\Omega)=1$ +14305,$\mathcal A_\rho= \{ X\mid \rho(X)\le 0 \}$ +14306,$\Delta Q_{gc}(a)$ +14307,$\kappa(\theta)=\log\int e^{\theta x} c(x)dx$ +14308,$(1\times m)\times (m\times n) \times (n\times 1)=(1\times 1)$ +14309,$X \prec_n^* Y$ +14310,$0 < x_j < 1$ +14311,$\mathsf{Q}_k$ +14312,$p=0.95$ +14313,$g(S(\alpha))=1$ +14314,$\mathsf{EPD}_\pi(X)$ +14315,$p\uparrow 1$ +14316,$\sum_k a_k I_{X=X_k}$ +14317,$\rho(X/n)=\rho(n(X/n))/n=\rho(X)/n$ +14318,$v=1$ +14319,$G:=T_1-1$ +14320,"$500/year HO insurance then I don't really notice it compared to upkeep, mortgage, property tax etc. It is just a sunk cost. But if I pay $" +14321,$\rho(X)=\mathsf{E}[X] + c\sigma(X)$ +14322,$750K xs \$ +14323,$(x-a)_+^\alpha$ +14324,"$u'>0, u''>0$" +14325,$\mathsf{CP}_2$ +14326,$6\times 10^{14}$ +14327,$\mathsf{E}[r] = \mu_r = M/K = 0.132$ +14328,$\rho(X) \ge -\rho(-X)$ +14329,$k= \mathsf{E}(X\wedge k) + (\rho_m(X) - \mathsf{E}(X\wedge k)) + (k-\rho_m(X))$ +14330,$p=3$ +14331,"$P(A,t)=P_t(A)$" +14332,$Z>\mathsf{E} Z$ +14333,$\mathsf{E}[q(U_X)1_{U_X\ge p}]$ +14334,"$\displaystyle\int_0^a \beta_i(x)g(S(x))\,dx$" +14335,$V(1)$ +14336,$\Delta_k^s = (1-\alpha)\Delta_k^u + \alpha\Delta_k^o$ +14337,$\mathbf {S}$ +14338,"$Z=\mathsf{CP}(\lambda,\text{gamma}(\alpha,\beta))$" +14339,$m({1})=0$ +14340,$O(dt)$ +14341,"$0,1,2,\dots$" +14342,$a(X+\Delta X)$ +14343,$\nabla^i\phi(X)$ +14344,$s_1=0$ +14345,"$435,801,207 | | -$" +14346,"$[0,1]\subset\mathbb R$" +14347,$x\in D(E)$ +14348,$\rho_m$ +14349,$\displaystyle\int_0^1 \text{AVaR}_\alpha(X)d\alpha$ +14350,$\bar P_2\ge (\rho(X_1)-\bar P_1) + \rho(\mathsf{E}[X_2\mid X_1])\ge \rho(\mathsf{E}[X_2\mid X_1])$ +14351,$1_{X>x_1}$ +14352,$n-5$ +14353,$\mathsf{E}[X]=\sum_{\omega\in\Omega} X(\omega)\mathsf{Pr}r(\omega)$ +14354,"$m_j=m([p_{j-1},p_j])$" +14355,$\int_0^s \phi(1-t)dt$ +14356,$z=x$ +14357,"$\rho(X+tY)=\langle \zeta_t, X+tY \rangle$" +14358,$\beta_1gdX$ +14359,$E[X_\tau] = 1$ +14360,$B\in\mathscr{G}$ +14361,$g:\text{thin layer risk}\mapsto\text{price}$ +14362,$\bar S_i(a) := \mathsf{E}[X_i(a)]$ +14363,$(x-\mu_x)^+$ +14364,$\mu+\lambda\sigma$ +14365,$5 \times 10^{14}$ +14366,$dp=g'(S(x))f(x)dx$ +14367,$\mathsf{E}_{\mathsf Q}[X_i \mid X]$ +14368,$c(S)= \rho\left( \sum_{i\in S} X_i \right)$ +14369,$-\int xd(g\circ S)=\int g(S(x))dx$ +14370,"$ED(\mu,\sigma^2)$" +14371,$g(s)=1-\check g(1-s)$ +14372,$2$ +14373,$R_0$ +14374,$j>0$ +14375,$A(-X)$ +14376,$\dfrac{1}{1-p^*}\displaystyle\int_{1-p^*}^1 q(s)ds = q(p) = 100.0$ +14377,$r=0.98$ +14378,$\mu_U = 1-p = 0.995$ +14379,$S(a)da$ +14380,"$\mathcal{P}_{X,c}=\mathcal{P}$" +14381,$M_i(x) = \beta_i(x)gS(x) - \alpha_i(x)S(x)$ +14382,$\partial Y/\partial x_i$ +14383,$\mathsf{E}[X_1]=\infty$ +14384,$z^n$ +14385,$\mathbf {Z_4}$ +14386,$m>n$ +14387,$\DeltaR = \betaR$ +14388,$g(s)=s^{1/4}$ +14389,$E[X\bar gS(X)]$ +14390,"$g(s) = \min(1, s / (1-p))$" +14391,$375-185=190 > 0$ +14392,$V(m)$ +14393,$j_n(x)=j(x)\wedge n$ +14394,"$8.2 billion loss for homeowners, $" +14395,$\rho(L) = q(1-g{-1}(1-p))\delta > \mathsf{E}(L)$ +14396,"${}^{[<148,67]}$" +14397,$\alpha_1(90) = (0.0909 \times 0.0625 + 0.1 \times 0.0625)/(0.0625+0.0625)=0.0955$ +14398,$\int_0^\infty j(x)dx$ +14399,$\int_a^{a+y} S(x)dx$ +14400,$x_i+y_{k(i)}$ +14401,$U\subset\mathcal{B}bb{R}^n$ +14402,"$p\in[0,1]$" +14403,$\kappa_j(x)/x > \alpha_j(x)$ +14404,$\mathsf{COH}+\mathsf{FAT}$ +14405,$S_{X_{-1}}(a)$ +14406,$j > 0$ +14407,$\log(1-1/n)<-1/n$ +14408,$\phi(X) = \mathsf{TVaR}_{0.99}(X) = \mathsf E[X\mid p>0.99]$ +14409,$R_1(t)-R_0(t)$ +14410,$\alpha=d_i$ +14411,$\{ \zeta>0 \} = \{ G>c(x) \}$ +14412,"$\lambda c(\lambda y, \lambda)$" +14413,$m^3V_{X_\lambda}(1/m)$ +14414,$NPV = Q - Q = 0$ +14415,$x=q(1-g^{-1}(1-p)))$ +14416,$H_k=H_{g_k}$ +14417,$S(x-)=0.1$ +14418,$1\le p<\infty$ +14419,$e^{-|x|}/(1+x^4)$ +14420,$0.1358$ +14421,$\alpha\to\infty$ +14422,"$X_{t,2}$" +14423,$t\pi/2$ +14424,$\mathsf{E}[(X\wedge a)\_2(a)]$ +14425,$1_{X\le a}$ +14426,$\hat\rho(X)= 1-\check g\left(\displaystyle\frac{1-s}{1-\omega_I} \right)\check g(1-\omega_I)<1-\check g(1-s) = g(s)$ +14427,"$a_{n\!\urcorner,i}=v + \cdots + v^n = (1-v^n)/i$" +14428,$\arg \min_{q \in \mathbb{Q}} E_q[U(a)]$ +14429,$t_0^* < t < t_1^*$ +14430,$\rho(X+Y)\ge \mathsf{E}_{\mathsf Q}[X+Y] = \rho(X) + \mathsf{E}_{\mathsf Q}[Y]$ +14431,"$\rho=0.5, x=1.5, M=1.5,\sigma=0.75, K=8$" +14432,"$5^{-1},5^{-2},5^{-3},\dots$" +14433,$\lim_{t\downarrow 0} R_1(t) \ge \mathsf{E}[X_1](1-g(1-p))$ +14434,"$\langle \mu,tX \rangle - \rho(tX) =t(\langle \mu,X \rangle - \rho(X))$" +14435,$^{*}$ +14436,$M(a)$ +14437,$y_i-\hat y_i$ +14438,$S\not=0$ +14439,$F_{\mathbf{x}}$ +14440,"$\delta = 34/39, \nu=5/39$" +14441,$-\epsilon(\mathsf{E}_q(X_2)-s)$ +14442,$E_2=0$ +14443,$Z=(1-p)^{-1}1_{X>q_X(p)}$ +14444,$\mathsf E[Q\mid \mathscr F_1]$ +14445,$Z_i=\mathsf{Pr}hi^{-1}(U_i)$ +14446,$p=0.74$ +14447,$2/17=0.118$ +14448,"$i=1,\ldots,n$" +14449,$\rho(X)\ge\rho(X+Y)\ge \rho(X)+\mathsf{E}[gY]$ +14450,$k_i=\mathsf{E}_Q(X_i)$ +14451,$\rho(A)\le \rho(N)\rho(X)$ +14452,"$B_1,B_2,\dots$" +14453,$\mathsf{E}[e^{tY}]=\sum \mathsf{E}[(tY)^n/n!]$ +14454,"$i=i^\star,\dots,m$" +14455,$\mathsf{TVaR}_{0.75}$ +14456,$Z_{a}(a)$ +14457,$X_s \le \mathsf{E}[X_t \mid \mathscr{F}_s]$ +14458,$p_j=\mathsf{P}(X=X_j)$ +14459,"$310 billion in premiums annually in California. Since 2011 the California Department of Insurance received more than 1,000,000 calls from consumers and helped recover over $" +14460,$\kappa'(\theta)=-\dfrac{1}{\sqrt{-2 \theta}}$ +14461,$\rho(X\wedge a)=\mathsf{E}_{\mathsf{Q}}[X\wedge a]$ +14462,"$u_j, j3(k+1)/n$ +14467,$R_2(t) 1-p \}$ +14489,$\mathbf {M_{2}}$ +14490,$v=1/(1+\iota)$ +14491,$\delta_k/k!$ +14492,$\mathsf{E}[u(P-X)]=0$ +14493,$D^f\rho_{X;\tilde X}(X_i)$ +14494,"$[l_c, r_c)$" +14495,$\mathsf E[f(X)]\le\mathsf E[f(Y)]$ +14496,$t>\tau$ +14497,$\rho^u$ +14498,"$[1,2]$" +14499,$\mathscr{G}amma$ +14500,"$136,366,904 | | $" +14501,$\mathsf Q\not\ll \mathsf P$ +14502,$F_U$ +14503,$\Sigma$ +14504,$\tau=0.156$ +14505,$t > T$ +14506,$a(\mathbf{v}) =\mathsf{TVaR}_p(X(\mathbf{v}))= (1-p)^{-1}\int_p^1 q_{\mathbf{v}}(s)ds$ +14507,$Y=X$ +14508,"$424) for the initial filing of each letter of credit utilized pursuant to subdivision (a). In addition, the commissioner shall require payment, in advance, of a fee of two hundred eighty-three dollars ($" +14509,$1/6 + 2 /6 + 4/2 + 9/6$ +14510,"$(4.5-\s, \s)$" +14511,$R^2=0.898$ +14512,$\log(X)$ +14513,$10^{-2} - 10$ +14514,$a = b$ +14515,$\mathsf{E}[X] + c\mathsf{E}[(X-\mathsf{E} X)^21_{X>\mathsf{E}[X]}]$ +14516,$\Delta Q(a)$ +14517,$m_0\in\mathcal{M}$ +14518,$\rho_{t+1}(X)=\rho_{t+1}(Y)\implies \rho_{t}(X)=\rho_{t}(Y)$ +14519,$a=\bar P(a)+\bar Q(a)=\bar S(a)+\bar M(a)+\bar Q(a)$ +14520,$g(0+):=\lim_{s\downarrow 0}g(s)$ +14521,$X_i(a)=X_i\dfrac{X\wedge a}{X}$ +14522,$p=0.5$ +14523,$1-P$ +14524,$g(s)=sv+d$ +14525,$\mathsf{WCE}_p(X) := \sup \{ \mathsf{E}[X \mid A] \mid \mathsf{Pr}r(A) \ge 1-p \}$ +14526,$c={c}$ +14527,$0 < p_0 = 1-s_0 < p_1 = 1 - s_1 < 1$ +14528,$l\ge 1$ +14529,$\Omega^X_{i_L-1} < c-c_r \le \Omega^X_{i_L}$ +14530,$\int_x^\infty$ +14531,$\mathsf Q(X>a)/\mathsf P(X>a)$ +14532,$1-2/3=1/3$ +14533,"$\mathsf E[X] \le \rho(\mathsf E[X\mid I], \mu) \le \rho(X)$" +14534,$\tau \ge 0$ +14535,$2.38\times 10^{24}$ +14536,"$(s_{R+1},1)$" +14537,$\sum_n x_n^2\lambda_n$ +14538,$-\rho(-H)$ +14543,$\mathsf{E}[(S_t-a)1_{\{S_t>a\}}$ +14544,$S=e^{\mu t}$ +14545,$\rho(X)=g(q)$ +14546,$A\wedge L$ +14547,$X \preceq_m Y$ +14548,"$\mathscr F_t=\sigma(U_1,\dots, U_t)$" +14549,$a < b$ +14550,$a = a(X)$ +14551,$p\mapsto q(\hat p)=q(1-g^{-1}(1-p))$ +14552,$S =$ +14553,$\bar P^a_g(X_i\subseteq X)$ +14554,$U=X\wedge a$ +14555,$\prec_2$ +14556,$\mathsf{E}[X_i/X\mid X]$ +14557,$\alpha_1+\alpha_2=\beta_1+\beta_2=1$ +14558,$F(a)=\mathsf{P}(1_{X\le a}=1)$ +14559,$g(x)/x\to\infty$ +14560,"$w_{p,p}=1$" +14561,$\rho(X)=\sum_n X(n)\mathsf{P}(n)$ +14562,"$105,000 | 0 | 0 | \ | \ | \$" +14563,$\mathsf{E}_\mathsf{P}[X']$ +14564,$\mathit{MV}_{gc}(a_{gc})=a_{gc}-\rho(X\wedge a_{gc})={{mv_gc}}$ +14565,$W = 0$ +14566,$\pi_X(t_{2j-1})\le \pi_Y(t_{2j-1})$ +14567,$ϕ$ +14568,$\mathsf PV$ +14569,$\lambda=\lambda_1 + \lambda_2$ +14570,"$(p,q(1-g^{-1}(1-p)))=(1-g(S(x)),x)$" +14571,$\nu_0$ +14572,$g(S(x))=g(S(x-))=1$ +14573,$\mathring\Theta$ +14574,$IL$ +14575,"$\partial\eta/\partial\beta_1=0,1,2$" +14576,$\mathsf{E}_\mu[\phi(\mathsf{E}_\pi u\circ f)]$ +14577,"$(Bob) + (0,-2.5)$" +14578,$\mathcal T(X)=\hat\rho(X) - \rho(X)$ +14579,$X=10$ +14580,$(1-p)^{-1}1_A$ +14581,$p \in P$ +14582,$\rho(X_i)\le 0$ +14583,$\zeta\in\mathcal{A}$ +14584,$\mathsf{E}(X_i\wedge x)$ +14585,$\bar P = a - \bar Q$ +14586,"$X \in (k,k+1]$" +14587,$\Omega_R^X=\Omega_m^X$ +14588,$A=f^{-1}(B)$ +14589,$A=\partial \rho(0)$ +14590,$\delta^{18}O$ +14591,"$X\tilde N(0,\sigma^2)$" +14592,$\mathsf{E}[f(X)]\le\mathsf{E}[f(Y)]$ +14593,$\partial a/ \partial v_i$ +14594,$\beta_k^i g(S_k)\Delta X_k$ +14595,$\iota(0.5)=\iota^*$ +14596,$\rho_{t+1}(X) \ge \rho_{t+1}(Y) \implies \rho_{t}(X) \ge \rho_{t}(Y)$ +14597,"$P(A, \cdot) = \mathsf{Pr}(A\mid \mathscr{F}_1)(\cdot)$" +14598,$\rho(X_1\mid \mathcal F_1)\le \rho(X_2\mid \mathcal F_1)$ +14599,$\lambda (\mu_U - \min(U)) = 1$ +14600,$S(x)=d/dx(\mathsf{E}[X \wedge x])$ +14601,$\mathsf{E}_\mathsf{Q}[X_i(a)]$ +14602,$P(A | \mathcal{G})$ +14603,$\mathsf{E}(X|X\ge a)$ +14604,"$[-0.5, 0.5]$" +14605,"$(1,\dots,1)$" +14606,$\mathsf Q\mid X$ +14607,$d(g(S(x))/dx=g'(S(x))f(x)$ +14608,$\rho_m(X)=r$ +14609,$\rho(X) = \max \{ \rho_\phi(X) \mid \phi\in A \}$ +14610,$0.0725$ +14611,$f_P=f$ +14612,"$p(dx,y)$" +14613,"$b_0, b_1,\dots, b_k$" +14614,$\mathsf{VaR}_{0.99}(X_2)=100$ +14615,$A_x$ +14616,"$(0.030, 0.138)$" +14617,$\delta(p)=1-\nu(p)=d+(\delta^*-d)\sqrt{(1-p)/p}$ +14618,"$\kappa_{0,t}(x):=\mathsf{E}[(1-t)X_0 \mid X_t]=(1-t)\mathsf{E}[X_0]$" +14619,$kg$ +14620,"$(2,3)$" +14621,$N\sim\text{Poisson}(1.74)$ +14622,$M(a)=\mathsf{E}(X\wedge a)+dN(a)+(\delta^*-d)\displaystyle\int_0^a \sqrt{F(x)S(x)}dx$ +14623,$E_g[X_2(a)]$ +14624,$X_{i} = \mathsf E[\mathsf E[X_{i+1}\mid I_i]]$ +14625,"$L^\infty(a, b)$" +14626,$\tau=2Z$ +14627,$dp$ +14628,"$\displaystyle\int_0^a \mathsf{E}[X_i\mid X=x]g'(S(x))f(x)\,dx + a\beta_i(a)g(S(a))$" +14629,$S(y_j-)-S(y_j) =\mathsf{Pr}r(X=y_j)$ +14630,$X=1$ +14631,$P=a - v(a-L)$ +14632,$\dfrac{(y-m)^2}{m^2 y}$ +14633,$P(x)=vEL(x) + da = vS(x) + d$ +14634,$\mathsf E[\exp(r_1Z_1)]\mathsf E[\exp(r_2Z_2)]$ +14635,$a_t:=\mathsf{VaR}_p(X_t)$ +14636,$\mathsf{Pr}(X_n\in A)\to\mathsf{Pr}(X\in A)$ +14637,$\mathsf{E}(X_i/X)$ +14638,"$4,225,340 \* 24.3% = \$" +14639,$_1$ +14640,$3^{20}$ +14641,$p < 1/2$ +14642,$Q_{act} = Q - F_0$ +14643,$\rho_e^s(X)$ +14644,"$F_1,F_2$" +14645,"$x\in(-\frac{1}{2}\mu(U), \frac{1}{2}\mu(U))$" +14646,$f(kb)$ +14647,$-stable distribution with Lévy density $ +14648,"$5,000 | \$" +14649,$a=Q+R$ +14650,$\mathsf{E}_Q[\cdot]$ +14651,"$\rho_g(X)= \sum_j X_j\,\Delta g(S_j)$" +14652,$\Delta Q_{ro}(a)$ +14653,$\rho=\sup$ +14654,$\delta=\rho\nu$ +14655,"$(2,\infty)$" +14656,"$s_j,g_j\in[0,1]$" +14657,$\mathcal{R}^c$ +14658,$-X$ +14659,$F(p)=p$ +14660,$q_{Z_k}$ +14661,$a-b_h<0$ +14662,$\delta=i/(1+\iota)$ +14663,$X(p)=q(p)$ +14664,"$30,000, an excess loss premium of \$" +14665,"$\bar M(a)=\bar S(a) -\bar P(a)=\int_0^a M(x)\,dx$" +14666,$F(x_1)=1-S(x_1)=p$ +14667,$X_i=\mathsf{E}[X_i\mid X]$ +14668,$X_i$ +14669,$X({\mathbf{v}})$ +14670,$a>1$ +14671,$. Let $ +14672,$g(1-t)= (1-k)+k(1-t) = 1-kt$ +14673,$x_{i-1}\le x'_i\le x_i$ +14674,$\mathsf{E}[X]=0.6$ +14675,"$\bar P(a)=\int_0^a g(S(x))\,dx$" +14676,$pl(p)$ +14677,$m=\kappa'(\theta)$ +14678,"$h(x):=H(x, 1, t)$" +14679,$\mathsf{VaR}_{0.98}$ +14680,"$b_{X,q}(Y)\le b_{X,r}(Y)$" +14681,$X(0)=X_0$ +14682,"$\{(1-\alpha)^{-1}1_A\mid \mathsf{Pr}r(A)=1-\alpha, X(\omega)\ge a,\ \forall \omega\in A \}$" +14683,"$a,b=\pm 1/n$" +14684,$N\times r$ +14685,$L_X$ +14686,$\sigma_L$ +14687,$\sqrt{p}$ +14688,$0 < p < 1$ +14689,$\bar F(a):=\int_0^a F(x)dx=a-\mathsf{E}[X\wedge a]$ +14690,$g(t) = \mathsf E[u(X-\pi(R+tQ) +R+tQ)]$ +14691,$m\in\mathcal{M}$ +14692,$q(g)$ +14693,$r_1$ +14694,"$i=1,\dots,k$" +14695,$\mathsf{TVaR}_p(X)-\mathsf{E}[X]$ +14696,$\mathscr{Q}$ +14697,$g''>0$ +14698,"$\mathsf{E}_{\mathsf{Q}}[Y]=\mathsf{E}[Y\,g'(S(X))]$" +14699,$f(x)/S(x)$ +14700,$E[X]=e^{\mu+\sigma^2/2}$ +14701,$\mathsf{E}[x_s \mid \mathscr{F}_t] = x_t$ +14702,$\mathscr{P} = \{P\}$ +14703,"$\mathsf{E}[X] \le c \le \mathrm{ess\,sup}(X)$" +14704,$f(x)=x^{-\alpha}L(x)$ +14705,$P'$ +14706,$r>0$ +14707,$\mathsf{Pr}r(X=x_i)=\lambda_i/\lambda$ +14708,"$\mathbf{B}:\left [0,1 \right ] \ni t \mapsto (x(t),y(t)) \in \mathbb{R}^2$" +14709,$\mathsf{Var}(e_i) = s^2(1-h_{ii})$ +14710,"$T_L \mathsf{biTVaR}_{1-s_L,1-s_1}^w$" +14711,${}^{[<144]}$ +14712,$f(w) = \exp(-w)$ +14713,$\iota K / (1-\tau)$ +14714,"$\{(s_i, g(s_i)) \mid i\in\mathrm{bonds} \}$" +14715,$N=365$ +14716,$b\le 1$ +14717,$x-\epsilon$ +14718,$M(x)$ +14719,$a-P$ +14720,$q(p)=S^{-1}(1-p)$ +14721,$X_0=\mathsf PX$ +14722,$1/X_t$ +14723,$-2$ +14724,$\min_{\eta\in \mathbb{R}} \eta + \alpha \mathsf{E}(X-\eta)_+ -\beta\mathsf{E}(X-\eta)_-$ +14725,"$H(z_1,\dots,z_k)$" +14726,$\mathsf{VaR}(X)$ +14727,$(1-w)\mathsf{TVaR}_{0} + w\mathsf{TVaR}_{1}$ +14728,$\sum_i \mathsf{Pr}hi_i(a) = a$ +14729,"$r(X_i,X_j)$" +14730,$f(x) < f(y)$ +14731,$=\mathsf{E}(X \mid X\le a)$ +14732,$F(x):=\mathsf{Pr}r(X\le x)$ +14733,$a_{d}$ +14734,"$\mathsf{TVaR}_p(Y)\le\mathrm{ess\,sup} Y<\infty$" +14735,$1-v$ +14736,"$f_x(x_i, \hat x_i) = f(x_i, \hat x_i) / f_X(x)$" +14737,$^2$ +14738,$\beta\ge 0$ +14739,$\mathcal D(X)\ge 0$ +14740,$P(\cdot)$ +14741,$g(s)=s^\lambda$ +14742,$\mathsf P(A)=0$ +14743,$X\mapsto aX+b$ +14744,$r=16$ +14745,$m_n$ +14746,"$(s^*,g(s^*))$" +14747,$\bar\alpha=-\alpha$ +14748,$cv=0.137$ +14749,$P_X(dx)$ +14750,$\kappa_i(t)/t$ +14751,$\dfrac{\partial \theta}{\partial \mu}=\dfrac{1}{V(\mu)}$ +14752,$u_i$ +14753,$x20+10t$ +14773,$a(X_i)$ +14774,$X=X\wedge a + (X-a)^+$ +14775,$R_i a_i$ +14776,$a=\mathsf{TVaR}(X)$ +14777,"$c(1,2)-c(2)$" +14778,"$(T,\mu)$" +14779,"$M_i^-=-\min(M_i, 0)\ge 0$" +14780,$e_i / (s_{(i)}\sqrt{1-h_{ii}})$ +14781,$\mathsf{E}_\mathsf{Q}[X_i] = \mathsf{E}_\mathsf{Q}[\mathsf{E}_\mathsf{Q}[X_i \mid X]] = \mathsf{E}_\mathsf{Q}[\mathsf{E}[X_i \mid X]]$ +14782,$(1-s)^{th}$ +14783,$A=\sum_i I_iX_i$ +14784,$Z(ω)$ +14785,$\mathsf{TVaR}_p(X)=c$ +14786,$(X-d)^+\wedge (a-d)$ +14787,$\sup\{ \mathsf{E}[Y\sigma(U)] \mid U\text{\ uniform} \}$ +14788,"$2,000,000,000), regardless of the frequency or severity of earthquake losses at any and all times subsequent to the creation of the authority. Once a participating insurer has paid, pursuant to this section, amounts equal to its percentage share of the authority's total gross written premium, multiplied by two billion dollars ($" +14789,$\bar p_i$ +14790,$R_2(t)= \bar P^a_2(t)/t$ +14791,$x_{\max{}} = x_{\min{}} + n$ +14792,$d\tilde p/dp = g'(1-p)=\tilde f(F^{-1}(p))/f(F^{-1}(p))$ +14793,$10$ +14794,$y=0$ +14795,$h(X)=X$ +14796,$p_n(a)$ +14797,$\omega < 1/n$ +14798,$1.7 \times 10^{-3}$ +14799,$h_j(x) = 0$ +14800,$ROE=(g-s)/(1-g)=m/(1-s-m)$ +14801,$aS(a)\not=0$ +14802,$4.7\times 10^{21} / 10^{19} \approx 8\text{mins}$ +14803,${}^{[>29]}$ +14804,$\mathsf{E}[e^{sX_1}]^{1/n}=\mathsf{E}[e^{sX_{1/n}}]$ +14805,$j(x)=x^{-3/2}$ +14806,$l(p):=\log(p/(1-p))$ +14807,$\lambda X_1 +(1-\lambda) X_2$ +14808,"${}^{[<89,94]}$" +14809,$m_{xx}=n^{-1}\sum_i v_i^2$ +14810,"$i=1,\dots,20$" +14811,$\int g(S) = \mathsf E[g'(S)X]$ +14812,$\sum S\Delta(X\wedge a)$ +14813,$E[X_t \mid \mathcal{F}_{t-1}] = X_{t-1}$ +14814,$-15$ +14815,$X\in L^\infty(\Omega)$ +14816,$\mathcal M'$ +14817,"$100 billion; and costs associated with providing regulated insurance paper, such as underwriting, product management, regulatory and compliance costs, taxes licenses and fees, billing, policy maintenance and policy issuance, of nearly 10% of premium, or $" +14818,$p=1.005$ +14819,$\forall\omega$ +14820,$r=0.038$ +14821,$X=X_s + X_l$ +14822,"$(-\epsilon, \epsilon)$" +14823,$E[s|t]=0.08353$ +14824,"$A=\mathbb Q\cap [0,1]$" +14825,$\mathsf{Pr}r(E')=1-\mathsf{Pr}r(E)$ +14826,$X_4=X_5=10$ +14827,$x^{\ast}$ +14828,$p_s=\mathsf{Pr}r(\Theta>0.5) = \mathsf{E}(R(\Theta_s))$ +14829,$X(t)=X(\mathbf{x})=(1-t)X_0 + tX_1$ +14830,$\hat f(t)$ +14831,$\bar y - \kappa(\theta)=0$ +14832,$\tau(a-\bar P_\tau(a))$ +14833,$P(a)=\mathsf{E}(Y\wedge a)+\rho K(a)$ +14834,$\iota(p)$ +14835,$A\le \dfrac{P-\mathsf{E}[U]}{\rho} + P\approx \dfrac{P-\mathsf{E}[X]}{\rho} + P$ +14836,$X_i=q(p_i)$ +14837,$2^n$ +14838,"$1,000. The member company should not get $" +14839,$\mathsf{E}[X\mid X > a]$ +14840,$+ \mathit{PV}_{r_f}(\text{Inv Inc tax})$ +14841,$\tilde X$ +14842,"$[0,1]\times [0,1]$" +14843,$-2< -\alpha<-1$ +14844,$t>\sup(X)$ +14845,$\mathsf{E}[X\_2]$ +14846,$\rho_1(X)(u_1)=\rho(X)$ +14847,$\mathsf{Pr}(B)=1$ +14848,$G$ +14849,$Z=d\mathbb{Q}/d\mathbb{P}$ +14850,"$(fun1a.south -| fun2a.east)+(\smlspc,-\smlspc)$" +14851,$\sum t_i=\infty$ +14852,$P(t)=(1-t)R_0(t)+tR_1(t)>tR_1(t)>P(1)$ +14853,$\mathcal{A}_y\supset \mathcal{A}$ +14854,"$r1,0(s)$" +14855,$\int_0^1 f(p)dp = 1 - \alpha < 1$ +14856,$\mathsf{E}[z^T]$ +14857,$x_1+x_2=x$ +14858,$\mathsf{var}(N)=\mathsf{E}(N)$ +14859,"$\mathsf E_g|\mathcal W(g, W)|$" +14860,$a = M(a)+Q(a)= \mathsf{E}(X\wedge a) + \delta N(a) + \nu N(a)$ +14861,$\rho_1(X) > X_1$ +14862,$e^{tX}=\sum (tX)^n/n!$ +14863,$D\rho(X_0)=\{Z \}$ +14864,$\mathsf{Pr}r(X>0)$ +14865,$g(s_I)$ +14866,$0\le t<\beta$ +14867,$\mathbf {X'p}$ +14868,$-\infty+\lambda=-\infty$ +14869,$\int udv = uv - \int vdu$ +14870,$\pi = \lambda \displaystyle\frac{\partial \alpha}{\partial v }$ +14871,$4.184×10^9$ +14872,"$\bar P_i(v_1, v_2, a) / v_i$" +14873,$\mathbf {\Sigma}$ +14874,$\mathbf {X'\Delta S}$ +14875,$X-V$ +14876,$tX_i$ +14877,$\alpha<2$ +14878,$vEL$ +14879,"$X_{t+2,1}$" +14880,$y-\kappa'(\theta)=0$ +14881,"$\|\{\alpha_0, \alpha_{01},\alpha_{02}, \alpha_{1}, \alpha_{11}, \alpha_{12}\}\| = 6$" +14882,$\mathsf{E}(X) = E(X_i \mid X\le a)F(a) + E(X_i \mid X > a)S(a)$ +14883,$\alpha>1$ +14884,$P=\rho(X \wedge a)$ +14885,$C_p = \frac{7}{2}R$ +14886,$N\times 1$ +14887,$0\ < p < 1$ +14888,$X=X_i + (X-X_i)$ +14889,$\alpha=\text{E}[X \mid X > F_u^{-1}(p)]$ +14890,"$(B.north east) + (-0.07mm,0)$" +14891,$g'(s-)\ge 0$ +14892,$S(a)=1-p$ +14893,"$139,466.21) = \$" +14894,$\exp(\mu t)$ +14895,"$(\Omega, \mathscr{G})$" +14896,$\mathsf{E}(X)=0$ +14897,$g \ge g^o_h$ +14898,$x \!\!\urcorner$ +14899,$g(s)/(1-g(s))$ +14900,$\mathsf{Q}_1$ +14901,$X=X' + X''$ +14902,$D_i$ +14903,$X(\mathbf{x}) = \sum_i x_iX_i(1)$ +14904,$\nu p$ +14905,$P(\omega)$ +14906,$g_k(s) = 1-(1-s)^k$ +14907,$s_0=0$ +14908,$\|Z\|_p = \mathsf{E}[| Z|^p]^{1/p}$ +14909,"$k \in \{0,\dots,m\}$" +14910,$g(s) = s^r$ +14911,$P_i+Q_i$ +14912,$\{n_s\}$ +14913,$\mathsf{E}_{\mathsf{Q}}[(X - a)^+] = \rho((X - a)^+)$ +14914,$\alpha_i(k)$ +14915,$F^{-1}(1-g^{-1}(1-p))$ +14916,$0\le \lambda_1 \le 1$ +14917,$G_i$ +14918,$d=2$ +14919,$n^{-1}\mathbf M^t\mathbf M=\mathbf E\mathbf E^t$ +14920,$\{X = x\}$ +14921,$\int c(x)dx=\infty$ +14922,$p=0.9973$ +14923,$L_\infty$ +14924,$p>2$ +14925,$1_{U0$ +14992,$d_i=d$ +14993,$\mathsf{TCE}_{p_j}=\mathsf{E}[X \mid X \ge x_j]$ +14994,$w_{0.75}$ +14995,$\mathsf{Z}=\mathsf{Y}\mathsf{C}$ +14996,$E_1$ +14997,$\mathsf{Pr}(T=n)$ +14998,$PV = nRT$ +14999,$X_2=X$ +15000,$Z_{a}(x)=g(S_X(a))/S_X(a))$ +15001,$X_1+X_2$ +15002,$\sigma=2.70$ +15003,$p^y := 1/N$ +15004,$w$ +15005,$g(t) = h_R + \phi_t(t-s_R)$ +15006,$Z^*$ +15007,"$Y_{2,1}$" +15008,$V(0)=1$ +15009,"$(1+t)(1), (1+t)(2),\dots,(1+t)(10)$" +15010,$\mathscr{F}_1:=\sigma(I_1)$ +15011,$\mathsf{E}(W \mid X\ge a)$ +15012,$1-2^{n-1}$ +15013,$0\le a-L_0^a\le a$ +15014,$F^{\times}_{359}$ +15015,$S\not=xf$ +15016,$g(x) = x^\alpha$ +15017,$m_1=m+b$ +15018,$\sigma(X)^2$ +15019,$C = 2.46$ +15020,$q(\hat p)=q(1-g^{-1}(1-p))$ +15021,$N=N(\bar x)$ +15022,$X \le Y$ +15023,"$\{1+\lambda(\zeta-\mathsf{E}\zeta) \mid \zeta\ge 0, \|\zeta\|_q\le 1 \}$" +15024,$F_Y(t)> F_X(t)$ +15025,"$R = 8.314 \, \text{J/(mol·K)}$" +15026,"$P(N, ω) = 1$" +15027,$\kappa_i(x)/x$ +15028,"$1,2,\dots, m$" +15029,$r_0$ +15030,$d=0.111$ +15031,${}^{[<340]}$ +15032,$x \mapsto -x$ +15033,$P^a$ +15034,$1 \le l \le n-1$ +15035,$kP$ +15036,$:=a$ +15037,$p_s=\mathsf{Pr}hi((\hat\theta_s-0.5)/\hat\sigma_s)>0.5$ +15038,"$\mathcal{W}(g, W)$" +15039,$q(U_X) > m$ +15040,$x < x^\ast$ +15041,${}_tq_x=1-{{}_tp_x}$ +15042,"$\displaystyle\int H\xi\,d\mathsf{Pr}$" +15043,$c=$ +15044,$\mu_{t+1}=\mu_t$ +15045,$\mathsf{P}(\{n\})>0$ +15046,$9/6$ +15047,$(X-a)^+=X-a$ +15048,$\rho(X_1) \ge P_1$ +15049,"$\int_0^\infty -z(x)\,dF(x)=-1$" +15050,$Y=e^{-X}$ +15051,"$\{X_k^1,\dots,X_k^m\}$" +15052,$F(x_0)= p_+>p_0$ +15053,$\mathsf{E}_{\mathsf{Q}}[X\wedge a] \le \rho(X\wedge a)$ +15054,$k_4$ +15055,$\mu(n)$ +15056,"$\langle \zeta_{\bar x}, X_i \rangle$" +15057,$a \ge a'$ +15058,$p+dp$ +15059,$L_1$ +15060,$X=40$ +15061,$\mathsf{E}(X_i(a)) = E(X_i \mid X\le a)F(a) + aE(X_i/X \mid X> a)S(a)$ +15062,$X(x)=\sum_i x_iX_i$ +15063,$\{2\}$ +15064,$\log(S) =\mu t$ +15065,$g(s)=\dfrac{r}{1+r}+\dfrac{s}{1+r}=d+sv$ +15066,$11$ +15067,$\log_2\ge 1$ +15068,$\phi_R-\phi_{m}^o$ +15069,$\mu\left(1+\dfrac{m}{p}\right)^2$ +15070,$\rho(Y)=\rho_m(Y)\le b_Y$ +15071,"$(\mathsf{Pr},\sigma(T))$" +15072,$\hat\rho$ +15073,$b < 1$ +15074,$\mathsf{E}_\mathsf{Q}[X\mid A]$ +15075,$d=1$ +15076,$(T\lambda)g=\mu g$ +15077,$\mathsf{Pr}(X_n=0)=1-1/n$ +15078,$1=S(x)+F(x)$ +15079,$\mathsf{E}[\phi] = 1$ +15080,$\mathsf{E}[X]=1$ +15081,$x\le y$ +15082,$F(X) - F(X-)$ +15083,$A=\{X>x\}$ +15084,$\lambda(F(x_2)-F(x_1))$ +15085,$S^i= \sum_{j=0}^a p_j \kappa_i(j)$ +15086,$P(X\wedge a)=\bar P(a)$ +15087,$0 < b \le 1$ +15088,$\iota=0.15$ +15089,$\epsilon\downarrow 0$ +15090,$Q(x)=1-P(x)$ +15091,"$[0,1)$" +15092,$X_2 =$ +15093,$\bar F(a)$ +15094,"$\rho_h(s,t)$" +15095,$\mathbf{x}$ +15096,$2 \times 10^{19}$ +15097,"$u_1, u_2>0$" +15098,"$Y=\{y_1,\dots,y_n\}$" +15099,$Z\not=0$ +15100,$(\lambda S(x))$ +15101,$a-Y$ +15102,$\mathsf{E}(X\mid X > a) = (\mathsf{E}(X)-\mathsf{E}(X\mid X \le a)F(a))/S(a)$ +15103,$\sum_{i}X_{i} = X$ +15104,$\Theta_i^X$ +15105,$\mathsf{Pr}hi(\mathsf{Pr}hi^{-1}(s)+\lambda)$ +15106,$g''$ +15107,$\theta=\log(p)<0$ +15108,$\rho(X_n)=0$ +15109,$u\mapsto \mathsf{E}[X_i/u\mid X(t)=u]$ +15110,$g(s) = \mathsf{Pr}hi(\mathsf{Pr}hi^{-1}(x) + \lambda)$ +15111,$\mathsf{Pr}r(X=\mathsf{VaR}_p(X))=0$ +15112,$\mathcal Q(X)$ +15113,"$\omega=(1,0,0,1,0,0,\dots)$" +15114,$c_1=c_2=0$ +15115,$y\not=z$ +15116,"$m_{p_i, p_j}$" +15117,"$\nu p\,da$" +15118,$p_j=\mathsf{P}(X=x_j)$ +15119,$g(t)=O(d)$ +15120,$\sqrt{F(x)S(x)}\approx \sqrt{S(x)}$ +15121,$\alpha=1.5952$ +15122,"$\rho=\mathrm{ess\,sup}=\mathsf{TVaR}_1$" +15123,$g=3$ +15124,$Q\circ T\in\mathcal{Q}$ +15125,$10^{19}$ +15126,"$\mu+h\sigma, \sigma$" +15127,$g^*$ +15128,$(100/(100+x))^{1.3}$ +15129,$\sqrt{dt}$ +15130,$X_1+\dots+X_n$ +15131,$r \ge i$ +15132,$a_1 < a_0-X_1$ +15133,"$[p, d+dp]$" +15134,$\rho_g(X)=\int xg'(S(x))f(x)dx$ +15135,$X_t=t-G_t$ +15136,$I=\iota_i$ +15137,$(X_N-a)^-$ +15138,"$n=(0.702, 1.163)$" +15139,$1+2c(Z-\tau)$ +15140,$\epsilon\mathsf{E}_q(X_1)$ +15141,$s^{-1}(\cdot)$ +15142,$\eta_i\ge 0$ +15143,$T^2$ +15144,$E[X \wedge x+a]-E[X \wedge a]$ +15145,$\lambda\mathsf{E}[X]$ +15146,$(x+b)$ +15147,$\mathsf{E}[X_iZ]=\rho_g(X)/2$ +15148,$s>0.1$ +15149,$\mathsf{Pr}_S(A\mid T)$ +15150,$B(b)$ +15151,$\mathsf{VaR}_{0.85}(X)=389$ +15152,$x\in\mathbb{R}^n$ +15153,$\mu=0$ +15154,$\mathit{MV}_{ro}(a) = a-P(X_{-1}\wedge a)$ +15155,$u_i \ge 0$ +15156,$1 \times 10^{23}$ +15157,$g(s)=d + vs$ +15158,$e^x-1$ +15159,$\psi^{-1}(p)$ +15160,$H(X) > -H(-Y)$ +15161,$1=\nu+\delta$ +15162,$\mathsf{E}_G(X)$ +15163,$t=0.55$ +15164,$\mu \cdot T_k$ +15165,$B(1_{X\le x})$ +15166,$\iff \rho$ +15167,"$\sigma=0.25, 0.375, 0.5$" +15168,$p+q=1=\nu+\delta$ +15169,"$X_i, Y$" +15170,$\epsilon^A$ +15171,$\Delta X_j^{Net}=0 < \Delta X_j^{Gross}$ +15172,$t>t^*$ +15173,$A/(A-P)$ +15174,$b^2 \mu_x /2$ +15175,$Z\in D\rho(X_0)$ +15176,$Z\circ T_X$ +15177,$B\subset Y$ +15178,$20+10t$ +15179,$g^mA^R=g^m(g^a)^R=g^{m+Ra}$ +15180,$\lambda \uparrow 1$ +15181,$\Delta_1=a_1'-a_1$ +15182,$s=S(x)=\mathsf{Pr}r(X>x)$ +15183,$f=1$ +15184,$E_{\mathcal{B}bb{Q}}[X] := E[Xg'(S(X))]$ +15185,$r_f = 0.02$ +15186,$g(s)=s^{1/\rho}$ +15187,$\beta_i/\alpha_i$ +15188,$\beta=d^\ast-d$ +15189,$\theta\to 0$ +15190,$2^{20}$ +15191,$-\theta - \lambda(e^{-\theta} - 1)) = y$ +15192,$\mathsf{Pr}r(\{\omega_1\})=1/3$ +15193,$\forall a\forall b\exists x[a\in x \wedge b\in x]$ +15194,$= 1 - T_\mathrm{sink} / T_\mathrm{source}$ +15195,$\phi F$ +15196,"$a,0\le a\le\infty$" +15197,"$\int_0^s g'(t)\,dt=\nu s$" +15198,$5.97 \times 10^{24}$ +15199,"$[\alpha_\epsilon,1]$" +15200,$V(\mu)=1/(\tau^{-1})'(\mu)=\mu^3$ +15201,$-(1-s)g''(1-s) + g(0+)\delta_1 + \sum_s s\Delta_s \delta_{1-s} + g'(1)\delta_0$ +15202,$g''(s)=-\phi'(1-s)\le 0$ +15203,$\alpha \mu(U_n) \le \mu(E\cap U_n)$ +15204,$s_l < s < s_u$ +15205,$p=1-g^{-1}(1-p)$ +15206,$\not =$ +15207,$\text{E}[X_i \mid X]$ +15208,$x\mapsto 1/x$ +15209,$B\in\mathscr{F}$ +15210,$\mathcal{P}$ +15211,$\bar P_g(a)=\rho(X\wedge a)$ +15212,$\rho(X)=\int_0^\infty g(S(x))dx$ +15213,$q_{\mathbf{x}}$ +15214,"ho=0.5, x=3, M=1.5,\sigma=0.85, K=8$" +15215,$\sigma=0.1246$ +15216,$1-p=S(s)$ +15217,$\mathsf{TVaR}_{p_1}(X)$ +15218,"$x=1000,2000,\ldots$" +15219,$\mathsf{E}_\mathsf{Q}[X1_A] / \mathsf{E}_\mathsf{Q}[1_A]$ +15220,$\mathsf{VaR}_1(X)$ +15221,$LR_{\mathsf{TVaR}}$ +15222,"$(x_{2,1}, x_{2,2})$" +15223,$(1-p)/p=1$ +15224,$p\to\infty$ +15225,$\nu_k$ +15226,"$\mathcal{Z}=\{ Z\in L^\infty\mid \mathsf{E}[Z]=0, \mathsf{E}[Z^2]\le 1 \}$" +15227,$k(s)$ +15228,$\bar\delta(a)$ +15229,$\mathsf{Pr}(L=l)$ +15230,$V(m)=m(m+a)$ +15231,$p_0 = 1 - p_1 - p_2- p_3$ +15232,$\mathsf{E}_{\mathsf{Q}}[X]$ +15233,$V(U)$ +15234,$(\bar a_x - \bar a_{b\!\urcorner})/\bar a_x$ +15235,$\sech$ +15236,"$\mathsf{biTVaR}_{p_i,p_j}^{w_{i,j}}(X) = c$" +15237,"$936,594,646 | 26.1% | $" +15238,$0\le p_0\le p^*\le p_1\le 1$ +15239,"$\beta, \kappa$" +15240,$a(f + (1-f)/q) -1$ +15241,$\mathbf {Z_\mathit{lift}}$ +15242,$\tilde X_1 = X_1 + \mathsf{E}[X_2]$ +15243,"$(\Omega, \mathcal F, \mathsf{P})$" +15244,$d/v=r$ +15245,$u_0={ef(K)}$ +15246,$g(1-S_k)$ +15247,"$[0, t_1]$" +15248,$g(S(M-))/S(M-)$ +15249,$t\to 0$ +15250,$\mathsf P^g X$ +15251,$\rho=\mathsf{VaR}_p$ +15252,$X\wedge a'$ +15253,$\mu\{ l>1\}=0$ +15254,"$R,S$" +15255,$0\le \lambda < 1$ +15256,$D-N = \sum_{i\in I} (D_i-N_i) - N_a$ +15257,$\bar P(x)=\int_0^x P(t)dt$ +15258,$\mathsf{Pr}r(X < x)\le 0.99 \le \mathsf{Pr}r(X\le x)$ +15259,$s>0$ +15260,"$(X_k, X_{k+1}]$" +15261,"$X:\Omega\to [0,\infty]$" +15262,$1-\tilde p=g(S(s)$ +15263,$y\in A$ +15264,$n=30$ +15265,$0\le x < X_1$ +15266,$(p_0 < p^\ast < p_1)$ +15267,$f(X)$ +15268,$\rho(X)\le \liminf \rho(X_n)$ +15269,$gS_t$ +15270,$\rho(X) = \mathsf{E}(X) + c\| X-\mathsf{E}(X) \|_p$ +15271,$g'(s)=bs^{b-1}$ +15272,$\mathsf{VaR}_p(X_1+X_2)\le \mathsf{VaR}_p(X_1)+\mathsf{VaR}_p(X_2)$ +15273,$\mathsf{Pr}(T\in A\mid\mathscr{G})(\cdot)$ +15274,$a+da$ +15275,"$\mathsf{Tw}_0(\mu,\sigma^2)$" +15276,$G_0$ +15277,$\hat\theta_s$ +15278,$\mathsf{E}[X] = E(X_i \mid X\le a)F(a) + \mathsf{E}[X_i \mid X > a]S(a)$ +15279,$\iota=\delta/\nu$ +15280,$s(X'X)^{-1/2}_{jj}$ +15281,$X'=\mathsf{E}[X\mid A]$ +15282,$Y_i=\partial Y/\partial x_i$ +15283,$\mu=\lambda/ \psi$ +15284,$X\circ\tau$ +15285,$M'(0)$ +15286,$Z=g'(S_X(X))$ +15287,$a(X_i;X)\ge \mathsf{E}[X_i]$ +15288,$g(s)=a^\alpha$ +15289,$\kappa'(\theta)=\theta=\mu$ +15290,$X^{-1}(A)\in\mathcal F$ +15291,$\lambda^Q_t = \lambda^Q\mu_t$ +15292,$w < s$ +15293,"$g(s)=\min(s/(1-p),1)$" +15294,"$(-\infty, \infty)$" +15295,$m_X(s)\to\mathsf E[X] + k$ +15296,$15$ +15297,$\bar P'$ +15298,$\mathsf{Pr}r(X>x)$ +15299,"$\int_0^a \alpha_2(x)F(x)\, dx$" +15300,$\hat{s}$ +15301,$\mathsf{E}(X \mid X\le a)$ +15302,$Z(\omega)>1$ +15303,$p_0=1$ +15304,$Z_X$ +15305,"$i=1,\dots,n$" +15306,$A(0)=0$ +15307,$Z(X(\omega))$ +15308,$U(1)=2$ +15309,$\iota = \dfrac{\delta}{1-\delta}$ +15310,$Z=Y-X$ +15311,$\rho_m(X)=\rho(X)$ +15312,$g(pq)=g(p)g(q)$ +15313,"$\rho(X)\le \mathrm{ess\,sup}(X)$" +15314,$\{\omega\mid X(\omega) = x_1\}$ +15315,"$Y_{0,2}$" +15316,$X_t=(1-t)X_0 +tX_1$ +15317,$t^a$ +15318,"$\mathsf{Pr}(B\cap A)=\int_B \mathsf{Pr}(A\mid\mathscr{G})\,d\mathsf{Pr}$" +15319,"$200,000 are \$" +15320,$\nu^*$ +15321,$\mathsf{E}[X_i]=14$ +15322,$\sigma_1=\cos(\pi\theta/2)$ +15323,$\hat S(x_i) = (i-1)/N$ +15324,$c=\lambda\mathsf{E}[Y]$ +15325,$Y_{1}$ +15326,$s = 1/M$ +15327,$10^{-1}$ +15328,$ is accumulated profit from an inflow of premium 1 per unit time and a cumulative claims process $ +15329,$L_2(\Omega)$ +15330,$g(0+)M$ +15331,"$M:=\mathrm{ess\,sup} X$" +15332,$X\le Y+\Vert X-Y\Vert$ +15333,"$(i,j)$" +15334,"$, S&P applies the premium risk to $" +15335,$\mathsf{E}_\mathsf{Q}(X_i \mid X=x)=\mathsf{E}(X_i \mid X=x)$ +15336,$F^{-1}(1-s)$ +15337,$\frac{1}{4}(1 + 8\mu-\sqrt{1+8\mu})$ +15338,${}^{[>35]}$ +15339,$A/L$ +15340,$X_n\downarrow X$ +15341,$\rho(X_1\mid \mathscr F_1)\le \rho(X_2\mid \mathscr F_1)$ +15342,"$50,057,600 | 30.7% | $" +15343,"$X_i\sim \mathsf{Tw}_p(\mu, \sigma/w_i)$" +15344,$\mathcal D$ +15345,$X \le A$ +15346,$\mathsf{E}(X\wedge k)$ +15347,$A_2$ +15348,$p^* =0.7501$ +15349,$0.495(r-i)$ +15350,$\sigma=0.225$ +15351,$\H$ +15352,$1 million punitive award is grossly excessive and unconstitutional in a case where Dr. Mann had only $ +15353,$\mathsf{LI}\iff\mathsf{SSD}$ +15354,$1-1_{X>a}=1_{X\le a}$ +15355,$\mathsf{E}[X_i \wedge a]$ +15356,$\mathsf{E}[kX]=k\mathsf{E}[X]$ +15357,"$(s_{i+1}, g(s_{i+1}))$" +15358,$\mu = \delta_{\alpha}$ +15359,$(1+\theta)\rho$ +15360,$EL=\mathsf E[X\wedge a]$ +15361,"$\{1+\lambda(f-\mathsf{E} f) \mid f\ge 0, \|f\|_q\le 1 \}$" +15362,"$\psi(S,T)$" +15363,$500 | \$ +15364,$p_j=\Delta S_j$ +15365,$\succeq$ +15366,$\mathcal{Q}=\{Q\mid \alpha(Q)<\infty\}$ +15367,$X=X(\mathbf{x})$ +15368,$\lambda=5$ +15369,$Y_n=-X_n$ +15370,$n\to \infty$ +15371,$\{3\}$ +15372,$\mu_X\ge r_f + a\sigma_X$ +15373,$P$ +15374,$X_1 + \cdots + X_N$ +15375,$c = 0.5(0.5)2.5$ +15376,"$2,000,000 and \$" +15377,$\beta_i(x)<\alpha_i(x)$ +15378,$G(x)=\mathsf{Q}(\{\omega\mid X(\omega)\le x\})$ +15379,$\tilde\Delta(s)$ +15380,"$Y=\max(X_1, \dots, X_N)$" +15381,$α$ +15382,$b-X\ge 0$ +15383,$\bar P_{1}$ +15384,$0\le\lambda \le 1$ +15385,$\forall X[\forall x\in X(x\not=\emptyset) \wedge \forall x\in X\forall y\in X(x=y\vee x\cap y=\emptyset)]\rightarrow\exists S\forall x\in X\exists !z(z\in S\wedge z\in x)$ +15386,$X^y := \sum_{i=1}^{n^y} X_i^y$ +15387,$\rho(X_0+\epsilon Y)=\mathsf{E}[(X_0+\epsilon Y)Z_\epsilon ]$ +15388,$\omega=\exp(2\pi i / n)$ +15389,$P_k+Q_k=1$ +15390,$X_{1}(a)$ +15391,$\bar S_i(3463)$ +15392,$p=100043$ +15393,$\alpha_1(98)=0.1$ +15394,$b$ +15395,$tX$ +15396,$\mathsf E_q[X]$ +15397,$\mu_d = (6-d)^2$ +15398,$A^c$ +15399,$\mathcal{G}\subset\mathscr{F}F$ +15400,$a=18000.0$ +15401,$\bar \iota$ +15402,$\rho(G) = \mathsf{E}_Q(G)$ +15403,$s/(1-p) \wedge 1$ +15404,$Q_k\ge \Delta X_k - P_k$ +15405,$\mathsf{E}[X_x] = 0$ +15406,$\eta$ +15407,$\Omega$ +15408,$f_X\sim cf_Y$ +15409,$a(X)\equiv a$ +15410,$Z'(s)=1/(\mathsf{Pr}hi'(Z(s)))=\sqrt{2\pi}\exp(Z(s)^2/2)$ +15411,$\sup_{\mu\in M} \int CTEd\mu$ +15412,$E_Q(X_i(a)\mid X)$ +15413,$s_I=(s-\omega_I)/(1-\omega_I)$ +15414,$1/(1-p) > 1$ +15415,$\check S$ +15416,$t_i>0$ +15417,$E[\tau]$ +15418,$p_k=Pr(X=k)$ +15419,$A_t = \mathsf E[X\mid\mathscr F_t]$ +15420,$\rho(aX+bY) = a\rho(X) + b\rho(Y)$ +15421,$y_{3}=7$ +15422,$x=\max(X)$ +15423,$\mathsf{E}(G)=a\theta$ +15424,$\mathsf{Var}(\pi)$ +15425,$\mathsf{FAT}'$ +15426,$\mathsf{Pr}r(X>\mathsf{VaR}_p(X))>1-p$ +15427,$\{1\}$ +15428,$y^+$ +15429,$\mathsf{Pr}hi(x):=\int_{-\infty}^x \phi(t)dt$ +15430,$t_* \le t \le t^*$ +15431,$\mathsf E[Y_i\mid S]$ +15432,$n=1$ +15433,"$S(x)=d/dx(\mathsf{E}[I_{(0, x]}(X)])$" +15434,$X=c$ +15435,$a=\sum_i a\alpha_i(a) = \sum_i\kappa_i(a)$ +15436,$\Delta X_j=X_{j+1}-X_j$ +15437,$\beta>\alpha$ +15438,$. Call $ +15439,$\epsilon x_1$ +15440,$\mathsf{M}$ +15441,$\sum_i \kappa_i'(x)=1$ +15442,$r_p$ +15443,$X^i$ +15444,$C=\mathrm{conv}(S)$ +15445,$\mathsf{Pr}(X_n=Y)=\mathsf{Pr}(X=Y)=0$ +15446,$\inf_x \{ x + \alpha\mathsf{E}[(X-x)^+] + \beta\mathsf{E}[(X-x)^-] \}$ +15447,$c(y;\lambda)$ +15448,$p\uparrow 2$ +15449,$X+W$ +15450,$\mathsf{Pr}(A\mid Y=y)=\mathsf{E}[1_A\mid Y=y]$ +15451,$x_{i-1}$ +15452,$a=a_0+a_1$ +15453,$\displaystyle\int_0^a S(x)dx$ +15454,$\mu=\tau(\theta)=ne^\theta/(1+e^\theta)=np$ +15455,$\rho(nX)= \rho(X+\cdots + X)=\rho(X)+\cdots +\rho(X)=n\rho(X)$ +15456,$n\mathsf{Pr}r(Y\le y_c)$ +15457,$\rho=P/L-1=M/L$ +15458,$(1-p)$ +15459,$x=1000$ +15460,$\tau$ +15461,$\sum u_iX_i$ +15462,$S(X)$ +15463,"$i_U \in \{i^\star,\dots,m \}$" +15464,$\mathsf{E}[X_i / X]$ +15465,$\mathsf{E}[X_2\mid X=x]$ +15466,"$(\Omega, \mathscr{F}, \mathsf{Pr})$" +15467,$\Delta_R=0$ +15468,$\check g(uv)=0 < \check g(u)\check g(v)$ +15469,$\rho^e$ +15470,$s_0=1$ +15471,$g'S_t=g'(S_t(X_t))$ +15472,$kX$ +15473,$\alpha=1.5$ +15474,$\int (r-\mu)f=0$ +15475,$\tau=\kappa'$ +15476,$P = \mathsf{VaR}_\pi(X)$ +15477,$\phi_i$ +15478,"$(B.south east) + (-0.07mm,0)$" +15479,$+\frac{1}{2}$ +15480,$0\le a\le 2^{256}$ +15481,$\mathsf{E}(G)=M_G'(0)=1$ +15482,"$\{\omega\mid P(\mathscr{G}amma\cap Z, \omega) = \frac{1}{2}1_\mathscr{G}amma(\omega) \ \forall\mathscr{G}amma\in\mathscr I\})$" +15483,"$\mathsf{biTVaR}_{p_0,p_1}^w(X)=\bar P$" +15484,$-1<\alpha<0$ +15485,$\gamma_a(x)$ +15486,$\mathsf{E}[ X_i \mid X(x) = q_{x}(p)]$ +15487,$x^n=e^{n\log(x)}$ +15488,$\kappa(\theta(\mu))=n\log(n/(n+\mu))$ +15489,$X\wedge l$ +15490,$\bar q_{X_1+X_2}(s) \ge \bar q(s/2)$ +15491,"$P(A, \mathscr{F})$" +15492,$\tilde Z=Z$ +15493,$2\mathsf{VaR}(X)$ +15494,$g = s/(1-f)$ +15495,$s_1=s_L$ +15496,"$\phantom{P}= \displaystyle\frac{1}{1+r}\,\mathrm{EL} + \displaystyle\frac{r}{1+r}\,a$" +15497,$dt=g'(1-s)ds$ +15498,$1 \times 10^{17}$ +15499,$1_A/\mathsf{Pr}r(A)$ +15500,$\mathcal Q(X)=\{ \mathsf Q\in\mathcal Q\mid \rho(X)=\mathsf{E}_\mathsf{Q}[X] \}$ +15501,$\times T=\rho T$ +15502,$S(x)=u$ +15503,$D = L^* - L$ +15504,"$\bar x\mapsto G\circ F(\bar x, \omega)$" +15505,$g(t)$ +15506,$0\le\theta_s\le 1$ +15507,$P_1(1)=P(1)$ +15508,$5 \times 10^5$ +15509,$6$ +15510,$\lim_{t\to 0} R_1(t)$ +15511,$p=2$ +15512,"$\mathcal{P}_{X,r}=\mathsf{var}nothing$" +15513,$p'_i$ +15514,$L_{250}^{\infty}(x)$ +15515,$X=X_1 + X_2$ +15516,$(1-r_0)δ_1$ +15517,$10^{-3}$ +15518,$T_\nu$ +15519,"$[0,1]\times[0,1]$" +15520,$g(p)=p^{1/2}$ +15521,"$F(\cdot, F)=\mathsf{Pr}(T\le x\mid \mathscr{G})(\omega)$" +15522,"$g(s)= \displaystyle\int_0^s g'(t)\,dt = (s/(1-p)) \wedge 1$" +15523,$X=q(U)$ +15524,$=\mathsf{E}[X \mid X\le a]$ +15525,$k>2$ +15526,$H:\Omega\to\mathbb{R}$ +15527,$S_T=S_0X_T$ +15528,$h_j(x) \le 0$ +15529,$y_1 < y_2$ +15530,$1-\bar\zeta_t^2$ +15531,$v=1-d$ +15532,$p=e^\theta$ +15533,$\alpha_X > 2$ +15534,$g^{ak}=(g^k)^a=K^a$ +15535,$X > k+1$ +15536,"$(-1,1)$" +15537,$x\to 0$ +15538,$16$ +15539,"$C_2(t) < \bar P^a(0, 1)$" +15540,$\lambda<1$ +15541,$g^u$ +15542,$1 = p(a) + (1-p(a))$ +15543,$ the rank of $ +15544,$s > s^*$ +15545,$tP(1)$ +15546,$v_s$ +15547,$A_k=A_0 + kN \ge A_0 + k'N = A_{k'}$ +15548,$\sup_\Omega |X_n - X| \to 0$ +15549,$\mathsf{TVaR}_{p^*}=C$ +15550,"$\{\rho^E(t_1,t_2) : 0 \le t_1 \le s_L, s_R \le t_2 \le 1\}$" +15551,$(dX_t)^2$ +15552,$\Theta_{i^\star+1}^X < c \le \Theta_{i^\star}^X$ +15553,$n=2^{\log_2}$ +15554,$1-1/c$ +15555,$-g'(1-p)<0$ +15556,$E_\mathsf{Q}[1]$ +15557,$v^{n-2}$ +15558,$X\wedge a=X$ +15559,$AR(2)$ +15560,$P=D=L/(1+R_L)$ +15561,$\beta_0+\beta_1+\beta_2$ +15562,$\rho_{\mathsf{Wang}}$ +15563,"$w_{i,j}$" +15564,$\mathsf{TVaR}(0)$ +15565,$F\subset E\in \H\implies F\in \H$ +15566,$0 < s < 1/4$ +15567,$c(k)=\binom{n+k-1}{k}$ +15568,$1-g(S(t))$ +15569,$P=\exp(R)$ +15570,$\{ \mathsf{E}[X] \}$ +15571,$\mathsf{E}[X] = \mathsf{E}[X\mid X \le a]F(a) + \mathsf{E}[X\mid X > a]S(a)$ +15572,$\cos(iz)=\cosh(z)$ +15573,$a_{gc}:=\mathit{VaR}_{p}(X)={{a_x}}$ +15574,$e^{\theta X_t -t\kappa_X(\theta)}$ +15575,$\kappa(\theta) = \int \kappa'(t)dt = \int m/V(m) dm$ +15576,"$(\nu,\delta)$" +15577,$\mathsf{Var}(\mathsf{Pr}i)$ +15578,$1 - \mathsf{Pr}r(Z>\mathsf{E} Z)$ +15579,$1-g^{-1}(U)$ +15580,"$\rho_g(X)=\int_0^\infty g(S(x))\,dx$" +15581,"$X\wedge a'=\min(X, a')$" +15582,$m>0$ +15583,$q=\Delta g(S)$ +15584,"$\mathsf{E}[W]=\sum_{d\ge 0} \mathsf{E}[Y_{-d,d}]$" +15585,$f(x)=1$ +15586,$\rho(X)=\displaystyle\int_0^1 q(p) \phi(p) dp$ +15587,$F(2)=0.75$ +15588,$\mathsf{E}[Z\mid X]=Z$ +15589,$\mathsf v$ +15590,$x_{i1}$ +15591,$p^*={pstar:.3f}$ +15592,$x_i / \sum_i x_i$ +15593,"$(\Omega, \mathcal F, \mathsf{Pr}r)$" +15594,"$b,-2b,b$" +15595,$r=d/(1-d)$ +15596,$q=g'(S(x))f(x)$ +15597,$x\mapsto x^{n}$ +15598,$p(1-p)/(v-l)^2$ +15599,$q(U_X) = m$ +15600,"$X\le \mathrm{ess\,sup}(X)$" +15601,$\mathsf{TCE} < \mathsf{TVaR}$ +15602,$B=\Omega$ +15603,$S\_{Total}(a)$ +15604,$\rho(0)=\rho(0.X)=0\rho(X)=0$ +15605,$Q=\Delta X-P$ +15606,$\lambda_t$ +15607,$iv$ +15608,$\nu=\mu_X-\mu_Y$ +15609,"$\mathcal W(g, W)$" +15610,$X_\infty(\omega)$ +15611,"$[-10, 10)$" +15612,$E[\tau] = 2$ +15613,$\H(R)$ +15614,$kρ(X)$ +15615,$\rho_t(X) \ge \mathsf E[\rho_{t+1}(X)\mid \mathscr F_t]$ +15616,$r_l$ +15617,$w_0^y := \mu(n^y)/\nu(n^y)$ +15618,$\sum_{m\in\mathbb Z} F((mn + k +1/2)b)-F(mn + k - 1/2)b)$ +15619,$\partial l/\partial \mu$ +15620,"$=\mathsf{E}[\min(X,a)]=\mathsf{E}[X\wedge a]$" +15621,"$\alpha_X\in(\alpha_Y+1/2, \alpha_Y+1)$" +15622,$\rho_g(X)=\bar P$ +15623,"$[\alpha_0,1]$" +15624,$v/i/d$ +15625,$-\inf -X_i \le -\rho(-X_i) \le D\rho_X(X_i) \le \rho(X_i)\le \sup X_i$ +15626,$\mathsf{TVaR}_0(X)=\mathsf E[X]$ +15627,"$(4,1)$" +15628,"$1,353*:*02 + \$" +15629,$a = a(\mathbf{v}) = a(X(\mathbf{v}))$ +15630,$m = n \times \text{molar mass} = 114.6g$ +15631,$p=(1-s)$ +15632,$\rho(X)\ge\limsup \rho(X_n)$ +15633,$\omega = 2\pi / T$ +15634,$1 \times 10^{14}$ +15635,"$x=x(A,L)=A/L$" +15636,$X > a$ +15637,$\mathsf{E}(X_i)$ +15638,$ag(S(a))$ +15639,$S(p)=1-p$ +15640,"$f(u,v)=(1 - w - s_0 )uv - s_0w(u+v) -ws_0^2$" +15641,$-g$ +15642,"$f(X,Y)$" +15643,$> 10^{19}$ +15644,$\mathsf{E} X + c{(X-\mathsf{E} X)^+}_p$ +15645,$F = M v^2 / r$ +15646,$p_n<1$ +15647,$\mathsf{E}_\mathsf{Q}[X_i] = \mathsf{E}_\mathsf{Q}[\mathsf{E}_\mathsf{Q}[X_i \mid X]]$ +15648,$\alpha_{Cat} \le \beta_{Cat}$ +15649,$I = i$ +15650,$\mathsf{Pr}(X_n\in A)=1$ +15651,"$\alpha_i(\mathbf{v}, x)$" +15652,$T(x)$ +15653,$\bar\iota = \dfrac{\bar M(a)}{\bar Q(a)}$ +15654,$X < a$ +15655,$\beta/\alpha < S/gS$ +15656,$Q=1-P$ +15657,$B(X)=-A(-X)$ +15658,"$g(s) = \min(1, s/(1-\alpha))$" +15659,"$F_n,F$" +15660,$x_2$ +15661,$(1-f)$ +15662,$g'\not=0$ +15663,$x_i^2\lambda_i$ +15664,$p=1.6$ +15665,"$3,000,000,000), regardless of the frequency or severity of earthquake losses at any and all times subsequent to the creation of the authority. Once a participating insurer has paid, pursuant to this section, amounts equal to the percentage share of the authority's total gross written premium attributable to that participating insurer's sales of authority insurance policies, as of April 30 of the immediately preceding year or the most recent full year for which premium data not more than one year old are available, multiplied by three billion dollars ($" +15666,$\kappa(\theta)=-n\log(1-e^\theta)$ +15667,$X\le_{st} Y$ +15668,$\Rightarrow$ +15669,$\mathsf{E}(\cdot)$ +15670,$S/P$ +15671,$x_1 \wedge x_2$ +15672,$\lambda = 2$ +15673,$t+dt$ +15674,$p(v_p-l_p)$ +15675,$x_i$ +15676,$\mathscr F_1=\sigma(X)$ +15677,$\mathsf{Pr}_T$ +15678,"$3,464 and \$" +15679,$B\mapsto\mathsf{Pr}(B\mid\mathscr{G})(\omega)$ +15680,$\kappa_i$ +15681,$\succeq^2$ +15682,"$i=1,\dots,m=3$" +15683,$\mathsf{E} X$ +15684,$a=\sum_i a^i$ +15685,"$x=1, M=1.5,\sigma=0.75, K=6$" +15686,$k(t)l(t)<\infty$ +15687,$k \ge k'$ +15688,$0=0$ +15689,"$s^{\ast}=1/2, \lambda^{\ast}=0$" +15690,$g_i=u_i^{1/b} < u_i$ +15691,$\mathsf{E}[\cdot\mid\mathscr{G}]$ +15692,"$\int_0^1 xj(x)\,dx=\infty$" +15693,$(\sum_i \iota^i Q^i)/Q$ +15694,$a={{a_x}}$ +15695,$r_U = (P-U)/K$ +15696,$x\to \infty$ +15697,$\rho^*= (1-\alpha-\beta)^{-1}-1$ +15698,$t < s^\star < s$ +15699,$(1/4)(1/3)=1/12$ +15700,$K''(0) = \kappa''(\theta) = \tau'(\theta)$ +15701,$s-i$ +15702,"$20,000,000 Owners\' equity: \$" +15703,$\mathsf{TVaR}_{0.95}(X)$ +15704,$\{X_t = a(t)\}$ +15705,$X\le a$ +15706,$\{\mathsf{E}_{\mathsf Q}[X_i] \mid \mathsf Q\in\mathcal Q(X)\}$ +15707,$X_2=x-t$ +15708,"$[0,\infty)\subset\mathbb{R}$" +15709,$N_i$ +15710,$E[X_2(a)]$ +15711,$P(X_{-1}(a))$ +15712,$R_f-R_L>0$ +15713,$r_a$ +15714,$\mathsf{E}[W]$ +15715,$c \le 0$ +15716,$(1+\epsilon)\mathbf{X}$ +15717,$\mathbf {F}$ +15718,$p \ge 0.9$ +15719,"$w_{p_1,p_2}$" +15720,$g \circ f$ +15721,"$\mu=0.1, \sigma=0.15$" +15722,$l(t)=\lambda_t \Omega$ +15723,"$s\in[0, 1-p]$" +15724,$X=Y$ +15725,$g(st)\le g(s)g(t)$ +15726,$\mathsf{SD}(X)$ +15727,$\kappa'(\theta)$ +15728,"$X_{i,j}\Delta g(S_j)$" +15729,$X(S(n))$ +15730,"$\mathcal E'_{X,r}\subset \mathcal R'_{X,r}$" +15731,$\hat \theta$ +15732,$ is defined on the dual $ +15733,$u_l=r_l$ +15734,$A\in\mathcal{A}$ +15735,$g'(s)=\nu$ +15736,"$(-\infty, 0)$" +15737,$\mathsf{E}[Y \mid U]$ +15738,$2\left( \log\frac{y}{m} + \frac{y}{m} - 1 \right)$ +15739,"$w_{p_1,p_2}\in [0,1]$" +15740,"$95,622.71 - \$" +15741,$P=\rho_{\mathsf{Wang}}(X)$ +15742,$\mathsf{E}[Z \tilde X]$ +15743,$\sum_\omega Z(\omega)\mathsf{P}(\omega)=\mathsf{E}[Z]$ +15744,$T_L \mathsf{TVaR}_1$ +15745,$\mathsf{Pr}hi(Z(s))=s$ +15746,$\mathsf E_p[ZX]$ +15747,$r_K$ +15748,$X_1+\cdots +X_n$ +15749,$\rho(X^{\oplus n})$ +15750,$s > 1/3$ +15751,$E\in\mathcal F$ +15752,$\phi(p)\ge 0$ +15753,$L^2$ +15754,$t \wedge \tau_n$ +15755,$q(p)$ +15756,$\mathbf X / l(\mathbf X)$ +15757,$X_0=1$ +15758,$\{ x | f(x)\le t \}$ +15759,$t^\star$ +15760,"$x_{1,1}$" +15761,"$(brR15 |- lee.south)+(-0.25,-0.25)$" +15762,$\bar p=(1-p)/p$ +15763,$\mathsf E[X_i|X]$ +15764,$\exp(\beta X_t-\lambda t)$ +15765,$R_2$ +15766,"$w_{p^*,p^*}=1$" +15767,$t<0.5$ +15768,$y^{p-1}e^{-y} / \mathscr{G}amma(p)$ +15769,$\beta_1$ +15770,$\mathsf{E}(A) = \mathsf{E}(N) \times \mathsf{E}(X) = \dfrac{\text{num events}}{\text{num years}} \times \dfrac{\text{sum losses}}{\text{num events}} = \dfrac{\text{sum losses}}{\text{num years}}$ +15771,$A\subseteq \mathbb{R}^n$ +15772,$\mathcal{N}_{X\wedge a}(X_i(a))$ +15773,$\mathbf {\Delta_{R+1}}$ +15774,"$(s,g(s))=({s0:.3g}, {gs0:.3g})$" +15775,$\delta/\nu=\rho$ +15776,$P(t)=\mathsf{E}[X_t]$ +15777,"$f(W_t,t)$" +15778,$\bar w_t = \zeta_1z_1 + \cdots + \bar\zeta_tz_t$ +15779,$M=\mathsf{var}nothing$ +15780,$\iota(p)\leftrightarrow g(1-p)$ +15781,$s_0/2^{n+1}$ +15782,$p=95.4\%$ +15783,$a_l \le 1$ +15784,$X+\rho(X)$ +15785,"$\forall\ E,F\in R$" +15786,$>x$ +15787,$dg/ds$ +15788,"$X^+=\max(X,0)$" +15789,$\mathsf{E}_g$ +15790,$\sum a_i=\mathop\square\rho_i(X)=\rho{\min(g_i)}(X)$ +15791,$5.184 \times 10^{19}$ +15792,"${}^{[<45,148]}$" +15793,$(\mathsf{E}[X_i]-\mathsf{E}[X\wedge a)]/\mathsf{E}[X_i]$ +15794,$\int_0^1 1-g(s)ds=1-\int_0^1 g(s)ds < 0.5$ +15795,$P(a)=g(S_X(a))$ +15796,$m(a-m)$ +15797,"$(s_L,g_L)$" +15798,$A_t=\mathsf E[X_0]=\mathsf E[X]$ +15799,"$\lambda,df$" +15800,$\rho=0.671$ +15801,$f(\omega)=(s/\omega-1)(1-\omega)=\omega + s/\omega - 1 -s$ +15802,$(a'-X)^+$ +15803,${}_{dt}q_{x+t}\approx dt\mu_{x+t}$ +15804,$\mathsf{E}_Q[X \mid \mathcal{G}]$ +15805,"$\mathsf{TVaR}_{p_i}(X),\mathsf{TVaR}_{p_j}(X)$" +15806,$i=2$ +15807,$a=150$ +15808,$f_{\max{}}=1/b$ +15809,"$I(q^\star,p)=I^\star$" +15810,$s=t$ +15811,"$\{0,1,2,\dots\}$" +15812,$-\rho(-X)\le \rho(X)$ +15813,"$\rho_e^U(X,s)$" +15814,$\iota_i$ +15815,$\mathsf{Y}$ +15816,$-\theta - \lambda(e^{-\theta}-1)=y$ +15817,"$50,000) for individual condominium units valued at more than one hundred thirty-five thousand dollars ($" +15818,"$459,798 | \$" +15819,$C=2\mathbb Z + 1 + \xi\mathbb Z$ +15820,$1<\alpha<\omega_c$ +15821,$\mu = t \nu$ +15822,$\tau(X)=r$ +15823,$\partial P/\partial a \times \partial a\partial x_i$ +15824,$m_{xx}$ +15825,$\rho(X)\ge \mathsf{E}[h_\epsilon X]$ +15826,$\mathsf x\mathsf{TVaR}_p(X):= \mathsf{TVaR}_p(X)-\mathsf{E}[X]$ +15827,$\mathsf{E}[X] + d(\max(X)-\mathsf{E}[X])$ +15828,$P_S$ +15829,$\cos\arctan(\mu)=1/\sqrt{1+\mu^2}$ +15830,$D_c$ +15831,$L(X)=k(X-\mathsf{E} X)$ +15832,$ where the last term is the CV of $ +15833,$e^{\theta_1 y}$ +15834,"$[0,\alpha)$" +15835,$2.09 \times 10^{14}$ +15836,$P(x)$ +15837,"$5,000 one year hence and \$" +15838,$c$ +15839,$x+n$ +15840,$c=0.5$ +15841,$g(S(x))\approx S(x)\approx 0$ +15842,"$\min(x_1,x_2)$" +15843,$f:L\to M$ +15844,"$B_1,B_2\in\mathscr{F}$" +15845,$\rho(X) \le \rho^\star(X)$ +15846,$q_\alpha\in B'\cap F_\alpha$ +15847,$1 \times 10^{18}$ +15848,"$T_{m_1}(Y)\ge a_{X,r'}(Y)$" +15849,"$\mathsf{E}(C_1(a,c))$" +15850,$af + a(1-f)/q$ +15851,"$\lambda\rho(X) + (1-\lambda)\rho(Y) \le \max(\rho(X),\rho(Y))$" +15852,$\rho(X_1)=\infty$ +15853,$p\approx 1$ +15854,$\delta(p)=\iota(p)/(1+\iota(p))=1-\nu(p)$ +15855,$00.9$ +15884,"$(0,0,0,0,0,5,0,0,0,5)$" +15885,$\mathbf R$ +15886,"$\theta\sigma_1=0,1$" +15887,$\rho(X-X)=\rho(X)+\rho(-X)=0$ +15888,$(\lambda)$ +15889,$\rho(X_0+Y) \ge \rho(X_0) + \mathsf{E}[YZ]$ +15890,$dQ/dP=g'(S(X))$ +15891,$(1-p)/(p(\nu_p-l_p)^2)$ +15892,$\mathsf{var}(A) = \mathsf{E}(N).\mathsf{var}(X) + (\mathsf{E}(X))^2 . \mathsf{var}(N)$ +15893,$\rho(X)=50$ +15894,$\mu_x$ +15895,$\rho_0$ +15896,$\leftrightarrow\mathcal P\rightarrow \rho_t(X)=\max_{P\in \mathcal P}\mathsf E_P[X\mid mathcal F_t]$ +15897,"$x_{1,i}, x_{2,i}$" +15898,$c(x)\ge 0$ +15899,$\mathsf{E}[X1_A] / \mathsf{E}[1_A]$ +15900,$x=\mathsf{VaR}$ +15901,$\mathsf{E}[Xe^{\pi X}]/\mathsf{E}[e^{\pi X}]$ +15902,$\tilde p$ +15903,$|e^{isy}|=1$ +15904,$\mathsf{E}[X_iZ]=500$ +15905,"$(p, x)$" +15906,$y\le q_C(p)$ +15907,"$x_i, y_i$" +15908,$c(z; \nu)$ +15909,$s_1=1$ +15910,$r\le 0$ +15911,"$X = u_{1,2}(X')$" +15912,"$\beta_i(a) = \dfrac{\sum_{j:X_j>a} (X_{i,j}/X_j) \Delta g(S_j)}{\sum_{j:X_j>a} \Delta g(S_j)}$" +15913,"$\mathsf E[(X-K)^+] \le \mathsf E[(Y-K)^+],\ \forall K\in\mathbb R$" +15914,"$\alpha_X\in(\alpha_Y, \alpha_Y+1/3)$" +15915,"$i=1,2,\dots,10000$" +15916,$\forall x\ [\exists y\ (y\in x)\rightarrow \exists y\ (y\in x \wedge \neg\exists z(z\in x \wedge z\in y))]$ +15917,$E_g[X_1(a)]$ +15918,$\alpha + F_1/a$ +15919,$R_1(t) > R_1(0)$ +15920,$r_i+a_i-\iota'$ +15921,$e=0.24$ +15922,$\rho(X)\le 0$ +15923,$e^{kx}S(x)\to\infty$ +15924,$\hat\rho(X)$ +15925,$R_2(t)>R_2(0)$ +15926,$p(1-p)/\nu^2$ +15927,"$\mathsf{VaR}_1=\mathrm{ess\,sup}$" +15928,"$X_n=1_{\{0,1,\dots,n-1\}}$" +15929,$\mathsf{E}(B(p))=p$ +15930,$X = \sum_i X_i$ +15931,$g(s_j)$ +15932,$(X_n)$ +15933,$\hat p_s$ +15934,$\hat q(p)=q(1-g(1-p))$ +15935,$\sum_i \mathsf{VaR}_i(p) = \mathsf{VaR}_{\mathrm{Total}}(0.996)$ +15936,$\rho_\alpha(X)=\mathsf{E}(X\mid X\ge q_\alpha(X))$ +15937,$r>i$ +15938,$\mathsf{E}[Z]=1$ +15939,$K(t; \theta)$ +15940,$j>i$ +15941,$v\in V_X$ +15942,"$r \approx 384{,}400 \ \text{km}$" +15943,$\leq$ +15944,$d\mathsf Q/d\mathsf P$ +15945,$P(X_{-1}\wedge a_{ro})=9196.39$ +15946,$l=a$ +15947,"$P_X(a,b] = F(b)-F(a)$" +15948,$\omega'$ +15949,$Z\cap J\not=Z\cap (J\setminus\{\omega\})$ +15950,$T_Z$ +15951,$\nu=1/(1+\iota)$ +15952,$F(x)=Pr(X ≤ x)$ +15953,$\mathsf{Pr}hi^{-1}(p)$ +15954,$\theta_r$ +15955,$g_i$ +15956,$t = 2$ +15957,$R_A=R_f$ +15958,$2.00 per $ +15959,$\mathbf{B}'(1) = -3\mathbf{P_2}+3\mathbf{P_3}$ +15960,$\mathsf{E}(A)$ +15961,"$\mathscr{G}amma(T)=\{ (\omega, T\omega) \mid \omega\in\Omega \}$" +15962,$\bar Q_{2}$ +15963,"$\Delta_t:=a_{0,t}'-a_{0,t}$" +15964,$R_f=0$ +15965,$p/q-1=(p-q)/q>0$ +15966,$A-A\mathsf{Pr}hi(d^*)=A\mathsf{Pr}hi(-d^*)$ +15967,$\sum_n 1/n$ +15968,$\mu + \lambda\sigma$ +15969,$X> 0$ +15970,$\phi_L$ +15971,$\prec X$ +15972,$(T\lambda)k = \mu kl < \infty$ +15973,$\mathsf{E}_{\mathsf Q_k}[X'']=\mathsf{E}_\mathsf{P}[X'']$ +15974,$\mathcal{A}=f^{-1}(\mathcal{B}B)$ +15975,"$\rho\in\mathcal R'_{X,r}$" +15976,$s > \mathsf{Pr}r(X > \alpha)$ +15977,"$ occurs, i.e., those with the value 1 in the $" +15978,$\rho(X\wedge a)=\mathsf{E}[(X\wedge a)Z(X)]$ +15979,"$B\subset [0,1]$" +15980,${}^2$ +15981,$D_g(\mathcal{B}bb Q\mid \mathcal{B}bb P)=E_{\mathcal{B}bb P}(g(d\mathcal{B}bb Q/d\mathcal{B}bb P))$ +15982,$\tilde Z$ +15983,$X = \sum X_i$ +15984,$\tilde X\wedge a$ +15985,$X_1(u_1) = X$ +15986,$\hat x > x$ +15987,$a(X_i;X) \ge 0$ +15988,$a=b=r$ +15989,$n=2^5=32$ +15990,$(\alpha-1)/\alpha=1/(2-p)$ +15991,$k(\theta)>0$ +15992,$N\sim\mathrm{Po}(\lambda)$ +15993,$\bar P(a) = \rho_g(X\wedge a)$ +15994,$\downarrow 0$ +15995,$\mathsf{Pr}r(H=1)=p$ +15996,$M_G(\zeta):=\text{E}(e^{\zeta G})$ +15997,$W_j$ +15998,$P_i$ +15999,$P_1(t)>P(1)$ +16000,"$(M, \mathcal{B})$" +16001,$\mathsf{E}_\mathsf{Q}[X_i]= \mathsf{E}_\mathsf{Q}[\mathsf{E}[X_i \mid X]]$ +16002,$\lambda(t)$ +16003,$T$ +16004,$\rho=0.9$ +16005,$\rho(X)=\rho_\phi(X):=\displaystyle\int_0^1 q(p)\phi(p)dp$ +16006,"$\mathcal{A} = \{ X\mid \exists \alpha\ge 0, \exists Y : \rho(Y)=0, X=Y+\alpha \}$" +16007,$\log(x)\le x-1$ +16008,"$(fun5a.south east)+(\medspc,-0.5*\medspc)$" +16009,$\frac{1}{2} \theta^2$ +16010,$\mathsf{A}$ +16011,$\partial P/\partial a=g(S(x))$ +16012,$\mathbf {\rho(X\wedge a)}$ +16013,$\rho(X)=\sup_{m\in \mathcal{M}} \rho_m(X)$ +16014,$\Delta \mathit{MV}_{ro}(a)$ +16015,$P = \mathsf{E}[X] + \pi \mathsf{E}[((X-\tau)^+)^p]^{1/p}$ +16016,$q/p$ +16017,$m_l$ +16018,$\log(1)=0$ +16019,$c > c_{max}$ +16020,"$((0, x), (1-p, p))$" +16021,"$968,000.00 - \$" +16022,"$i \in \{1,\dots,4\}$" +16023,"$S:(L,\mathcal{A})\to ??$" +16024,"$j=1,\dots,n$" +16025,"$(fun2a.south -| fun4a.east)+(\ , -\ )$" +16026,$s:=8.75=10.5 / 1.2$ +16027,$(1+r) = (1+rP)(1+m)$ +16028,$f(y;\mu)=\dfrac{1}{\sqrt{2\pi y^3}}e^{-(y-\mu)/(2\mu^2y)}$ +16029,$e^{{{slant:.2f}x}}X$ +16030,$\mathsf{Pr}hi_i(y)=\mathsf{E}(X_i \mid Y = y)$ +16031,$r-r_L$ +16032,$wg_1+(1-w)g_2$ +16033,$\int g(S)$ +16034,$f(z)=E[e^{zN}]=\exp(\lambda(e^t-1))$ +16035,$t 0$ +16067,"$p:\Omega\times\mathscr{F}\to [0,1]$" +16068,$x \urcorner$ +16069,$\bar K$ +16070,$h(x)=-d/dx(\log(S(x)))$ +16071,$a<\max(X)$ +16072,$p_-$ +16073,$\sigma(X_d)$ +16074,$\{ X(\mathbf{x}) \le a\}$ +16075,$D\rho_{X_n}(X_c)$ +16076,"$x_{2,1}$" +16077,$10^{-35}$ +16078,$K_\theta(t)=\kappa(\theta+t)-\kappa(\theta)$ +16079,$\mathsf{E}[X_i(v_i)]=v_i\mathsf{E}[X(1)]$ +16080,$\tilde\rho(X)\le\liminf\tilde\rho(X_n)$ +16081,$g_\min$ +16082,$f'_-$ +16083,$\sum_i x_i^2$ +16084,$\mathsf{TVaR}_0( X )=\mathsf{E}[X]$ +16085,$S(x_5)$ +16086,$\mathbf {\Delta S}$ +16087,$y_s$ +16088,"$k=1,\dots, n-1$" +16089,$g(s)=1 + k(s-1)=1-k + ks$ +16090,"$1,376.27, which is \$" +16091,$h=1+\lambda(\zeta-\mathsf{E}\zeta)$ +16092,$\mathsf{Pr}hi_i(a) = \int_0^a \phi_i(t) dt$ +16093,$X^n\to X$ +16094,$\mathscr{S}(X)$ +16095,$aggfft$ +16096,$((1-\alpha)\phi_R + \alpha\phi_m^o) - ((1-\alpha)\phi_m^e + \alpha\phi_m^o)$ +16097,$f:\Omega\to\mathbb{R}$ +16098,$D_n$ +16099,$> 0.98$ +16100,$\eta_i \ge 0$ +16101,$G=const_j$ +16102,$D\rho_{X}(Y) \subset D\rho_{X\wedge a}(Y)$ +16103,$\mathsf{E}[X]=\mathsf{E}[Y]$ +16104,$\{ p \mid q^-(p) \le x \}=\{ p \mid p \le F(x) \}$ +16105,"$X_2=(0,1,2,3,4,8,6,4,0,9)$" +16106,$p_i < p_j$ +16107,$a \ge 1$ +16108,$X({\mathbf{x}})$ +16109,$M(a)=\beta(a) g(S(a))-\alpha(a)S(a)$ +16110,$T+1$ +16111,"$I=[x_{\min{}}, x_{\max{}}]\subset\mathbb R$" +16112,$\mathsf{E}_\mathsf{P}[X]$ +16113,$\pm 3\sigma$ +16114,$n \times r$ +16115,$t=0.4$ +16116,$a^{\star}(X)-a(X)$ +16117,$p_2$ +16118,$Y = h(Z)$ +16119,$\mathcal R^b$ +16120,$S_T$ +16121,$\mathsf{Q}(A)=2\mathsf{P}(A\cap B)$ +16122,$c(\alpha)x^\alpha g(x)$ +16123,$\alpha(\mathsf Q)=\infty$ +16124,$E[M_0]=0$ +16125,$\bar P_0$ +16126,$\{u_j\}$ +16127,$\tau(\omega_I)=\hat\rho(X)-\rho(X)$ +16128,$s = m / P$ +16129,$x^2 - 2 = 0$ +16130,$\approx 0.999999999$ +16131,$s_0/2^n$ +16132,$g(s)-\hat g(s)$ +16133,$\mathsf{Pr}r(Z=0)$ +16134,$\mathsf{E}_Q[X_i\mid X]=\mathsf{E}[X_i\mid X]$ +16135,$x^3$ +16136,$2^{-72}=1/4.7\times 10^{21}$ +16137,$-norm by integrating against a function with $ +16138,$\mathbf {pK}$ +16139,$\theta\mathsf{TVaR}_{p_0}+(1-\theta)\mathsf{TVaR}_{p_1}$ +16140,$V(m)=1-m$ +16141,$\mathsf{E}[F_1] > \mathsf{E}[F_0]$ +16142,$\mathsf{E}_{\mathsf Q}[Y]=\mathsf{E}[Yg'(S(X))]$ +16143,"$1_A:\Omega\to \{0,1\}$" +16144,$F\le s$ +16145,$(-t)^a=|t|^a$ +16146,"$\omega\mapsto (\omega, T\omega)\in \Omega\times M$" +16147,$v_{res}$ +16148,$S_j$ +16149,"$\rho(X)=\int g(S(t))\,dt$" +16150,$Z=\tilde Z$ +16151,$\rho_1(X) = \rho_2(X)=c$ +16152,$g(s) = 1 - (1 -s)^\beta$ +16153,$\mathsf{Q}(A)=\mathsf{E}[1_AZ]=0$ +16154,$w_l=1-c\gamma$ +16155,$P/E[U]-1$ +16156,$\mathsf{E}[X_i\mid \{X=X(\omega)\}]$ +16157,$\{Q\}= \partial\rho(X)$ +16158,$s_L=s_0=0$ +16159,$g(s)=s^r$ +16160,"$\int_0^1 a'(tx)\,dt=\int_0^1 a(1)\,dt = a(1)=a'(x)$" +16161,$\mathsf{E}[X] + c\mathsf{E}[(X-\mathsf{E} X)_+^2]$ +16162,$\kappa^i(x)$ +16163,$\mathsf{Pr}(L\ge l)$ +16164,$\xi=(\alpha-2)/(\alpha-1)$ +16165,$\mathsf{POS\ LOAD}$ +16166,$1_G(ω)$ +16167,$\mathbf {\iota}$ +16168,$Gn$ +16169,"$\mathsf{E}[(X-a)^+]= p\,\mathsf{E} X$" +16170,$\bar S_i$ +16171,$\mathsf{Pr}r(V=v)$ +16172,$Xp$ +16173,$Q_i=a_i-P_i$ +16174,$X-\pi(X)$ +16175,$\zeta_t-\zeta$ +16176,$J_n$ +16177,$M_i = \beta_ig-\alpha_iS$ +16178,$\mathrm{CV}(X_t)=\mathrm{CV}(X_1)/\sqrt{t}\to 0$ +16179,$\sum_j x_j L_{ij} - \zeta - \eta_i\le 0$ +16180,$\bar\iota : 1$ +16181,"$p:E\times \S\to [0,1]$" +16182,$Q(x)$ +16183,$F(a-)=\lim_{x\uparrow a} F(x)$ +16184,$\rho_{m}(Z)\le \rho_{m'}(Z)$ +16185,$j < \mathsf{j}(a)$ +16186,"$\min(\delta, \max(X-x))$" +16187,$V=(a-X)^+$ +16188,$O(mn^2)$ +16189,"${}^{[<81,78]}$" +16190,$1/x^3$ +16191,"$L_a=[a, a+da]$" +16192,$\mu=\tau(\theta)=-\dfrac{\alpha}{\theta}$ +16193,$S_k=Pr(X > k)$ +16194,$g(t) = r_0 + (1-r_0)t$ +16195,$\log(0)=-\infty$ +16196,$0.05$ +16197,$\mathit{ROL} = \mathsf{Pr}hi( \alpha_{0} + \alpha_{1} \mathsf{Pr}hi^{- 1\ }( \mathit{AEL} ) )$ +16198,"$\phi(p) = (1-\alpha)^{-1}1_{[1-\alpha, 1)}(p)$" +16199,$\mathsf{E}[X_1\mid X=a]$ +16200,$\mathsf{TVaR}_p(X) = c$ +16201,$\int c=1$ +16202,$k E[X]$ +16203,$-l$ +16204,$d(y;\mu)=d^*(y-\mu)$ +16205,"$\partial\rho(X+\epsilon X_i)=\{Q_{i, \epsilon} \}$" +16206,"$\mathcal E_{X,r}$" +16207,$0.98$ +16208,$\mu\in\mathbf R$ +16209,$X_\lambda$ +16210,$U_i=X_i$ +16211,$\alpha_2$ +16212,$\mathsf{P}(X=0)=0.4$ +16213,$\$ +16214,$-1\le \beta\le 1$ +16215,$m_X(s)=\mathsf E[X\mid S=s]$ +16216,$\mathsf{E}[Z^*\mid X] = n^{-1}\sum_{T\in\mathscr{S}(X)} Z^*\circ T = n^{-1}\sum_i \alpha_i \sum_T Z\circ T_i\circ T = n^{-1}\sum_i \alpha_i \sum_T Z\circ T=\sum_i \alpha_i\tilde Z =\tilde Z$ +16217,"$(x_1, \dots, x_n)$" +16218,"$(\Omega, \mathcal{B})$" +16219,$\text{VIF}_j = 1/(1 - R^2_j)$ +16220,$\mathsf{TVaR}_{p^\star}$ +16221,$X_0 =\mathsf E[X]$ +16222,$d_x$ +16223,$-5$ +16224,"$(\omega^{kl})_{k,l}$" +16225,$S_t=S_0 X_t$ +16226,"$p_j, j=1,\dots,n=7$" +16227,$\bar F(a)=\int_0^a F(x)dx = a-\bar S(a)$ +16228,$=\dfrac{s}{g(s)}$ +16229,$P_1(t)<\rho(t X_1)$ +16230,"${}^{[<2,77]}$" +16231,$\delta=\delta_i=0$ +16232,$y\not=\mu$ +16233,$r_f/(1+ r_f) = 0.0196$ +16234,"$\Delta_{i,\epsilon}$" +16235,$1-p < s$ +16236,$l_i$ +16237,$(X(\omega_1)-Y(\omega_1))(X(\omega_2)-Y(\omega_2))\ge 0$ +16238,"$5,000) to as much as \$" +16239,$\mathsf{E}[1_{U_X\ge p}]=\mathsf{E}[B]$ +16240,$\sum Y_i=S$ +16241,$k= \mathsf{E}(X) + (\rho_m(X) - \mathsf{E}(X)) + (k-\rho_m(X))$ +16242,$F_X\ge F_Y$ +16243,$g(S_{X\wedge a'}(x))$ +16244,$\mathsf{Pr}$ +16245,"$\mathcal E_{X,r}(Y)=\{\rho(Y)\mid \rho\in\mathcal E_{X,r} \}$" +16246,$v^T\mathsf P_0X + \sum_t v^{T-i}\Delta_i(X)$ +16247,$X^m$ +16248,$E_Q(Y)$ +16249,$X_j^i$ +16250,$=E(X_i \mid X\le a)$ +16251,$\mathsf{Pr}(E\mid A) = \mathsf{Pr}(E\cap A) / \mathsf{Pr}(A)$ +16252,$\mathsf{CTE}$ +16253,$\lambda = 1 / \sigma$ +16254,$m(1+m)$ +16255,"$n_{\text{water, initial}} = \frac{P_{\text{water}}V}{RT} = \frac{4240 \times 1}{8.314 \times 303.15}$" +16256,$F^{(-2)}\int_0^p F^{-1}$ +16257,"$\rho(X)=\langle \zeta, X \rangle$" +16258,$\rho_L^E(t)$ +16259,$\lambda = L/P$ +16260,$\hat\rho(X_1)\le\hat\rho(X_2)$ +16261,$\alpha_i(x)<\kappa_i(x)/x$ +16262,"$209,265,158 | 16.0% | $" +16263,"$[p_{-},p_{+}]$" +16264,$\kappa'(\tau^{-1}(\mu))=\mu$ +16265,$\mathsf{Pr}r(X_t\le 0) = 1-p$ +16266,$N=6$ +16267,"$\beta_{1,t}(x)$" +16268,$GF(\bar x)$ +16269,$\mathsf{E}(X-k)_+$ +16270,$226.13 Shapley Value \$ +16271,$\rho(X)=\rho_{m_X}(X)=r$ +16272,$P_i/x_i$ +16273,"$50 of the amount allowed on each claim in the classes under subs. (3) to (6), except for claims of the federal government under subs. (3) and (3c), shall be deducted from the claim and included in the class under sub. (8). Claims may not be cumulated by assignment to avoid application of the $" +16274,"$74,500 | $" +16275,$n\to\infty$ +16276,$\mu(\{p_j\})$ +16277,$c(S)\le c(T)$ +16278,$f \mapsto e^{\theta X} f$ +16279,$\rho(A+B)\not=\rho(A)+\rho(B)$ +16280,$\forall X\ \exists P\ \forall z\ [z\subset X\rightarrow z\in P]$ +16281,"$(V,\Theta)$" +16282,$R'_1(1)$ +16283,$\alpha_iS\Delta X$ +16284,$E\triangle F=(E\setminus F)\cup (F\setminus E)$ +16285,$\alpha_0 \not= 0$ +16286,$10^{-13}$ +16287,$p'>p$ +16288,$g(Q)$ +16289,$P(0)=P_0(0)=R_0(0)$ +16290,$h(1_{X\le a})$ +16291,$\mathsf{T}=\mathsf{M}\mathsf{F}^{-1}\mathsf{C}$ +16292,$2aw$ +16293,$c+\mathsf{E}(X-c)_+ = E(X-c)_- + \mathsf{E}(X)$ +16294,$\iota a + \mathsf{E}_Q(X-a)^+$ +16295,$f(x_i)$ +16296,"$[0,1]\times \{y\}$" +16297,"$k \in \{R, R+1, m\}$" +16298,$c_x/c_{\text{Nov 1}}-1$ +16299,$g'>0$ +16300,$\mathsf{E}_F(h(X))$ +16301,"$\displaystyle\int_0^a \alpha_i(x)\,dx$" +16302,$Y(\omega)=0$ +16303,$Z_k$ +16304,$\mathcal Q_2$ +16305,$K_{X+Y} =\log M_{X+Y}=\log (M_{X}M_Y)=\log (M_{X}) + \log(M_Y)=K_X+K_Y$ +16306,$F(X)$ +16307,$Q=100$ +16308,$\mathsf{E}[L\wedge A]$ +16309,$f=f(s)$ +16310,$i\times(n-i)$ +16311,$a\mapsto \sum_i a_iX_i$ +16312,$M_0^i$ +16313,$t+d$ +16314,$X\circ T(\omega)=X(T(\omega))$ +16315,$q^-$ +16316,$P/\Delta X=g(S)$ +16317,$P/A$ +16318,$Y\succeq X$ +16319,$\phi'(p)=-g''(1-p)>0$ +16320,$e^{-rt}S_t$ +16321,$\check S\implies \ \check g$ +16322,"$h(p)=p/(1+\iota(p))=\nu(p)\, p$" +16323,$\tpx^{(\tau)}$ +16324,$\iota_U$ +16325,$|X_n - X_{n-1}| \le K$ +16326,$\mathsf{Pr}r(X\le a)=F(a)$ +16327,"$X\ge 0,(\tilde X-X)\ge 0$" +16328,$\bar A_{x+b} - \bar P_{x+b}\bar a_{x+b}=0$ +16329,$(1/2x_0)\mathbb Z$ +16330,$\sum \rho_k$ +16331,$1-\mathsf{Pr}hi(x)=\mathsf{Pr}hi(-x)$ +16332,$\rho_\phi(X)=\displaystyle\int_0^1 q(p)\phi(p)dp$ +16333,$P(x):=g(S(x))$ +16334,$\phi(r_2)$ +16335,$1-p=S(x)$ +16336,"$p\in [1,\infty)$" +16337,$\sigma\in L_\infty$ +16338,$P(1-e)$ +16339,$\alpha(\mathsf Q)=0$ +16340,$\mathbb{R}^\infty$ +16341,"$\rho(X) = \max(\mathsf{E}_{\mathsf{Q}_1}(X), \mathsf{E}_{\mathsf{Q}_2}(X))$" +16342,$1 = m(a)+\nu F(a) = (S(a) + \delta F(a)) + \nu F(a)$ +16343,$a=a[X]$ +16344,$\lambda=0.421$ +16345,$B_i^c$ +16346,$\mathsf{E}(N^2)=$ +16347,$\zeta:=1/(1-\alpha)1_A$ +16348,$\zeta_1 Z_1 + \cdots \zeta_T Z_T$ +16349,"$17,130,000 | 23.8% | $" +16350,"$\mathsf{cov}(X_1, g'(S(X(t)))$" +16351,$p_j=1/10$ +16352,"$1.075 billion, equal to $" +16353,"$\{2, 3\}$" +16354,$P(A)=1-p$ +16355,$\mathsf{E}[X_i\sum_j w_jZ_j]=\sum_iw_j\mathsf{E}[X_i Z_j]$ +16356,$2X_{t-1}\epsilon_t$ +16357,$1/(1-p)$ +16358,$\rho(X_1+X_2) \le \rho(X_1)+\rho(X_2)$ +16359,$\Delta_1=0$ +16360,"$\mathsf{Tw}^*(\lambda, \alpha, \beta) = \mathsf{CP}(\lambda, \mathsf(Ga(\alpha,\beta))$" +16361,$q_{\mathbf{w}}(p)$ +16362,$6/6$ +16363,$dt\to 0$ +16364,"$(a,b] \subset [0,1]$" +16365,$Z_5$ +16366,$19.117-18.7=0.417$ +16367,$\bar P(a)=\mathsf{E}_\mathsf{Q}(X\wedge a)$ +16368,$α_1 < α_2$ +16369,"$(fun3a.south -| fun4a.south east)+(\smlspc,-\smlspc)$" +16370,$S_{\mathbf{x}}(t)=\text{Pr}(X({\mathbf{x}})>t)$ +16371,$P = \mathsf{E}[X] + \pi \mathsf{Var}(X)$ +16372,$1B then IAL of $ +16373,$1500$ +16374,$\sigma=\sqrt(\log(v^2+1))$ +16375,$x=q(\hat p)$ +16376,$\rho(0)=\rho(0 \times X)=0\times \rho(X)=0$ +16377,$\DeltaX$ +16378,$m'(1) = -m_2/(1-s_2)$ +16379,"$\forall\,\omega,t\in M$" +16380,$r_i = (P_i-U_i)/K_i$ +16381,$10Y + 50$ +16382,$S(x)=s$ +16383,"$25,000 is paid and \$" +16384,"$10,000) plus \$" +16385,"$(0,t_2]$" +16386,$e^{-rt}$ +16387,$\theta=0.87$ +16388,$1_A(x)=0$ +16389,$\rho_{\mathsf{PH}}$ +16390,$/$ +16391,$3^{30}=2.06\cdot 10^{14}$ +16392,$g(s)=3s$ +16393,$4\nu$ +16394,"$(s,g)$" +16395,$\mathsf{Pr}r[X > a]$ +16396,$\mathsf{Pr}r(E)$ +16397,"$(s_j, g_j)$" +16398,$\mathsf{E}[X|A]= n^{-1}\sum_T X\circ T$ +16399,$\{ j:X_{0}^{j} \geq A\}$ +16400,$\mathsf E[X_i \mid X](\omega) = \mathsf E[X_i \mid X=X(ω)]$ +16401,$q_k = g(S_{k-1}) - g(S_{k})$ +16402,$X(\omega_1) > Y(\omega_1)$ +16403,$R_i$ +16404,${}^{[>340]}$ +16405,"$\mathrm{PV}(\Delta\mathrm{EQ}, r) = \mathrm{PV}(\Delta\mathrm{TCF}, i)$" +16406,$S_k<0.5$ +16407,$O$ +16408,$\phi(\cdot)$ +16409,$Q+P>\Delta X$ +16410,$d=iv=1-v$ +16411,$X=h(Z)$ +16412,"$\mu=7.4, \sigma=1.9$" +16413,"$q^-(p) = \int_0^M I(F(x) < p)\,dx$" +16414,$\mathsf{E}[X_1]=4.75$ +16415,$9+1$ +16416,$\displaystyle\int_0^a xf(x)dx \not= \displaystyle\int_0^a S(x)dx$ +16417,$\mathcal X$ +16418,$q_C\le q_A$ +16419,$x_1=x_2$ +16420,$\mathsf{COHERENT}$ +16421,$X_{-1}+X_{0}$ +16422,$\rho_g(X)=g(s)$ +16423,$F_M$ +16424,$\mathsf{TVaR}_0$ +16425,$t_1^*-\epsilon/2$ +16426,$1+Z-\mathsf{E} Z$ +16427,$\rho_g(1)=1$ +16428,$\mathsf{Pr}(L\ge 270)=77.1\%$ +16429,$g(s) = \mathsf{Pr}hi(\mathsf{Pr}hi^{-1}(s)+\lambda)$ +16430,$\rho(Y)\le\rho(0)=0$ +16431,"$P(\omega, B)$" +16432,$W(1-f)$ +16433,"$\alpha,\beta$" +16434,$p=0.6$ +16435,"$\mathrm N(0, \sigma_n^2)$" +16436,$\mathsf{E}[(X-\mu)^n]$ +16437,$s=0.5$ +16438,$g(S(x))=0$ +16439,$\rho(-X) \ge -\rho(X)$ +16440,$dG(x)=g'(S(x))dF(x)$ +16441,$\sum f_i$ +16442,$C_1(t) > C_1(t)$ +16443,$\mathsf{E}_{\mathsf{Q}}[X_i\mid X\le a](1-g(S(a))) + a\mathsf{E}_{\mathsf{Q}}[X_i/X\mid X >a]g(S(a))$ +16444,"$f(x)=\int_0^1 f'(tx)\,dt$" +16445,$tX_1 \dfrac{X_t\wedge a_t}{X_t}\le tX_1$ +16446,$\rho_m(X) = \mathsf{E}(X) + (\rho_m(X) - \mathsf{E}(X))$ +16447,$T'$ +16448,$\mathsf{E}[X_i\wedge x]$ +16449,$p-1$ +16450,$\mathsf{E}_{\mathsf Q}[X\mid \mathcal F]=\mathsf{E}[XZ\mid \mathcal F]/\mathsf{E}[Z\mid \mathcal F]$ +16451,$1=1_{X\le a}+1_{X>a}$ +16452,$NPV = Q - Q_{act} = F_0$ +16453,$\Theta_i$ +16454,"$(Y,\mathscr{G},\mathbb{Q})$" +16455,$G = C + \sum N_i$ +16456,$1m limit over a \$ +16457,$\mu-\sigma^2/2=0.0992$ +16458,$\delta F(a)$ +16459,$p=0.7$ +16460,$c=\lambda$ +16461,$\kappa_i(x)/t$ +16462,"$p\not=1,2$" +16463,$V_\lambda$ +16464,$-\rho(X-Y)\le \rho(Y)-\rho(X)$ +16465,$E_g[Y] = \int g(S_Y(t))dt$ +16466,"$\rho(L_a^{a+l}(X)=\displaystyle\int_a^{a+y} g(S(x))\,dx$" +16467,$E[X_\tau] = E[X_0]$ +16468,$\mathsf{E}[g(-Y)]\ge 0$ +16469,$-(1-s)(1-r_0)\delta_1$ +16470,$X_n=n1_A$ +16471,$X_k^3$ +16472,$\mathsf{E}[(S_T-a)1_{\{S_T>a\}}$ +16473,$\mathbb R^+$ +16474,$\mathsf{Pr}r(\{X_a=a\})\ge 1-p'>1-p$ +16475,$p={p}$ +16476,$g(s)=0.9s + 0.1$ +16477,$f_P$ +16478,$a(\mathbf{x}) =\mathsf{TVaR}_p(X(\mathbf{x}))$ +16479,$\mathscr Q$ +16480,$\mathcal R^c\subset \mathcal R^b$ +16481,$\mathsf{E}[X]=\sum_i x_i$ +16482,"$\displaystyle\int g(S_X) = \sup\{ E_Q(X) \mid Q(A)\le g(P(A)), \forall A\in \mathscr{F}F) \}$" +16483,$=\mathsf{E}(X_i \mid X=q(\alpha))$ +16484,$\mathsf{E}[X \mid X \ge x] = \mathsf{E}[X 1_{X \ge x}] / \mathsf{Pr}r(X \ge x)$ +16485,$t=-ln(1-p)$ +16486,$k'(h)=\mathsf{E}[X_h]=:\mu_h$ +16487,$E_i\cap E_j = \mathsf{var}nothing$ +16488,$1< \alpha<2$ +16489,$X\sim$ +16490,$R^2=$ +16491,"$w, 1-w$" +16492,$X^{\oplus n} -\mathsf E[X] \succeq_2 X^{\oplus n-1}$ +16493,"$\rho=\mathrm{ess\,sup}$" +16494,$g(S(x))-S(x)$ +16495,$l_Q(X)=\mathrm{E}_Q(-X)$ +16496,$\rho(X)=\rho(X-Y+Y)\le \rho(X-Y) + \rho(Y)$ +16497,"$(2,2)$" +16498,"$9,4,4,4,2,1$" +16499,$\mathsf{E}_g(X_i(a))$ +16500,$\mathsf E[Z]=1$ +16501,$\mathsf{E}[X\wedge a(X)]$ +16502,$=1/(1-p)$ +16503,$750K on the \$ +16504,$R^2$ +16505,$\mathit{NPV}_1 = F_0$ +16506,$B_t\le x$ +16507,$\bar\theta_s>0.5$ +16508,$\bar x=n^{-1}\sum_i x_i$ +16509,$\sigma(\C)$ +16510,$\nu s+\delta$ +16511,"$X\wedge \alpha(X):=\text{min}(X, \alpha(X))$" +16512,$\mathcal F_1=\sigma(X)$ +16513,$X_i = F(e_i)$ +16514,$V(\mu)=\kappa''(\tau^{-1}(\mu))=1/(\tau^{-1})'(\mu)$ +16515,$S_t(a_t)=1-p$ +16516,$\text{Maximum number of electrons in a shell} = 2n^2$ +16517,$\mathscr F_1 = \sigma(N)$ +16518,$X\le Y$ +16519,$p^*=p_i$ +16520,$R = P - EL$ +16521,$\{ X=x\}$ +16522,$ is different from the contact function $ +16523,"$\mathsf E[\rho(X, \mathsf P_I)]$" +16524,$\mathsf{E}[X\mid t]$ +16525,"$V_s=(1-p_s,0,0,\dots,p_s,0,\dots,0)$" +16526,$85.45) \$ +16527,$g(S(X))$ +16528,$q=1$ +16529,$\mathsf{Pr}r(X < x)=p=\mathsf{Pr}r(X\le x)$ +16530,$U(\omega)=p$ +16531,$A(-X)=-B(X)\not=-A(X)$ +16532,$m\in\mathcal{B}bb R$ +16533,$log(x)$ +16534,$\mathcal R^c$ +16535,$k''(h)$ +16536,$g(s) = d + sv$ +16537,$1.5 billion layer of coverage attaching at $ +16538,$N'/aw$ +16539,$\mathsf{E}_\mathsf{P}[X_1\mid X=4]=(3+2)/2=5/2$ +16540,$\rho(X)= \mathsf{E}_{\mathsf{Q}_X}[X]$ +16541,$g(s)=s^\alpha$ +16542,$\mathsf{E}[\mathsf{E}[X\mid \mathscr{G}_2]\mid \mathscr{G}_1]=\mathsf{E}[X\mid \mathscr{G}_1]$ +16543,$\rho(X)=\mathsf{E}[Xe^{kX}]/\mathsf{E}[e^{kX}]$ +16544,$D\rho_X(X_i)=D\rho_i = x_i\dfrac{\partial\rho}{\partial x_i}$ +16545,$m\in \mathbb R$ +16546,$ so $ +16547,$u^{iv}<0$ +16548,$LR = L/P$ +16549,$T_n=T\wedge n$ +16550,$x=1.5$ +16551,${}^nS(t) = \displaystyle\int_t^\infty {}^{n-1}S(u)du$ +16552,"$(1,1,\dots,1,1)$" +16553,$1-F(x)$ +16554,$Q\in \mathcal{Q}$ +16555,$\square \phi_i$ +16556,$u=a$ +16557,$\sigma^2/2$ +16558,$t=1$ +16559,$t>t_0$ +16560,$X_1=\mathsf{E}[X\mid \mathcal F_1]$ +16561,"$\mathsf{biTVaR}_{1-s,1-t}^v$" +16562,"$(0.5, 0.5)$" +16563,$\mathsf{X}$ +16564,$\tilde p_i$ +16565,$g(S(x))=F(x)=e^{-\mu x/b}$ +16566,$\|Y\|_{\sigma}=\int_0^\infty \tau_\sigma(F_{|Y|}(y))dy$ +16567,$g^{ak} = (g^a)^k$ +16568,$X\le Y\implies f_t(X)\le f_t(Y)$ +16569,$-g''(1-p) = \phi'(p) = (1-p)^{-1}f(p)$ +16570,$v=1/(1+d)$ +16571,$M_G(\zeta) = (1-\theta\zeta)^{-a}$ +16572,$dN(a)=d(a-\mathsf{E}(X\wedge a)$ +16573,$g(s)=s^2$ +16574,$r_f>0$ +16575,$X=1_{U < s}$ +16576,$X_1=t$ +16577,$\mathsf{j}$ +16578,$52.6 \times 10^6$ +16579,$\iota_K$ +16580,$f(x)dx$ +16581,$\pm 3$ +16582,$\tilde \rho_t = \rho_t(-\tilde\rho_{t+1})$ +16583,"$t\in[0.12, 0.25]$" +16584,$E[(X-qp)^+]$ +16585,$ω$ +16586,$\lambda_i$ +16587,$\mathsf{CP}_n$ +16588,$X\wedge 10$ +16589,"$\tilde X:[0,\infty)\to[0,\infty)$" +16590,$\rho_m(X)\le\rho(X)=r$ +16591,$a_i=\mathsf{Pr}hi^{-1}(i/(n+1))$ +16592,$v^n$ +16593,$\rho(X+Y)\ge$ +16594,"$(0+, T_L)$" +16595,$g’(1)>0$ +16596,$\delta(p)=\iota(p)/(1+\iota(p))=1-\iota(p)$ +16597,$-\pi/2\le\phi\le\pi/2$ +16598,$q(p)=F^{-1}(p)$ +16599,$a'=a(1+r)$ +16600,$V(m)=mQ(m)$ +16601,$\mathsf{E}[\mathsf{Pr}i]$ +16602,$Z\succeq_2 \mathsf{E}[Z\mid X]$ +16603,"$\bar P_i(\mathbf{v},a)$" +16604,$g(x)=1$ +16605,$t=t^*$ +16606,$s_f$ +16607,$\hat\sigma=0.0557$ +16608,$h=1$ +16609,"$\phi(s)= g'(1-s) = \frac{1-w}{1-p_0}1_{[p_0, 1)}(s) + \frac{w}{1-p_1}1_{[p_1, 1)}(s)$" +16610,$h(0.05) = 1-g(1-0.95) = 0.0203$ +16611,$g'(S(x))f(x)dx$ +16612,${}^{[<23]}$ +16613,$P_t\{T=t\}=1$ +16614,$q_\alpha$ +16615,"$p\not=1,2,\infty$" +16616,$(\mathsf{E}(X_i)-\mathsf{E}(X_i(a))/\mathsf{E}(X_i)$ +16617,$f=f_0$ +16618,"$80,000/\$" +16619,$\mathsf{Pr}r(X<0)>0$ +16620,$F_g$ +16621,"$\int_1^\infty xj(x)\,dx$" +16622,$f_t(X+m)=f_t(X)+m$ +16623,$\sum X_i(a)\Delta g(S)$ +16624,${}^{10}$ +16625,"$\rho_{a,\tau}(X)=v\rho(X\wedge a) + da$" +16626,$\rho(\tilde X)=34/9$ +16627,$\mathsf{E}_\mathsf{Q}[X]\le \mathsf{E}_\mathsf{Q}[Y]$ +16628,$\mathbf {\mathsf{VaR}_p(X_1)}$ +16629,$\nu-l$ +16630,$U=\mathsf{Pr}hi(W)$ +16631,$tt$ +16632,$1000$ +16633,$\bar P(a)=\bar S(a) + \bar\delta\bar F(a)$ +16634,$(r1 + r2) / 2$ +16635,"$\Omega=\{0,\dots,99\}$" +16636,$t\le 0.5$ +16637,$\int S(x)dx = \int xdF(x)$ +16638,"${3*(4-3)}*(1,0.5)$" +16639,$(a-X)^+=a-(X\wedge a)$ +16640,$\mu = \lambda \alpha \beta$ +16641,$q(U)$ +16642,$s_s$ +16643,$E[X^i/X|X > k]$ +16644,"$a_m,b_m$" +16645,$\mathcal{A}_4$ +16646,$V_3$ +16647,$R_1(t) > P(1)$ +16648,$X'_2$ +16649,$1 \le p \le 2$ +16650,$1+bf$ +16651,$m(1+\frac{m^2}{p^2})$ +16652,$\lambda F$ +16653,$\bar\alpha+1=1/(p-1)$ +16654,$f(x)=\exp(-x/\mu)/\mu$ +16655,"$(X_0,X_1)$" +16656,$1-B_p=B_{1-p}$ +16657,$a_i=a(X_i; X)$ +16658,$()_+$ +16659,$\displaystyle\int_0^1 X(p)dp$ +16660,$c(S)=\rho(\sum_{i\in S} X_i)$ +16661,$\exp(4t^2\sigma^2/2)$ +16662,$\lim_{s\downarrow 0} g_\tau(s) = \tau / (1+\tau)$ +16663,"$q_X, q_Y$" +16664,"$3,000,000,000), regardless of the frequency or severity of earthquake losses at any and all times subsequent to the creation of the authority. Once a participating insurer has paid, pursuant to this section, amounts equal to the percentage share of the authority's total gross written premium attributable to that participating insurers sales of authority insurance policies, as of April 30 of the immediately preceding year or the most recent full year for which premium data not more than one year old are available, multiplied by three billion dollars ($" +16665,$n=2^4=16$ +16666,$\phi_1$ +16667,"$[0,a]$" +16668,$8.617 \times 10^{5}$ +16669,$s_I= s / \omega_I$ +16670,"$\mathsf{LI},\mathsf{COH},\mathsf{COMON}$" +16671,$\mathsf{Pr}(X_i=0)>0$ +16672,"$D^n_{\rho,X}(Y)$" +16673,$x_{i0}$ +16674,$\hat{\mathsf a} = \mathsf F\mathsf a$ +16675,$id\times\pi$ +16676,$\kappa\ge K(n)=\sum_s n_s(1-g(s))k(s)$ +16677,$\mathsf{TVaR}_{0.95}=(x_{96} + x_{97} + x_{98} + x_{99} + x_{100})/5$ +16678,$W_t$ +16679,$Pr(N=0)$ +16680,$X_2 = \mathsf{E}[X\mid \mathscr{F}_2]=X$ +16681,$P^T_S$ +16682,$\rho(X)=\mathsf{E}_\mathsf{Q}[X]-\alpha(\mathsf Q)$ +16683,$d(y;m)$ +16684,"$S, S^{-1}$" +16685,$\mathsf{E}[X_i\mid X=x]f_X(x)/x$ +16686,"$X_{t-1,3}$" +16687,$(1-p + pe^t)^n$ +16688,$D_i>4/n$ +16689,$R_1(1)=P(1)$ +16690,$X^n$ +16691,$\mathbf{n}$ +16692,"$y,z\in X$" +16693,$P^i = \sum_{j=0}^a q_j \kappa_i(j)$ +16694,$t_* < t < t^*$ +16695,$\int f(-x)dx=-\int f(y)dy=-F(y)=-F(-x)$ +16696,$\{\omega\mid X(\omega)=x\}$ +16697,$a-EL$ +16698,"$(1-t,t)$" +16699,$T_s$ +16700,"$M(X_1, a_1)+M(X_2, a_2)=M(X_1+X_2, a_1+a_2)$" +16701,$P=\nu L + (1-\nu) a = a - \nu(a-L)$ +16702,$0\le w\le 1$ +16703,$\beta^\alpha x^\alpha e^{-\beta x}/\mathscr{G}amma(\alpha)$ +16704,$P_t B$ +16705,$a=a(X)$ +16706,$\mathsf{E}(X-x)_+$ +16707,$. Therefore $ +16708,$\mathsf{E}[X(1_{U_X\ge p}-B)]=\mathsf{E}[(X-m)(1_{U_X\ge p}-B)]\ge 0$ +16709,$p=0.01$ +16710,$1-(\text{ErrorSS}/df) / (\text{TotalSS}/df) = 1- (\text{ErrorSS}/(n-(k+1))) / (\text{TotalSS}/(n-1))=1- s^2 / s^2_y$ +16711,$chi^2$ +16712,$X=X_k$ +16713,$\Delta X-P$ +16714,$\{\omega\in\Omega \mid X(\omega) \le x\}\in\mathcal F$ +16715,$r_p X` lines (by simply using the same or more `extra` space). This property makes binary search on the *minimum possible maximum lines* an efficient solution. @@ -1386,13 +1421,12 @@ class GT(object): The function returns the `optimal_max_lines` and the `best_allocation` dictionary, mapping each column name to the minimal `extra_space` it needs to achieve that optimal height. Why this approach is effective: - ------------------------------ + --------------------------------- * **Optimal Solution:** The binary search guarantees finding the absolute minimum possible `max_lines` because it systematically explores the entire solution space. * **Efficiency:** The `check` function runs in time proportional to the number of columns times the average number of `extra` options per column. The binary search itself performs `log(range_of_num_lines)` iterations. This makes the overall complexity efficient for typical table sizes. * **Flexibility:** It does not assume any particular mathematical function relating `extra` space to `num_lines`. It works with arbitrary discrete relationships provided in the input DataFrame, as long as `num_lines` is non-increasing as `extra` increases (which is the natural expectation for this problem). - """ # Pre-processing unique_cols = input_df['col'].unique().tolist() @@ -1465,10 +1499,14 @@ class GT(object): return optimal_max_lines, best_allocation - @staticmethod - def estimate_column_widths(df, target_width, nc_index, scale, equal=False): + def estimate_column_widths(self): """ - Estimate sensible column widths for the dataframe [in what units?] + Estimate sensible column widths for the dataframe in character units. + + Used by HTML and TeX output. returns tikz_colw used by TeX output to print + the tikz (no impact on output, just makes the produced TeX align nicely), + tabs and scaled_tabs (reflecting scale). These three columns are added + to the column_width_df. Internal variables: mxmn affects alignment: are all columns the same width? @@ -1480,18 +1518,28 @@ class GT(object): :param nc_index: number of columns in the index...these are not counted as "data columns" :param config.equal: if True, try to make all data columns the same width (hint can be rejected) :return: - colw affects how the tex is printed to ensure it "looks neat" (actual width of data elements) + tikz_colw affects how the tex is printed to ensure it "looks neat" (actual width of data elements) tabs affects the actual output """ - # this + # local variables (conversion from global method) + df = self.df + target_width = self.config.max_table_width + nc_index = self.nindex + scale = self.config.tikz_scale + equal = self.config.equal + # tabs from _tabs, an estimate column widths, determines the size of the table columns as displayed # print(f'{nc_index=}, {scale=}, {config.equal=}') - colw = dict.fromkeys(df.columns, 0) + # without tex adjustment + tikz_colw = dict.fromkeys(df.columns, 0) + # with tex adjustment + tex_colw = dict.fromkeys(df.columns, 0) headw = dict.fromkeys(df.columns, 0) tabs = [] + scaled_tabs = [] mxmn = {} if df.empty: - return colw, tabs + return tikz_colw, tabs, scaled_tabs nl = nc_index for i, c in enumerate(df.columns): # figure width of the column labels; if index c= str, if MI then c = tuple @@ -1527,21 +1575,26 @@ class GT(object): try: lens = df.iloc[:, i].map( lambda x: GT.text_display_len(str(x))) - colw[c] = lens.max() + tex_colw[c] = lens.max() mxmn[c] = (lens.max(), lens.min()) + raw_lens = df.iloc[:, i].map(len) + tikz_colw[c] = raw_lens.max() except Exception as e: - logger.error( - f'{c} error {e} DO SOMETHING ABOUT THIS...if it never occurs dont need the if') - colw[c] = df[c].str.len().max() - mxmn[c] = (df[c].str.len().max(), df[c].str.len().min()) + raise + # logger.error( + # f'{c} error {e} DO SOMETHING ABOUT THIS...if it never occurs dont need the if') + # tikz_colw[c] = df[c].str.len().max() + # mxmn[c] = (df[c].str.len().max(), df[c].str.len().min()) else: lens = df.iloc[:, i].map(lambda x: GT.text_display_len(str(x))) - colw[c] = lens.max() + tex_colw[c] = lens.max() mxmn[c] = (lens.max(), lens.min()) - # print(f'{headw[c]=}, {colw[c]=}, {mxmn[c]=}, {c=}') + raw_lens = df.iloc[:, i].map(len) + tikz_colw[c] = raw_lens.max() + # print(tikz_colw) # now know all column widths...decide what to do # are all the data columns about the same width? - data_cols = np.array([colw[k] for k in df.columns[nl:]]) + data_cols = np.array([tex_colw[k] for k in df.columns[nl:]]) same_size = (data_cols.std() <= 0.1 * data_cols.mean()) # print(f'same size test requires {data_cols.std()} <= {0.1 * data_cols.mean()}') common_size = 0 @@ -1552,7 +1605,7 @@ class GT(object): for i, c in enumerate(df.columns): if i < nl or not same_size: # index columns - tabs.append(int(max(colw[c], headw[c]))) + tabs.append(int(max(tex_colw[c], headw[c]))) else: # data all seems about the same width tabs.append(common_size) @@ -1574,11 +1627,14 @@ class GT(object): if data_width and data_width / target_width < 0.9: # don't rescale above 1:1 - don't want too large rescale = min(1 / scale, target_width / data_width) - tabs = [w if i < nl else w * rescale for i, w in enumerate(tabs)] + scaled_tabs = [w if i < nl else + int(w * rescale) for i, w in enumerate(tabs)] logger.info(f'Rescale {rescale} applied; tabs = {tabs}') + else: + scaled_tabs = tabs # print(f'Rescale {rescale} applied; tabs = {tabs}') - # print(f'{colw.values()=}\n{tabs=}') - return colw, tabs + # print(f'{tikz_colw.values()=}\n{tabs=}') + return tikz_colw, tabs, scaled_tabs @staticmethod def text_display_len(s: str) -> int: @@ -1672,6 +1728,14 @@ class GT(object): .greater-table > table {{ display: inline-table; }} */ + /* try to turn off Jupyter and other formats for greater-table + all: unset => reset all inherited styles + display: revert -> put back to defaults + #greater-table * {{ + all: unset; + display: revert; + }} + */ /* tag formats */ #{self.df_id} caption {{ padding: {2 * padt}px {padr}px {padb}px {padl}px; @@ -1774,19 +1838,8 @@ class GT(object): idx_header = bit.iloc[:self.nindex, :self.ncolumns] columns = bit.iloc[self.nindex:, :self.ncolumns] - colw, tabs = GT.estimate_column_widths( - self.df, self.config.max_table_width, nc_index=self.nindex, scale=1, equal=self.config.equal) - if self.config.debug: - print(f'Make html Input {self.tabs=}\nComputed {tabs=}') - if self.tabs is not None: - if len(tabs) == len(self.tabs): - tabs = self.tabs - elif len(self.tabs) == 1: - tabs = self.tabs * len(tabs) - else: - logger.error( - f'{self.tabs=} must be None, a single number, or a list of numbers of the correct length. Ignoring.') - # print('HTML ' + ', '.join([f'{c:,.2f}' for c in tabs])) + # figure appropriate widths + tabs = self.column_width_df['tabs'] # set column widths; tabs returns lengths of strings in each column # for proportional fonts, average char is 0.4 to 0.5 em but numbers with @@ -1926,7 +1979,7 @@ class GT(object): return self.df_html def clean_style(self, soup): - """Minify CSS inside