From 863ee43b43c3bf7bb7888f6c3c79867357a9ec56 Mon Sep 17 00:00:00 2001 From: Stephen Mildenhall Date: Sat, 14 Jun 2025 13:25:26 +0100 Subject: [PATCH] Working draft of 3.0.0 Added robust testdf --- greater_tables/data/__init__.py | 0 greater_tables/data/tex_list.csv | 5849 +++++++++++++++++++++++++ greater_tables/data/tex_list.py | 119 + greater_tables/{ => data}/words-12.md | 0 greater_tables/gtconfig.py | 8 +- greater_tables/gtcore2.py | 6 +- greater_tables/hasher.py | 8 + greater_tables/testdf.py | 68 +- greater_tables/tex_list.csv | 5804 ------------------------ greater_tables/tex_svg.py | 188 + greater_tables/tex_svg2.py | 133 + pyproject.toml | 3 + 12 files changed, 6361 insertions(+), 5825 deletions(-) create mode 100644 greater_tables/data/__init__.py create mode 100644 greater_tables/data/tex_list.csv create mode 100644 greater_tables/data/tex_list.py rename greater_tables/{ => data}/words-12.md (100%) delete mode 100644 greater_tables/tex_list.csv create mode 100644 greater_tables/tex_svg.py create mode 100644 greater_tables/tex_svg2.py diff --git a/greater_tables/data/__init__.py b/greater_tables/data/__init__.py new file mode 100644 index 0000000..e69de29 diff --git a/greater_tables/data/tex_list.csv b/greater_tables/data/tex_list.csv new file mode 100644 index 0000000..436e08c --- /dev/null +++ b/greater_tables/data/tex_list.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/tex_list.py b/greater_tables/data/tex_list.py new file mode 100644 index 0000000..e329e87 --- /dev/null +++ b/greater_tables/data/tex_list.py @@ -0,0 +1,119 @@ +""" +Find and process blobs of TeX. + +Change target directory to find other blobs. +""" + +from pathlib import Path +import re +import subprocess + +import pandas as pd + + +class TeXMacros(): + """ + A class for dealing with TeX macros. + + made out of PublisherBase in blog_tools.py + from great2.blog + """ + + _macros = r""" +\def\AA{\mathcal{A}} +\def\atan{\mathrm{atan}} +\def\AVaR{\mathsf{AVaR}} +\def\bbeta{\mathbf{\beta}} +\def\bb{\mathbf b} +\def\bm{\mathbf } +\def\biTVaR{\mathsf{biTVaR}} +\def\corr{\mathsf{Corr}} +\def\cov{\mathsf{cov}} +\def\cp{\mathsf{CP}} +\def\CTE{\mathsf{CTE}} +\def\CVaR{\mathsf{CVaR}} +\def\dint{\displaystyle\int} +\def\dsum{\displaystyle\sum} +\def\ecirc{\accentset{\circ} e} +\def\ecirc{\accentset{\circ} e} +\def\EPD{\mathsf{EPD}} +\def\ES{\mathsf{ES}} +\def\E{\mathsf{E}} +\def\FFF{\mathscr{F}} +\def\FF{\mathcal{F}} +\def\HH{\mathbf{H}} +\def\kpx{{{}_kp_x}} +\def\MM{\mathcal{M}} +\def\NN{\mathbb{N}} +\def\nudge{2} +\def\norm{} +\def\OO{\mathscr{O}} +\def\PPP{\mathscr{P}} +\def\PP{\mathsf{P}} +\def\Pr{\mathsf{Pr}} +\def\QQ{\mathsf{Q}} +\def\RR{\mathbb{R}} +\def\SD{\mathsf{SD}} +\def\TCE{\mathsf{TCE}} +\def\TVaR{\mathsf{TVaR}} +\def\Var{\mathsf{Var}} +\def\var{\mathsf{var}} +\def\VaR{\mathsf{VaR}} +\def\WCE{\mathsf{WCE}} +\def\ww{\mathbf{w}} +\def\XXX{\mathcal{X}} +\def\xx{\mathbf{x}} +\def\XX{\mathbf{X}} +\def\yy{\mathbf{y}} +\def\ZZZ{\mathcal{Z}} +\def\ZZ{\mathbb{Z}} +""" + + @staticmethod + def process_tex_macros(text): + """Expand standard general.tex macros in the text.""" + m, regex = TeXMacros.tex_to_dict(TeXMacros._macros.strip()) + return re.sub(regex, lambda x: m.get(x[0]), text, flags=re.MULTILINE) + + @staticmethod + def tex_to_dict(text): + """ + Convert text, a series of def{} macros into a dictionary + returns the dictionary and the regex of all keys + """ + smacros = text.split('\n') + smacros = [TeXMacros.tex_splitter(i) for i in smacros] + m = {i: j for (i, j) in smacros} + regex = '|'.join([re.escape(k) for k in m.keys()]) + return m, regex + + @staticmethod + def tex_splitter(x): + """ + x is a single def style tex macro + """ + x = x.replace('\\def', '') + i = x.find('{') + return x[:i], x[i + 1:-1] + +def find_tex_snippeets(in_dir='\\S\\TELOS\\PIR\\docs', + out_file='tex_list.csv'): + """Ripgrep / TeX macro expand list of TeX snippets.""" + result = subprocess.run( + ['rg', '-N', '-o', '--no-filename', '-g', '*.md', r'\$.+?\$', in_dir], + capture_output=True, + text=True, + check=True + ) + output_text = result.stdout + tm = TeXMacros() + txt = tm.process_tex_macros(output_text) + tex = txt.split('\n') + stex = set(tex) + stext = [i for i in stex if len(i) and i.find('\\PP') < 0 and i.find('$$') < 0] + df = pd.DataFrame({'expr': stext}) + if out_file != '': + p = Path(__file__).parent / out_file + print(p) + df.to_csv(p, encoding='utf-8') + return df diff --git a/greater_tables/words-12.md b/greater_tables/data/words-12.md similarity index 100% rename from greater_tables/words-12.md rename to greater_tables/data/words-12.md diff --git a/greater_tables/gtconfig.py b/greater_tables/gtconfig.py index 91256a6..034f1ff 100644 --- a/greater_tables/gtconfig.py +++ b/greater_tables/gtconfig.py @@ -62,11 +62,11 @@ class GTConfigModel(BaseModel): table_vrule_width: int = Field( 1, description="Width of vertical rule separating index from body" ) - hrule_widths: Optional[tuple[int, int, int]] = Field( - (0, 0, 0), description="Tuple of three ints for horizontal rule widths (for multiindex use)" + hrule_widths: Optional[tuple[float, float, float]] = Field( + (0, 0, 0), description="Tuple of three floats for horizontal rule widths (for multiindex use)" ) - vrule_widths: Optional[tuple[int, int, int]] = Field( - (0, 0, 0), description="Tuple of three ints for vertical rule widths (for multiindex columns)" + vrule_widths: Optional[tuple[float, float, float]] = Field( + (0, 0, 0), description="Tuple of three floats for vertical rule widths (for multiindex columns)" ) sparsify: bool = Field( diff --git a/greater_tables/gtcore2.py b/greater_tables/gtcore2.py index 5c51ff7..300e4a9 100644 --- a/greater_tables/gtcore2.py +++ b/greater_tables/gtcore2.py @@ -261,7 +261,11 @@ class GT(object): base_config = GTConfigModel() # access through config - self.config = base_config.model_copy(update=overrides) + # update and validate; need to merge to avoid repeated args + merged = dict(base_config.model_dump(), **overrides) + self.config = GTConfigModel(**merged) + # no validation + # self.config = base_config.model_copy(update=overrides) # deal with alternative input modes for df: None, DataFrame, Series, markdown text table if df is None: diff --git a/greater_tables/hasher.py b/greater_tables/hasher.py index 2e9b8b4..7fb318a 100644 --- a/greater_tables/hasher.py +++ b/greater_tables/hasher.py @@ -26,3 +26,11 @@ def df_short_hash(df, length=12): hash_str = base64.b32encode(hash_bytes).decode("utf-8").rstrip("=") # Trim padding return f"T{hash_str[:length]}" # Prefix with 'T' to ensure a valid ID + + +def txt_short_hash(txt): + hasher = hashlib.md5() + hasher.update(txt.encode('utf-8')) + hash_bytes = hasher.digest() + hash_str = base64.b32encode(hash_bytes).decode("utf-8").rstrip("=") # Trim padding + return hash_str[::2] diff --git a/greater_tables/testdf.py b/greater_tables/testdf.py index a41657f..ed67677 100644 --- a/greater_tables/testdf.py +++ b/greater_tables/testdf.py @@ -5,7 +5,8 @@ GPT from SJMM design. """ from datetime import datetime, timedelta -from itertools import cycle +from importlib.resources import files +from itertools import cycle, chain from math import prod from pathlib import Path from typing import Optional, Union @@ -81,21 +82,39 @@ class TestDataFrameFactory: self._index_namer = cycle(nwl) # read words and create cycler - p = Path(__file__).parent / 'words-12.md' - assert p.exists() - txt = p.read_text(encoding='utf-8') + data_path = files('greater_tables').joinpath('data', 'words-12.md') + with data_path.open('r', encoding='utf-8') as f: + txt = f.read() word_list = txt.split('\n') temp = word_list[:] random.shuffle(temp) self._word_gen = cycle(temp) # read tex expressions and create cycler - tex_list = pd.read_csv(Path(__file__).parent / - 'tex_list.csv')['expr'].to_list() + data_path = files('greater_tables').joinpath('data', 'tex_list.csv') + with data_path.open('r', encoding='utf-8') as f: + tex_list = pd.read_csv(f, index_col=0)['expr'].to_list() + # trim down slightly tex_list = [i for i in tex_list if len(i) < 50] random.shuffle(tex_list) self._tex_gen = cycle(tex_list) + self.simple_namer = { + 'd': 'date', + 'f': 'float', + 'h': 'hash', + 'i': 'integer', + 'l': 'large_float', + 'm': 'yr-mo', + 'p': 'path', + 'r': 'ratio', + 's': 'string', + 't': 'time', + 'v': 'extreme_float', + 'x': 'tex', + 'y': 'year', + } + # lengths of index (word count) sampled from: self.index_value_lengths = [1]*10 + [2] * 4 + [3] @@ -113,13 +132,14 @@ class TestDataFrameFactory: l log float (greater range than float) m year - month p path (filename) + r ratio (smaller floats, for percents) sx string length x t time + v very large range float x tex text - an equation y year - - Args: + Args: rows: Number of rows. columns: Column type spec (int for all float cols, or string type codes). index: Index level types (int for RangeIndex or string like 'ti'). @@ -168,7 +188,7 @@ class TestDataFrameFactory: ['d', 'f', 'i', 's3', 'l', 'h', 't', 'p'], size=self.rng.integers(3, 7)) missing = round(float(self.rng.uniform(0, 0.15)), 2) index = ''.join(self.rng.choice( - ['t', 'd', 'i', 's2'], size=index_levels)) + ['t', 'd', 'y', 'i', 's2'], size=index_levels)) col_index = ''.join(self.rng.choice( ['s', 's2', 's2', 's3'], size=column_levels)) return self.make(rows=rows, columns=''.join(col_types), index=index, col_index=col_index, missing=missing) @@ -182,18 +202,22 @@ class TestDataFrameFactory: else: col_types = self._parse_colspec(columns) # if col_index is an int then use all strings of that depth - if isinstance(col_index, int): - col_index_types = ['s'] * col_index + if col_index == 'simple': + col_idx = map(self.simple_namer.get, [i[0] for i in col_types]) + col_idx = pd.Index(col_idx, name='simple') else: - col_index_types = self._parse_colspec(col_index) + if isinstance(col_index, int): + col_index_types = ['s'] * col_index + else: + col_index_types = self._parse_colspec(col_index) + col_idx = self._make_index(col_index_types, len(col_types)) if isinstance(index, int): index = ['s'] * index else: index = self._parse_colspec(index) - print(index) + # print(index) # col names are a transposed index. df = pd.DataFrame(index=range(rows)) - col_idx = self._make_index(col_index_types, len(col_types)) for dt, c in zip(col_types, range(len(col_idx))): df[c] = self._generate_column(dt, rows) df.columns = col_idx @@ -210,15 +234,23 @@ class TestDataFrameFactory: return pd.Series([" ".join(self.word() for i in range(max_words)) for j in range(n)]) if dtype == 'f': return pd.Series(self.rng.normal(loc=100000, scale=250000, size=n)) + if dtype == 'r': + return pd.Series(self.rng.normal(loc=0.5, scale=0.35, size=n)) if dtype == 'l': # log float (greater range) return pd.Series(np.exp(self.rng.normal(loc=-4 / 2 + 4, scale=4, size=n))) + if dtype == 'v': + # log float (greater range) + sc = 5 + return pd.Series(np.exp(self.rng.normal(loc=-sc**2 / 2 + 10, scale=sc, size=n))) if dtype == 'i': return pd.Series(self.rng.integers(-1e4, 1e6, size=n), dtype='int64') if dtype == 'd': start_date = TestDataFrameFactory.random_date_within_last_n_years( 10) return pd.Series(pd.date_range(start=start_date, periods=n, freq='D')) + if dtype == 'y': + return pd.Series(random.sample(range(1990, 2031), n)) if dtype == 't': start_dt = datetime.now() - timedelta(days=365 * 2) return pd.Series([ @@ -347,7 +379,11 @@ class TestDataFrameFactory: for w, k in zip(level_value_lengths, level_choices)] x = [[next(j) for j in r] for i in range(rows)] names = random.sample(name_word_list, levels) - idx = pd.MultiIndex.from_tuples( - random.sample(x, rows), names=names).sort_values() + if levels == 1: + idx = pd.Index( + list(chain.from_iterable(random.sample(x, rows))), name=names[0]).sort_values() + else: + idx = pd.MultiIndex.from_tuples( + random.sample(x, rows), names=names).sort_values() assert idx.is_unique return idx diff --git a/greater_tables/tex_list.csv b/greater_tables/tex_list.csv deleted file mode 100644 index 2c5365c..0000000 --- a/greater_tables/tex_list.csv +++ /dev/null @@ -1,5804 +0,0 @@ -,expr -0,$\bar M$ -1,$m(1)=m_3=0$ -2,$X_2=2$ -3,$a=1$ -4,$\mathbf{M_{1}\Delta X}$ -5,$U < s$ -6,$n \le pN < (n+1)$ -7,"$\mathsf{TI,\ MON}$" -8,$\log(g')$ -9,$(.*?)\$ -10,$\rho(X)=\infty$ -11,$F(x-) = \lim_{t\uparrow x} F(t)$ -12,"$\mathsf{MON,\ TI,\ PH}$" -13,$\mathsf E_Q\left[\dfrac{X_i}{X}(X\wedge A)\right] + \delta A \mathsf E_Q[X_i/X\mid X > a]$ -14,$Y\succeq Z$ -15,$|S|$ -16,$\mathsf{CONVEX}$ -17,$\Pr(X < x)\le \Pr(X\le x)$ -18,$\mathsf E_{\mathsf Q}[\kappa_i(X)]$ -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,$\sigma^2 t$ -28,$\uparrow\uparrow$ -29,$F(x)=1-e^{-x/\mu}$ -30,$g(S(X))$ -31,$0<\rho\le 1$ -32,$\bar Q_{0}=a_{0}-\bar P_{0}$ -33,$s\downarrow 0$ -34,$X=\frac{1}{n}\sum_i X_i$ -35,$>(s_0/2^{n+1})2^n\bar q(s_0)=s_0\bar q(s_0)/2$ -36,$\mathsf E_Q[X]$ -37,$\rho(X)>\max(X) g(0+)=\infty$ -38,$\lambda\to\infty$ -39,$\mathsf{j}(a)=6$ -40,"$g(s)=w+(1-w)s, s>0$" -41,$\mathsf{TVaR}_{0.65}$ -42,$\Pr(X = q(p)) > 0$ -43,$c(S\cup\{i\})=c(S)+c(i)$ -44,$\mu(\{p_j\})$ -45,$q(Y)$ -46,$Z_A$ -47,$\mathcal D(X)\ge 0$ -48,$p=\text{Pr}[L^* > A]$ -49,$X_{t+dt}=X_t + \mu dt + \sigma dW_{dt}$ -50,$\mathsf E[X] + \pi\mathsf E[(X-\mathsf E X)^+]$ -51,$u(x)=-v(-x)$ -52,$g(x)=1$ -53,$F_{\mathbf{v}}(x)=s$ -54,${n}-X_2$ -55,$U_X > p$ -56,$b_i$ -57,$\rho(\nu Z) \le \nu\rho(Z)$ -58,$\Phi(x):=\int_{-\infty}^x \phi(t)dt$ -59,$\rho(U)=\mathsf E_\mathsf Q[U]$ -60,$U = A$ -61,$X\le l$ -62,$U_X < p$ -63,$g'(1-p) \frac{q\wedge \alpha}{q}$ -64,$rpq$ -65,$c>0$ -66,$Y=0$ -67,$1-p_0$ -68,"$(p, 1-g^{-1}(1-p))=(p,\hat p)$" -69,$\mathit{MV}(a)$ -70,$Z_4$ -71,"$\kappa_i(\mathbf{v}, x)$" -72,"$x=A,L,S$" -73,$c(S)=\rho(\sum_{i\in S} X_i)$ -74,$F:\mathbb{R}^n \to \XXX$ -75,$S_X(a)$ -76,$\mathsf E[X\mid t]$ -77,"$a,b=\pm 1/n$" -78,"$x_{1,i}, x_{2,i}$" -79,$1_{X>a}$ -80,"$\int_0^\infty -z(x)\,dF(x)=-1$" -81,$k\mapsto k\rho(X)$ -82,$\rho_g(X)=\mu+\lambda\sigma$ -83,$\hat q$ -84,$F_X^{-1}(V)=q_X(V)$ -85,$Y=\mathsf E[Z\mid\mathcal G]$ -86,$0\le\beta<1$ -87,$p>S(x^*)$ -88,$a\le X\le b$ -89,$P(x)=A(1_{X>x})=g(S(x))$ -90,$g(S)\Delta X'$ -91,$1<\lambda=k+f$ -92,$1./16=0.0625$ -93,"$\alpha>1,0\le\beta\le 1$" -94,$P=(1+r)\lambda\mathsf E[X]$ -95,$g''(s)\le 0$ -96,$S(x_{max})=0$ -97,$\{X=x\}$ -98,$\mathsf{TVaR}_p(X)=\TCE_p(X)=\mathsf E[X\mid X \ge \mathsf{VaR}_p(X)]$ -99,$\rho_g(X\wedge a)$ -100,$x\mathsf E[X_i/X\mid X>x]$ -101,$Z=(1-p)^{-1}1_{\tilde X>q_{\tilde X}(p)}$ -102,$\mathsf E_\mathsf{Q_r}[X_j]$ -103,$G(x)=\mathbb{Q}(\{\omega\mid X(\omega)\le x\})$ -104,"$X_{t-1,1}$" -105,$Z_1$ -106,"$X_{t,3}$" -107,$X_2(10)$ -108,$\mathsf E[X_1]=\mu$ -109,$X\le x$ -110,$r = (g(s)-s)/(1-g(s))$ -111,$\mathsf{TVaR}_1(X)$ -112,$\rho(Y)=\rho(X)g(p)=g(q)g(p).$ -113,$\mathbf{s_0}$ -114,$M(x)=g(S(x))-S(x)$ -115,$Y_{1}$ -116,$g(s)-s$ -117,$-U$ -118,$X_n(\omega)\to X(\omega)$ -119,$^{***}$ -120,$\bar S(a)$ -121,$\sum (X\wedge a)p$ -122,"$\{1,2,\dots, N\}$" -123,$D\rho_{X_g}(X_c)$ -124,$\mathbf s$ -125,$(g(s)-s)/(1-g(s))=\iota$ -126,"$P_X(a,b] = F(b)-F(a)$" -127,$k > 0$ -128,$\mathsf EPD_p(X)$ -129,$X_n\downarrow X$ -130,$\mathsf E[X\mid X>x]/\Pr(X>x)$ -131,$x\to \infty$ -132,$\Phi(Z(s))=s$ -133,$q^-(p) = \inf\ \{ x\mid F(x) \ge p\}$ -134,$Y(\omega_1)\le Y(\omega_2)$ -135,$v(A)\le v(B)$ -136,$\alpha_i(a) S(a)$ -137,$\ge \mathsf E[X]$ -138,$\hat{\tilde p}=1-g^{-1}(1-[1-g(1-p)])=p$ -139,$\pi(X)=\log(m_X(\alpha)) / \alpha$ -140,$E[s|W=t]$ -141,$S(x)\gg 0$ -142,$1-\beta_i(x)g(S(x))$ -143,$\mathsf E[X_i\mid X=q(p)]$ -144,$S_X(x)=\Phi(-(x-\mu)/\sigma)$ -145,$\pi(X) = \rho(X\wedge \alpha(X))$ -146,$a(\mathbf{v}) =\mathsf{VaR}_p(X(\mathbf{v}))= q_{\mathbf{v}}(p)$ -147,$\mathsf Q \in \mathcal Q$ -148,$a=D+S$ -149,"$\bar P_{t,0}$" -150,"$0, 8, 10$" -151,$Q(x)/(1-S(x))$ -152,$p=1/6$ -153,$\rho=\mathsf{TVaR}_{0.95}$ -154,$\mathsf E_{\mathsf Q}[X\mid \mathcal F]=\mathsf E[XZ\mid \mathcal F]/\mathsf E[Z\mid \mathcal F]$ -155,$f(S_t)=\log(S_t)$ -156,$\int_0^\infty xdF(x) =\int_0^\infty xf(x)dx$ -157,$u_j(x)$ -158,$f_{xx}=-1/S_t^2$ -159,$X$ -160,$t+2$ -161,$n\ge m$ -162,"$\{1+\lambda(f-\mathsf E f) \mid f\ge 0, \|f\|_q\le 1 \}$" -163,$|f|$ -164,$b$ -165,$g'(S(x))$ -166,$r_l$ -167,"$\rho(Y_{2,0})$" -168,$1+\iota^*=(1+\iota)(1+\tau)$ -169,$r_f/(1+r_f)$ -170,$L^r$ -171,$u(0)=0$ -172,$(ng)$ -173,$E[X|X>qp]$ -174,$\mathbf{S\Delta X'}$ -175,$1-g(S)$ -176,$a_{0}$ -177,$\rho_g(X \wedge a)$ -178,$\rho(0)=\rho(0 \times X)=0\times \rho(X)=0$ -179,$-\rho(-X)\le \mathsf E[X]$ -180,$\rho_g(X)$ -181,"$n={{n}}, p=1/{{p}}={{pf}}$" -182,$\mathsf E[Xe^{hX}]/\mathsf E[e^{hX}]$ -183,$\Delta Q_{gc}(a) = a_{gc}-P(X_{0}(a_{gc}))-a$ -184,"$\bar S_i = \sum_{j} X_{i,j}p_j$" -185,$\mathcal G\subset\mathcal F$ -186,$10^{-12}$ -187,"$x\in[0,\infty)$" -188,$F_0 = \bar P_{act}-\bar P = R-\bar M$ -189,$X_{-3}$ -190,$\bar\delta$ -191,$t>0$ -192,$\mathit{LGD}$ -193,$\mu_c$ -194,$\mathsf E_{\mathsf Q}[X]=\mathsf E[XZ]$ -195,$p<0.5$ -196,$a_h=2-a_l<2-b_l=b_h$ -197,"$F(p)=\mu([0,p])$" -198,$\lambda dt\to 0$ -199,$0 < p_0 < p_1 < 1$ -200,$p\mapsto g'(1-p)$ -201,$\omega=0.\omega_1\omega_2\dots$ -202,$BCD$ -203,$\beta_i(x)<\alpha_i(x)$ -204,$\nu=\nu(p)$ -205,$a_1 = a(Y_{1})$ -206,$\mathit{NPV}_{\infty}=2\times 2.5=5$ -207,$dG/dF$ -208,$M = P - \mu_U= 0.505$ -209,$H_k(X)=H_k(Y)$ -210,$l(p)$ -211,$\bar Q$ -212,$L_0^{l_1} + L_{l_1}^{l_1+l_2} = L_0^{l_1+l_2}$ -213,$X''$ -214,$\mathsf{VaR}_{0.7}(X)=2.439 > 2 \times 1.204=2.408$ -215,$\mathsf{CTE}^+$ -216,$\mathbf{p}$ -217,$0 < p < 1$ -218,$\displaystyle\int_0^\infty xg'(S_X(x))dF_X(x)$ -219,$\pi=0$ -220,$h(p)=1-g(1-p)=1-(1-p)^{1/3}$ -221,$\alpha(\mathsf Q)=\infty$ -222,$\gamma$ -223,$c\ge \mathsf E[cZ]$ -224,$x\in A$ -225,"$F_n,F$" -226,$\rho(\lambda X)=\lambda\rho(X)$ -227,$\mathbf{pK}$ -228,$\mathbf{\Delta S}$ -229,$A(1_{X>x})$ -230,$g(s)=(\iota+s)/(\iota+1)$ -231,"$\max(x, 0)$" -232,$x\mapsto x^{n}$ -233,$E[G]=1$ -234,$\Lambda = \dfrac{E( r_{U} ) - r_{f}}{\sigma_{r_{U}}}$ -235,"$\{90,\dots,99\}$" -236,$g(s) \ge s$ -237,$P = 3.103$ -238,$\mathsf{MONETARY}$ -239,$p(\omega)=0$ -240,$a(X_i;X) = \lim_{t\to 0} (\rho(X+tX_i)-\rho(X))/t$ -241,"$\mathbf{X'\,\Delta g(S)}$" -242,$\sigma_{U} = \sqrt{1 - 2p - p^{2}} = 0.973$ -243,$\sigma_A$ -244,$\beta$ -245,$\mathit{NPV}_1 = \bar Q - \bar Q = 0$ -246,"$X_4, X_5$" -247,"$g:[0,1]\to[0,1]$" -248,$\mathbf{Z_2}$ -249,$X+Y$ -250,$Y=1-X$ -251,$A\subset\Omega$ -252,$g'(s)\ge 1$ -253,$K_h(t):=k(h+t)-k(t)$ -254,$\rho(X_0)\ge \mathsf E[X_0 Z_\epsilon]$ -255,$\mathscr{E}_i$ -256,$\rho_2$ -257,$\mathsf E[X\mid \mathcal F']$ -258,$y_c$ -259,$1-F(q(p));\alpha)$ -260,$w(X)=1_{X>X_p}$ -261,$\delta=0$ -262,$q(0)$ -263,$|x|$ -264,$Y_n$ -265,$X_1+({n}-X_2)$ -266,$w=0.06405$ -267,$\sum_j Y_j = 0$ -268,"$P_X(a,b]=\mathsf P(X\in (a,b])=F(b)-F(a)$" -269,$e^{kx}S(x)\to\infty$ -270,"$f(\cdot, \omega)$" -271,$N_i$ -272,$\lambda S(x)$ -273,$\rho(X)\ge \mathsf E[X]$ -274,$t=2$ -275,$\rho=\mathsf E$ -276,$\Pr(X=1)=s$ -277,$0\le s\le 1$ -278,$\mathsf{Var}^+(X) = \int_{\mathsf E[X]}^\infty (x-\mathsf E[X])^2 f(x)dx$ -279,$\rho(X) \le 0$ -280,$x_{i-1}$ -281,$Y_{0}$ -282,$\infty-\infty$ -283,$\mathsf{j}(a) = \max\{j:X_j < a \}$ -284,$s \ne s^\ast$ -285,$\mathsf E_{\mathsf{Q}}[X] = \rho(X)$ -286,$\sigma_d^2$ -287,$P=L + \iota Q = \nu L + \delta a=L(1+\rho)$ -288,$\rho(X)=x_p$ -289,"$\mu=7.4, \sigma=1.9$" -290,$\bar q(s/2)\le 2\bar q(s)$ -291,$Q_1=0.125$ -292,"$D_n, D_n^*$" -293,$a>b_h$ -294,$\sum_t Q_t$ -295,$0\le \lambda < 1$ -296,$-u''(w)/u'(w)$ -297,$q(p)=-\log(1-p)\mu$ -298,$1=v+d$ -299,$n=2$ -300,$\mathsf E[X] + \pi\mathsf E[((X-\mathsf E[X])^+)^2]^{1/2}$ -301,$X=U$ -302,$X(\omega') = \sum_\omega X(\omega)1_\omega(\omega')$ -303,$a'$ -304,$U_i$ -305,"$\bar P_{0,1}$" -306,$g_i=u_i^{1/b} < u_i$ -307,$\mathbf{D^n\rho_{X\wedge 30}(X_1)}$ -308,$\rho(X\wedge a)=\bar P(a)$ -309,$E(X\wedge a)=\bar S(a)$ -310,$1-g(0^+)$ -311,$\alpha\not\equiv 0$ -312,"$[0,1]\times [0,1]$" -313,"$X_{i,j}\Delta g(S_j)$" -314,$c_i=\displaystyle\sum_{i\not\in S\subset\Omega}\dfrac{|S|!(N-|S|-1)!}{N!}\times$ -315,"$\mathit{MV}(X, a) = a - \rho(X\wedge a)$" -316,$u'(0)=1$ -317,$S(x)=0.1$ -318,$\mathsf E X + c{X-\mathsf E X}_p$ -319,$s=0.01$ -320,$\int_a^{a+y} g(S(x))dx$ -321,$\sum X_i(a)p$ -322,$\beta(x)\le \alpha(x)$ -323,$X_1=18$ -324,$\bar P_i(a)=\mathsf E_{\mathsf{Q}}[X_i(a)]=\mathsf E[X_i(a)g'(S(X))]$ -325,$g(s)$ -326,$Z'(s)=1/(\Phi'(Z(s)))=\sqrt{2\pi}\exp(Z(s)^2/2)$ -327,$D/L$ -328,"$S\,\Delta X$" -329,$a=11$ -330,$\log(1-1/n)<-1/n$ -331,$P_i=\mathsf E_\mathsf{Q}[X_i]$ -332,"$, which he describes as the standard way to obtain the $" -333,$\phi(p) = g'(1-p)$ -334,$\mathsf{VaR}_p(X_1+X_2)\le \mathsf{VaR}_p(X_1)+\mathsf{VaR}_p(X_2)$ -335,$P(X_i(a_{gc}))$ -336,$n$ -337,$t > 1/3$ -338,"$(lee.west |- lee.north)+(0,-2.5)$" -339,$g'(S(x))f(x)$ -340,$\mathsf{Var}(\pi)$ -341,"$D^n\rho_X(X_{i,\cdot})$" -342,$-x^2$ -343,$\Pr(\{\omega \})= 1/100$ -344,$X_n\to X$ -345,$r_f/(1+ r_f) = 0.0196$ -346,$\mathbf{f}$ -347,"$\mathsf{biTVaR}_{0,1}^w(X)=(1-w)\mathsf E[X]+w\sup(X)$" -348,$D\rho_{X_n}(X_c)$ -349,$\mathsf E[F_1] > \mathsf E[F_0]$ -350,$f_{opt} =(pb - q)/b$ -351,$\{n\mid X(n)\not =0\}$ -352,$\ge 1$ -353,$n-3$ -354,$Q = C + lg$ -355,"$(1-p, 1]$" -356,$\tilde X-X$ -357,$\Delta Q_{ro}(a)$ -358,$\lim_{x\to\infty}F(x)=1$ -359,$g^{-1}$ -360,$p=0.9973$ -361,$M=P-s$ -362,$f(x_i)$ -363,$a\mathsf E_{\mathsf{Q}}[...]$ -364,$\mathcal F'_0\subset\mathcal F_0$ -365,$M/EL$ -366,$a(c_1;X) = c_1$ -367,$\mathit{EER}$ -368,"$\delta = 34/39, \nu=5/39$" -369,$\rho(X) = \mathsf E_{\mathsf{Q}}[X] = \mathsf E_{\mathsf{Q}}[X\wedge a + (X-a)^+] = \mathsf E_{\mathsf{Q}}[X\wedge a] + \mathsf E_{\mathsf{Q}}[(X-a)^+] \le \rho(X\wedge a) + \rho((X-a)^+) = \rho(X)$ -370,$A(X)-B(X)$ -371,$\rho(X\wedge a) = \sum\rho(X_i(a))$ -372,$q(0)=0$ -373,$k=c/(e^c-1)$ -374,$\Lambda = \dfrac{M - K r_f}{\sigma_U}$ -375,$\nu < 1$ -376,$\rho_g(X) = \infty$ -377,$U''(x)<0$ -378,$M = P \mu_U = 0.3$ -379,$\bar S_i(a)$ -380,$y=$ -381,$g'(S(x))=v$ -382,$\rho(X)=\mathsf E_{\mathsf{Q}}[X]=\mathsf E_{\mathsf{Q}}[\sum_i X_i]=\sum_i \mathsf E_{\mathsf{Q}}[X_i]$ -383,$\bar Q(a)$ -384,$\mathsf{j}(a)=4$ -385,$\mathsf{TVaR}_{0.8}(X)$ -386,$L/P$ -387,$\bar P(a+da)-\bar P(a)$ -388,$t+d$ -389,$\mathsf E[X]=\int_0^\infty S(x)dx$ -390,$g(0+)M$ -391,$Z(\omega)\mathsf{P}(\omega)$ -392,$t > 0$ -393,$g'(S(x))f(x)dx$ -394,$\mathsf E[h(X_i)L(X)]$ -395,$\rho$ -396,$\hat p = F(x) = 1-g^{-1}(1-p)$ -397,"$\min(x_1,x_2)$" -398,${\mathsf{Q}}$ -399,$0=\rho(0)=\rho(X-X)\le \rho(X) + \rho(-X)$ -400,$f'_-(x)\le f'_-(y)\le f'_+(y)$ -401,$\mathsf E[X_i\mid X](\omega)$ -402,$\rho(X)=\mathsf E_\mathsf{Q}[X]=\mathsf E[XZ]$ -403,"$(x_{1,1}, x_{1,2})$" -404,$\sum_n 1/n$ -405,"$\displaystyle\int_0^a \alpha_i(x)S(x)\,dx$" -406,"$\beta(X,M)=\mathsf{cov}(X,M)\sigma_M^2$" -407,$X_{-1}$ -408,$\mathcal Q=\{\mathsf Q\mid \alpha(\mathsf Q)=0 \}$ -409,$A_i$ -410,"$a(X,p)$" -411,$r\lambda\mathsf{E}[X]$ -412,"$(s,\iota)$" -413,$a-L_0^a(X)$ -414,$\mathbf{X'}$ -415,"$[p_{-},p_{+}]$" -416,$y=x$ -417,$af$ -418,$M$ -419,$\mathsf{TVaR}_{p^\ast}$ -420,$\mu=0.107$ -421,$E(X_{-1}(a))$ -422,$g'(S_X)$ -423,$j > 0$ -424,$a=\sum_i a\alpha_i(a) = \sum_i\kappa_i(a)$ -425,$\mu=0$ -426,$\mathsf E[X\wedge 0]=0$ -427,$x>1$ -428,$F(p)=p$ -429,$X_i$ -430,$q_{\tilde X}$ -431,$\omega\in \Omega$ -432,"$\var(W)=\sum_{d\ge 0} \var(Y_{-d,d})$" -433,$Y_c=(Y\mid Y > y_c)$ -434,$(m_1-m_0)/s_1$ -435,$q_B(p)=\sup B$ -436,$M_1\Delta X$ -437,"$(a,b]$" -438,$\rho(m)=\rho(0)-m$ -439,$\mathbf v$ -440,"$\omega=(1,0,0,1,0,0,\dots)$" -441,$g(S(x))=1$ -442,$0 < s < 1/4$ -443,$r_h$ -444,$X\ge a$ -445,$Q$ -446,$p\delta_p$ -447,$y^{\ast}$ -448,$\nu=1/(1+\iota)$ -449,$\mu=0.1$ -450,$s_1=0$ -451,$p=0.4$ -452,$g(S_{X}(x))$ -453,$\bar F(a):=\int_0^a F(x)dx=a-\mathsf E[X\wedge a]$ -454,$\mathsf E_{\mathsf Q}[Y]=\mathsf E[Yg'(S(X))]$ -455,$m(t^\star)=3m/4$ -456,$n_s(1-g(s))$ -457,"$g,h:[0,1]\to [0,1]$" -458,$x_{(j)}-x_{(j-1)}$ -459,$\mathsf{SRM}$ -460,$v\in V_X$ -461,$a(X_i)$ -462,$A/L$ -463,$a_{2}$ -464,$\rho_g(X)=\bar P$ -465,$\arg \min_{q \in \mathbb{Q}} E_q[U(a)]$ -466,$\Pr(X\wedge a > a)=0$ -467,$X=X_1+X_2$ -468,$\mathbf{M_{2}\Delta X}$ -469,"$n=(0.702, 1.163)$" -470,$\sum_i$ -471,$\phi'(p)$ -472,"$(X_{1,j},\dots,X_{m,j})$" -473,$E(X\wedge a)$ -474,$1/6$ -475,"$\Omega=\{\omega_1,\omega_2,\omega_3,\omega_4\}$" -476,$\nu = 1/\lambda$ -477,$\alpha \le 1$ -478,$n\times m$ -479,$\mathsf{Q}$ -480,$\mathsf E[Z]\ge 1$ -481,${6 \choose 2}=15$ -482,$\sup(\lambda X)=\lambda \sup(X)$ -483,$P+Q=a$ -484,$k=2$ -485,$f(x) \to 0$ -486,$X=1$ -487,$v_1X_1(1)$ -488,$\mathsf E[Z_1]=\mathsf E[Y]$ -489,$\pi=\Pi/p\nu(p)$ -490,$\mathcal{N}_X(X_i(a))$ -491,$\mathcal B_p$ -492,"$(p, \mathsf E[X_i\mid X=q(p)])$" -493,$S(x)\le s^*$ -494,$q_A \le q_B$ -495,"$A_2=[\epsilon, \epsilon]$" -496,$X=\sum_i X_i$ -497,$K = A - P$ -498,"$(1-g(s), 1-s)$" -499,"$r=1,2,3,4$" -500,$0=x_0a\}}$ -502,$\mathsf{Pr}(E\mid A) = \mathsf{Pr}(E\cap A) / \mathsf{Pr}(A)$ -503,$P=a - v(a-L)$ -504,$S(M-)$ -505,"$X_{t+1,2}$" -506,$7$ -507,$\nu F(a)$ -508,$\mathcal D(X)=c\mathsf{TVaR}_p(X-\mathsf E[X])$ -509,$\mu_d$ -510,"$[0,1]\to[0,\infty)$" -511,$\mathsf{SA}$ -512,$Y\le X+\Vert X-Y\Vert$ -513,$Y_1$ -514,$X=g(Z)$ -515,$\mathsf E[X_ig'(S(X))]$ -516,$\sup X=\mathsf E[XZ]=\int XZ$ -517,$Y\mid Y > y_c$ -518,$a_1' = a_0-X_1$ -519,"$X_{t-1,3}$" -520,$\mathbf{B}(t)$ -521,$\mathsf Q\in\mathcal Q(X)$ -522,$g''<0$ -523,$g(w s_1 + (1-w)s_2) \le w g(s_1) + (1-w) g(s_2)$ -524,"$k=1,\dots,m$" -525,$S_t=S_0 X_t$ -526,$\mathsf E[X\wedge a] = (1-e^{-a\beta})/\beta$ -527,$\rho(-X)$ -528,"$[s_1,1]$" -529,"$[0, 1-p]$" -530,$T = \min\{ t:U(t)\le 0 \}$ -531,$X(\omega)=1-\omega$ -532,$1-g(S(x))$ -533,$x_0=q^-(p_0)$ -534,"$\beta_i(t\mathbf{v}, x)$" -535,$\lambda=g(\lambda_{obj})$ -536,"$[-2\pi, 2\pi]$" -537,$X(\lambda\mathbf{v})$ -538,"$\bar P_{t,0} = D\rho_{W_t}(Y_{t,0})$" -539,$a>1$ -540,$a=R+Q$ -541,$k-L_0^k$ -542,$p\ge 0$ -543,$\int g(S)$ -544,$\mathsf E[X\tilde Z]$ -545,$0\le f<1$ -546,"$I(q,p)=0$" -547,$1_{X < q(1-s)}$ -548,$g - s$ -549,$x_i=1$ -550,$x\ge q(1-s^*)=:x^*$ -551,$X\succeq Z$ -552,$\Pr(X < x) \le 0.1 \le \Pr(X\le x)$ -553,$0\le w\le 1$ -554,$\mathsf{CTE}$ -555,$\iota = \dfrac{\delta}{1-\delta}$ -556,$X=x$ -557,$g^{-1}(s)$ -558,$U(0)=2$ -559,$\alpha = 0.642.$ -560,$s>1-p$ -561,$M_i := \beta_ig-\alpha_iS$ -562,${}^2$ -563,$C_c$ -564,$ROL = a + b\ \mathit{EL} + c \ C(t)$ -565,$X_2=0$ -566,$M=\delta a'$ -567,$\alpha(x) S(x)>\beta(x) g(S(x))$ -568,$P(X_{-1}(a_{gc}))$ -569,$L = \text{E}[L^*\wedge A]$ -570,$c(S)$ -571,$A\cap B\subset B$ -572,$g(s) = 1 - (1 - s)/(1 + r_f + Ck(s))$ -573,$X-b\le 0$ -574,$f(x)=(\sqrt{2\pi}x)^{-1}\exp(-(\log(x)-\mu)^2/2\sigma^2)$ -575,$r_f=0$ -576,$\mathsf{VaR}_p(X)-f(\mathsf{VaR}_p(X))$ -577,$MX$ -578,$\mathsf E_{\mathsf Q}[X_i(a)]=\mathsf E[X_i(a)g'(S(X))]$ -579,"$\displaystyle\int_0^{1-g(S(a))} \kappa_i(q(1-g^{-1}(1-p)))\,dp + a\beta_i(a)g(S(a))$" -580,$X(\omega)=\exp(10 + 2\Phi^{-1}(\omega))$ -581,$g(s)=\nu s + \delta$ -582,$W$ -583,$1_A$ -584,$f=f_x=f_{xx}$ -585,$\wedge$ -586,$g'(s)$ -587,$a$ -588,$\mathbf{Q_{1}\Delta X}$ -589,$X\wedge l$ -590,"$X_{t-d,d}$" -591,$\alpha(\mathsf Q)=0$ -592,"$\mathsf E[W]=\sum_{d\ge 0} \mathsf E[Y_{-d,d}]$" -593,$\bar q_{X_1+X_2}(s) \approx \bar q(s/2)$ -594,$X_2$ -595,"$(s,g(s))=(0.2,0.36)$" -596,$P = \mathsf E[X] + \pi\mathsf E[X]$ -597,$ \& $ -598,$\inf_x\{ x + c{(X-x)^+} \}$ -599,$P(X\wedge a)$ -600,$1-g(S(a))$ -601,"$Y_{1,0}$" -602,$s=S(x)=\Pr(X>x)$ -603,$\nu^{\ast}$ -604,$A(\lambda X)=A(\lambda X)$ -605,$dF$ -606,$\downarrow\downarrow$ -607,$\rho_2(X_1)=1$ -608,$-X$ -609,"$[x_1, x_2]$" -610,$\kappa_i(x)$ -611,$\mathsf E[(X-m)(1_{U_X\ge p}-B)]\ge 0$ -612,$r-r_L$ -613,$\alpha_i(x) S(x)$ -614,$(g(s_0)-g_0)/s_0 = g'(s_0)$ -615,"$\mathbb{Q} = \left \{ q:I(q,p) \le I^* \right \}$" -616,$\rho=0$ -617,$\mathbf{D^n\rho_{X\wedge 30}(X_2)}$ -618,$s=f'(x_0)$ -619,$\rho(X)=\sup(X)$ -620,$g(0+)>0$ -621,$\inf_x \{ x + \alpha\mathsf E[(X-x)^+] + \beta\mathsf E[(X-x)^-] \}$ -622,"$s_g, s_b$" -623,$S(x)=e^{-\beta x}$ -624,$1000$ -625,$da>0$ -626,$u'''\ge 0$ -627,$0\le \lambda_1 \le 1$ -628,$P_X$ -629,$x_1+x_2=x$ -630,$=\mathrm{MV}(X\wedge a)$ -631,$M_i(x)+Q_i(x)=\alpha_i(x)F(x)$ -632,$\delta = \iota/(1+\iota)$ -633,$a_1'=a_0-X_1$ -634,$X=\sum X_i$ -635,$X\le b$ -636,$\delta=\iota/(1+\iota)$ -637,$(\delta_p - il_p)/(\nu_p-l_p)$ -638,$x=\mathsf{VaR}_p(X)$ -639,$1200/1800=0.667$ -640,$\sigma_0=\sigma_1$ -641,$a(f + (1-f)/q) -1$ -642,$g \cdot dX$ -643,$\beta_i(a)/\alpha_i(a) < 1$ -644,$Q_{1}\Delta X$ -645,$X_g$ -646,"$X=X(x_1,\dots,x_n)=x_1X_1 + \cdots + x_nX_n$" -647,$s\leftrightarrow 1-s$ -648,$\mathcal Q_i(X)$ -649,$\mathbf{\Delta g(S)}$ -650,$V_j$ -651,$X'=X\wedge a$ -652,$20+8t$ -653,$\Delta_{2}$ -654,$\alpha_{2}$ -655,"$(1,1)$" -656,$4$ -657,"$Q_{i,j} = M_{i,j}/\iota_j$" -658,$L^\infty$ -659,$f(1)=1$ -660,"$0,10,40$" -661,$\rho(X+c)=\rho(X)+c$ -662,$H[Y_j]$ -663,$Z=(1-p)^{-1}1_A$ -664,$\mathsf E[p]=1$ -665,$\beta_i(x)g(S(x))$ -666,"$A_3=[0, \epsilon-k]$" -667,"$dx,dt,ds$" -668,$\mathsf{TVaR}_{0.95}$ -669,$f(\omega)\ge 0$ -670,$\beta=0.57$ -671,$(X\wedge a)$ -672,$X < a$ -673,$\lambda<1$ -674,"$X_{0,1}$" -675,$\omega'\not=\omega$ -676,$X_0< X_1 < \dots < X_m$ -677,$P = \mathsf E[X] + \pi \mathsf{SD}(X)$ -678,$\tilde X_1 + \tilde X_2 = X_1 + X_2$ -679,"$\{f' \in L_q \mid f'=1+f-\mathsf E f,\ \|f\|_q\le c \}$" -680,$\mathsf{VaR}\_p(X\_0)$ -681,$-(1-s)g''(1-s) + g(0+)\delta_1 + \sum_s s(g'(s-)-g'(s+))\delta_{1-s} + g'(1)\delta_0$ -682,$a>a_{ro}$ -683,$g'(0)=\infty$ -684,$(X\wedge a)/X$ -685,$\mathsf E[r] = \mu_r = M/K = 0.132$ -686,$\rho_g(V)= g(F(x^*)) \ge F(x^*)=\mathsf E[V]$ -687,$P_g\ll P_X$ -688,$Z\le (1-p)^{-1}$ -689,$F_g$ -690,$\bar P(x)$ -691,$\Pr\{a-X\le 10\}$ -692,$d^*=(\log(A/L) + (r_h-\mu_L + \sigma^2/2))/\sigma\sqrt{t}$ -693,$\mathsf E[\kappa_i(X)g'(S(X))]$ -694,"$g(s)= \displaystyle\int_0^s g'(t)\,dt = (s/(1-p)) \wedge 1$" -695,"$(s_j=0,g_j>0)$" -696,$P'<\rho(W_1\wedge a_1)$ -697,$\mathsf E[\mathsf E[X_iZ\mid X]]\not=\mathsf E[\mathsf E[X_i\mid X]\mathsf E[Z\mid X]]$ -698,$\mathsf E[S_t]=e^{\mu t}$ -699,$\mathsf{COHERENT}$ -700,$\Delta g(S_0)=1-g(S_0)$ -701,$\rho_g(V)$ -702,$X_t$ -703,$\mathsf E_{\mathsf Q}$ -704,$X_1+X_2=X=x$ -705,$v_1$ -706,$X_n\uparrow X$ -707,$\Pr(X_i>\bar q(s))=s$ -708,$m=1$ -709,$a\ge 10$ -710,$\gamma=0.633$ -711,$r=0.038$ -712,$1000(1+t)$ -713,$f(0)=0$ -714,$p(\nu(p)-l(p))$ -715,$B(X)$ -716,$h(0.9)/0.9 = 0.76$ -717,"$\int_{[0,p]} \dfrac{\mu(dt)}{1-t}$" -718,$\mathsf{TVaR}_{0.5}(X_1)=9$ -719,${}^nS(t)$ -720,$Q(a)=\nu F(a)$ -721,$\rho(X_i)$ -722,$S(x_5)$ -723,$h_x$ -724,$Y\le 0$ -725,$(I/a + U/R)$ -726,$v=1/1.1<1$ -727,$0 < r \le 1$ -728,$\{ p \mid q^-(p) \le x \}=\{ p \mid p \le F(x) \}$ -729,"$(s,g(s))$" -730,$R_f=0$ -731,$\alpha_i'(x)>0$ -732,$\lim_{s\downarrow 0} g_\tau(s) = \tau / (1+\tau)$ -733,$\mathit{NPV}_1=0$ -734,$X\wedge a\Delta S$ -735,$\mathsf{TVaR}_{0.75}(X_2)=90$ -736,$K = A-P$ -737,$A\in\mathcal F'$ -738,$\le 0$ -739,$Z'(g(s))g'(s)=Z'(s)$ -740,"$\sum_i a(X_i, p^*)=a(X)$" -741,$a_{gc}:=\mathit{VaR}_{p}(X)=18000.0$ -742,$v=1/(1+i)$ -743,"$\alpha, \beta, \kappa$" -744,$S_{X\wedge a}(x) = S_X(x)$ -745,$W_0=Y_{0} + W_1$ -746,"$s_0, s_1, s_2$" -747,$AR$ -748,$S_j:=S(X_j)$ -749,$f'_-$ -750,$\gamma=\Pr(X>\mathsf E[X])$ -751,"$ is average invested assets, equal to $" -752,$\mathsf{VaR}_{0.99}(X_2)=100$ -753,$q(F(x))$ -754,$a_i$ -755,$q=ps_g$ -756,$X_1=t$ -757,$X>Y$ -758,$M=g(S)-S$ -759,$X=1800$ -760,$g_2(s)=s^{0.5}$ -761,$xS(x)|_0^\infty$ -762,$x_h(1-p)$ -763,$v(\varnothing) =0$ -764,$\nu+\delta=1$ -765,$\rho_i$ -766,$\mathsf{SSD}$ -767,$X_i\dfrac{X\wedge a}{X}$ -768,$\varnothing$ -769,$\mathbf{X'p}$ -770,$r(X)=g'(S(X))$ -771,$X\wedge d$ -772,$1_{X>x_1}$ -773,"$\int g(S(x))\,dx$" -774,"$c(1,3)-c(3)$" -775,$\mathsf E[(X-\mu)^n]$ -776,$0.5$ -777,$A(\lambda X)=\lambda A(X)$ -778,$c=(1-\alpha)^{-1}$ -779,$\mathsf E[X_{d}]$ -780,$\mathbf{Z_1}$ -781,$M_2dX$ -782,"$(\mathsf x*0.65, 3.75*2)$" -783,$\mathit{EGL}_{ro}(a)=P(X_{-1}\wedge a) - P(X_{-1}\wedge a_{ro}) \ge 0$ -784,$2\le x\le 8$ -785,$\mathsf{CTE}_p$ -786,$\mathsf E_\mathsf{Q}\left[\dfrac{X_i}{X}(X\wedge a)\right] + \tau a \mathsf E_\mathsf{Q}[X_i/X\mid X > a]$ -787,$f(\mathsf{VaR}_p(X))$ -788,$X_n=X$ -789,"$Y_{t',d}$" -790,$D\rho_X(X_i)=D\rho_i = x_i\dfrac{\partial\rho}{\partial x_i}$ -791,$a(X)\le a(Y)$ -792,$g'(s)<1$ -793,$\mathsf E[(A-L)^+]/\mathsf E[L]$ -794,$\beta > \alpha$ -795,$\bar\iota=\iota$ -796,$\int_a^{a+y} S(x)dx$ -797,$0.125 \cdot 8 = 1$ -798,$\rho_c(X)=\mathsf E[X]+c\sigma(X)$ -799,$P = \mathsf E[Xe^{\pi X}]/\mathsf E[e^{\pi X}]$ -800,$\bar\delta(x)$ -801,$\mathsf EPD$ -802,"$\mathsf E[(X_i-\mathsf E X_i)(X-\mathsf E X)]/\mathsf{SD}(X)=\mathsf{cov}(X_i,X)/\mathsf{SD}(X)$" -803,$P_{act}-P$ -804,"$\rho(X, p^\star)=a(X)$" -805,$q(0.75)$ -806,$\mathbf{t+3}$ -807,$s=S_X(y)$ -808,$\rho l = \iota C$ -809,$\mathbf{a=1}$ -810,$\alpha(1-\alpha)(1-s)^{\alpha-1} + \alpha\delta_0$ -811,$Y_s$ -812,$\eta\nu$ -813,$(g_j-s_j)/(1-g_j)$ -814,$Z=g'(S_X(x))$ -815,$\Pr(X=x)=0$ -816,$\Delta S_5$ -817,$\mathsf E[X^k] \le \mathsf E[Y^k]$ -818,$F(x)$ -819,$D=(X-a)^+$ -820,$\sigma^2/2$ -821,$i=1$ -822,$h(p)\le p$ -823,$b = g/(1-g)$ -824,"$d=d(X_1,\dots,X_n)$" -825,$X=\max(X)$ -826,$v$ -827,$F(q(p))=p$ -828,$g(0+)=\mu(\{1\})$ -829,$X_i(a)$ -830,$p=0.999$ -831,$m\ge 1$ -832,$X_1(a)$ -833,$\Delta_s=g'(s-)-g'(s+)$ -834,$\mathsf Q \ll \mathsf P$ -835,$k/n$ -836,"$X_{t-1,2}$" -837,$d=1-v$ -838,"$f(t)=a(tx_1,\dots, tx_n)=ta(x_1,\dots, x_n)$" -839,$\partial a/ \partial v_i$ -840,$-g''$ -841,$g'(1)=0$ -842,$P(a)=g(S(a))\ge S(a)$ -843,$x\mapsto x$ -844,$x^{\ast}=\mathsf{VaR}_p(X)$ -845,"$(1,\dots,1)$" -846,$Y=-X$ -847,$\lim_{y\downarrow x} f(y)$ -848,$\iota=0.1$ -849,$A_Y = 2.155$ -850,$\Pr(S_t > a)=\Pr(X_t > a/S_0)=1-\Phi\left([\log(a/S_0)-(r-\sigma^2/2)t]/\sigma\sqrt{t} \right)=\Phi(d^*-\sigma\sqrt{t})$ -851,$g(S)=1$ -852,$X:=Y$ -853,$0.05$ -854,$\mathsf E[p] \le 1$ -855,$\Pr(E)$ -856,$xS(x)\vert_0^\infty =\lim_{x\to\infty} xS(x)=0$ -857,$k!$ -858,$602.6 billion and converted to net premium based on $ -859,$q(p)\phi(p)\times dp$ -860,$B_t$ -861,$ABC$ -862,$\lim_{x\to-\infty}F(x)=0$ -863,$\mathsf E[X^n]$ -864,$a = 0.6565$ -865,$\mu(ds)$ -866,$\mathsf E[YZ]$ -867,$p<\infty$ -868,$X_n(2/3)$ -869,$X_s$ -870,$x=q(p)$ -871,$q_X(p)=\mu+\sigma z_p$ -872,"$Y_{0,t}:=\sum_{d>t} X_{0,d}$" -873,$Z_{a}(a)$ -874,$\le p$ -875,$dx$ -876,"$G=\mathrm{cl}\{\, (\mathsf E_\mathsf{Q}[X_i], \mathsf E_\mathsf{Q}[X]) \mid \mathsf Q\in\mathcal Q \, \}$" -877,$A = 8.14864$ -878,$L(X)=1_{X=x_p}(X)/f(x_p)$ -879,$\mathbf{\mathsf E[X_i(a)]}$ -880,$\rho(X+tY)\ge \mathsf E_{\mathsf Q_X}[X+tY]$ -881,"$\{0, 8, 10\}$" -882,$P = \mathsf{TVaR}_\pi(X)$ -883,$w=w f(1)=w f(1)+(1-w)f(0) \le f(w 1 + (1-w)0)= f(w)$ -884,$Z_\mathit{lin}$ -885,$X_t=\mu t + \sigma W_t$ -886,$\alpha S$ -887,$\tilde X_1 = X_1 + \mathsf E[X_2]$ -888,$f(x)=\sin(x)$ -889,"$\Omega=\{\omega_1,\dots,\omega_n\}=\{\text{Ada}, \text{Bernhard}, \dots, \text{Zeno} \}$" -890,$\alpha(1+fg/(1-g))$ -891,$s > s_1$ -892,$t=2/3$ -893,$\int_0^s \phi(1-t)dt$ -894,$H_k(X) \le H_k(Y)$ -895,$\mathsf E[X_i/X \mid X > x]$ -896,$X\preceq Y$ -897,"$\beta_H:=\mathsf{cov}(r_H, r_M)/\var(r_M)$" -898,$1-1/c$ -899,$0 < s < 1$ -900,$\infty$ -901,$q(\hat p)$ -902,$\mathbf{\iota=M/Q}$ -903,$Z=g'(S(X))$ -904,$P=L/(1+R_L)$ -905,$n+1=N$ -906,$\rho(X_n)\not\to \rho(X)$ -907,$X'\Delta g(S)$ -908,${X}_p=\mathsf E[|X|^p]^{1/p}$ -909,$\bar M(a) = \bar P(a) - \mathsf E[X\wedge a]$ -910,$\beta_i(X_4)$ -911,$s>0.2$ -912,$\mathsf E[X1_{U_X\ge p}]\ge \mathsf E[XB]$ -913,$q_{X+c}(p)=c+q_X(p)$ -914,$X=q(F(X))$ -915,$\Pr[X > a]$ -916,$0.2 < s < 1$ -917,$t>0.5$ -918,$0 \le t \le 1$ -919,$\mathbf{Z_6}$ -920,"$\mathsf{TVaR}_p(X(x_1,x_2))=(x_1 + x_2)\mathsf{TVaR}_p(Y)$" -921,$X_1\le X_2\implies a(X_1;X)\le a(X_2;X)$ -922,$\rho_c(X)=\mathsf{TVaR}_{0.8}(X)=8.5$ -923,$\Pr(Y_m > y) = 1 - (1 - \Pr(X > y))^n$ -924,$V_X$ -925,$\mathbf{a_2'}$ -926,$\rho(1)=1$ -927,"$(3,2)$" -928,$a_2'$ -929,$\mathsf Q(A)=\mathsf E[Z1_A]$ -930,$x_{i-1}\le x'_i\le x_i$ -931,$\mathsf{TVaR}_p(X)=(12(0.9-p) + 2.5)/(1-p)$ -932,$V$ -933,"$D^f\rho_{W_t\wedge a, W_t}(Y_{0})$" -934,$\mu$ -935,$y=(\log(x)-\mu)/\sigma$ -936,$\sup(X)<\infty$ -937,$+\infty$ -938,$p=F(x)=\Pr(X\le x)$ -939,$\mathsf E[N]=2.0$ -940,$F^{-1}(p)=q(p)$ -941,$\mathbf{\max a}$ -942,$Z(y_j)$ -943,$\bar Q_{d}=a_{d}-\bar P_{d}$ -944,$\rho(X_n) \uparrow \rho(X)$ -945,$S(a)$ -946,"$\mathsf E[(X-a)^+]= p\,\mathsf E X$" -947,$(1-g(s))(1-q)$ -948,$\Delta \mathit{MV}_{gc}(a)$ -949,"$X_1,\dots,X_m$" -950,$da1_{X>x}$ -951,$g_1F$ -952,"$\bar P_{0,t}:=\rho(Y_{0,t})$" -953,$x_0+x_1+x_2$ -954,$\rho(X)=\mathsf E_{\mathbb{Q}}[X]=\mathsf E_{\mathbb{Q}}[\sum_i X_i]=\sum_i \mathsf E_{\mathbb{Q}}[X_i]$ -955,$\bar S(a)=\displaystyle\int_0^a S(x)dx$ -956,$S(X_j)>0$ -957,$f(s)=\alpha(1-\alpha)(1-s)^{\alpha-1}$ -958,"$1_A:\Omega\to \{0,1\}$" -959,$g(S(\infty))=0$ -960,"$\alpha_i(a) = \dfrac{\sum_{j:X_j>a} (X_{i,j}/X_j)p_j}{\sum_{j:X_j>a} p_j}$" -961,"$P_i,M_i, Q_i$" -962,$C'_i$ -963,$l_i$ -964,$A(c)=c$ -965,$I$ -966,$X\preceq_m Y$ -967,"$\rho(X),\rho(Y)\le 0$" -968,"$X_{0,t}$" -969,$a-X\le 0$ -970,$m_3=0$ -971,$\mathsf E[X_ie^{kX}]/\mathsf E[e^{kX}]$ -972,$\rho(W_1\wedge a_1 \wedge a_1')$ -973,"$\mathsf{CONVEX,LI}$" -974,$1_{X>x}$ -975,$\tau a$ -976,$E\in\mathcal F$ -977,$a/Q = 1 + R/Q$ -978,$\mathsf E_{\mathbb{Q}}[X_i]$ -979,$\mathsf E[X_2]=22.75$ -980,$F_Y$ -981,$X(T(U))$ -982,$\le 1/(1-p)$ -983,$\kappa_j(x)\approx \mathsf E[X_j]$ -984,$0\le \lambda\le 1$ -985,$r\times 1$ -986,$P = \mathsf E[X] + \pi \mathsf E[((X-\tau)^+)^p]^{1/p}$ -987,"$(0,1,2,3,4,5,6,7,8,9)$" -988,"$\mathsf E[X_i\,\mathsf E[Z\mid X]]$" -989,"$(3,1)$" -990,$\mathcal F_0\subset\mathcal F_1\subset \cdots\subset \mathcal F_N$ -991,$\dots$ -992,$R_C$ -993,$k = 3.3 s^{0.82}$ -994,"$X_n=1_{\{0,1,\dots,n-1\}}$" -995,$X(\omega)=x$ -996,$R_L$ -997,$D\rho_X(X_i)=\mathsf E_{\mathsf{Q}_X}[X_i]$ -998,$c(\varnothing)=0$ -999,$\mathsf E[Z\mid X]=Z$ -1000,$Q_i$ -1001,$X=10$ -1002,$P(a)$ -1003,$\rho(X)\ $ -1004,$U(1)=1$ -1005,$g(S_{X\wedge a'}(x))$ -1006,"$ occurs, i.e., those with the value 1 in the $" -1007,$\Delta X_m$ -1008,"$(0,0,0,0,0,0,0,5,0,5)$" -1009,$D=1$ -1010,$\rho(X)=\max_i \rho_i(X)$ -1011,$\mathsf E[1_{U < s}]=s$ -1012,$a_h=2-a_l$ -1013,$0 < \alpha \le 1$ -1014,"$i=1,\dots,N$" -1015,$-norm equal to 1. (Note that $ -1016,$g(0.1)=\sqrt{0.1}=0.316$ -1017,$\rho_g(X)=\mu+\lambda$ -1018,$0.5 + U/2$ -1019,$\mathsf E[Y_i\mid X_n]$ -1020,$-g'(S(x))f(x)$ -1021,$1-(p_R+p_Y)$ -1022,$\sum\mathsf E[C_i^2]=\sum m_i(1+v_i^2)$ -1023,$\Pr(X < x)\le 0.99 \le \Pr(X\le x)$ -1024,$(1+\epsilon)v_1$ -1025,$\Vert X-Y\Vert := \sup_{\omega\in\Omega} |X(\omega) - Y(\omega)|$ -1026,"$(\partial a/\partial x_1)(tx_1,tx_2)= 3tx_1 /a(tx_1, tx_2) = 3x_1 /a(x_1, x_2)=\partial a/\partial x_1$" -1027,$\beta_i(x)=\mathsf E_\mathsf{Q}\left[ \dfrac{X_i}{X}\mid X > x\right]$ -1028,$\mathsf{TVaR}_p$ -1029,$U\le u$ -1030,$-dS=f(x)dx$ -1031,$\mathbf{X_{1c}}$ -1032,$\mathsf{COM}$ -1033,$1_\omega$ -1034,$\alpha=0.5$ -1035,"$\mathsf{biTVaR}_{p_0,p_1}^w(X)=\mathsf{TVaR}_{p^\ast}(X)$" -1036,$\mathbf{x}=\mathbf{1}$ -1037,$\beta_i(x)/\alpha_i(x)$ -1038,$d^*$ -1039,$\mathsf E[q(U_X)1_{U_X\ge p}]$ -1040,$X_2-X_1$ -1041,$q_{X_i}(p)=\Phi^{-1}(p)$ -1042,$Z_a$ -1043,$\mu(\{p_0\}) = 1-w$ -1044,$Z(\omega)> 0$ -1045,$r=0.045$ -1046,$\sup_\mathsf{Q} (\mathsf E_\mathsf{Q}[X] - \alpha(Q))$ -1047,$h(s)=s^m$ -1048,$X\_{1}$ -1049,$cv=0.557$ -1050,$du = -g'(S(x))dF(x)$ -1051,$g(0)=r_0$ -1052,$M_i=\beta_ig(S)-\alpha_iS$ -1053,$j$ -1054,$g-s$ -1055,$\max_\mathsf{Q} \mathsf E_\mathsf{Q}[X]$ -1056,$w_u=1+c(1-\gamma)$ -1057,$a:=\rho(X)$ -1058,$g\Delta X \wedge a$ -1059,$\Pr(X < x)$ -1060,$M=rQ$ -1061,"$X,X_i$" -1062,$Y_c$ -1063,$($ -1064,$S_{X\wedge a}$ -1065,$\rho(1_A)$ -1066,$g_4(s)=s^{0.9}$ -1067,"$(4,1)$" -1068,$f(L)=0$ -1069,"$(-\mathsf x, 2)$" -1070,$E[(X-qp)^+]$ -1071,$I/a + U/R > 0$ -1072,$g'(S(x))=(1-p)^{-1}$ -1073,$a\le 1$ -1074,$a-b_h<0$ -1075,$\mathsf{TVaR}_p(X) := (1-p)^{-1}(T_1+T_2)/N$ -1076,$0.417 < p < 0.791$ -1077,$1-\nu p$ -1078,$\sqrt{0.9}=0.95$ -1079,"$c(1,2)-c(2)$" -1080,$\lambda X$ -1081,$r_A$ -1082,$\dfrac{\iota}{1+\iota} p$ -1083,$\rho_g(X)=452.98$ -1084,$a < \infty$ -1085,$\alpha(\mathsf Q) = 0$ -1086,$\mathsf{VaR}_p$ -1087,$Q_t=\rho(\mathsf E[X\mid t])$ -1088,$X_1=X_2=Y$ -1089,$P = 1.5$ -1090,$S(x)dx$ -1091,$L_a^{a+y}$ -1092,$\mathsf E_{\mathsf Q}[Y]=\mathsf E[YZ]$ -1093,"$\mathsf P,\mathsf Q_2,\dots,\mathsf Q_r$" -1094,$F(t)$ -1095,"$P((1+\epsilon)v_1, v_2, a+da)=P^a((1+\epsilon)v_1, v_2)$" -1096,$\mathsf E[Z(X)]=1$ -1097,$\Pr(X\le x)$ -1098,$\sum_\omega \mathsf Q(\omega) =\mathsf E[Z] / \mathsf E[Z]=1$ -1099,$\tau=0+d$ -1100,$Y=f(X)$ -1101,$a_1 = 5.991$ -1102,$\{\mathsf E_{\mathsf Q}[X_i] \mid \mathsf Q\in\mathcal Q(X)\}$ -1103,$X=X\wedge a + (X-a)^+$ -1104,"$s\wedge p=\min(s,p)$" -1105,$a=30$ -1106,$1_{U_X\ge p}$ -1107,$g(s)\ge s$ -1108,$\mathsf Q(A)>0$ -1109,$\mathsf{COH}$ -1110,$D f(x_0)$ -1111,$r_H$ -1112,$d=iv$ -1113,$U>p$ -1114,$p<0.1$ -1115,"$\mathsf{biTVaR}_{0,0.9}^{0.3138}$" -1116,$\mathsf{TVaR}_0(X)=\mathsf E[X]$ -1117,$(g(s)-s)/(1-s)$ -1118,$P/L$ -1119,$j=7$ -1120,$\mathsf E[XZ(X)]$ -1121,$\mathbf{v}'$ -1122,$0< p <1$ -1123,$\mathsf P(A)=0$ -1124,$X_{-1}=x$ -1125,$\mathbf{X'\Delta S}$ -1126,$x=q^-(p)$ -1127,$(\lambda S(x))$ -1128,$Q=1-g(S)$ -1129,$1^+$ -1130,$X \wedge a$ -1131,$\delta(s)$ -1132,"$[x, y]$" -1133,$\mathsf E X + c\mathsf E[((X-\tau)^+)^p]^{1/p}$ -1134,$\mathsf E_{\mathsf{Q}}[X] \le \rho(X)$ -1135,$\omega>0$ -1136,$K = \mathsf E[\exp (\lambda x)]^{-1}$ -1137,"$t \in (0,1)$" -1138,$1=1_{X\le a}+1_{X>a}$ -1139,$\rho(X_n)$ -1140,$Y\equiv 1$ -1141,$(dt)^{3/2}$ -1142,$m_0=0$ -1143,$\iota=\dfrac{M}{Q}$ -1144,$X\circ f$ -1145,$g(s)=s^\lambda$ -1146,$P\ge (\mathsf E[X] + \iota a)/(1 + \iota)$ -1147,"$\mathsf{MON,\ NORM}$" -1148,$\sum_i \kappa_i'(x)=1$ -1149,$ax$ -1151,$p'\ge p$ -1152,$\mathsf E[Xe^{\pi X}]/\mathsf E[e^{\pi X}]$ -1153,$\bar P_i(a)$ -1154,$Np=67.45$ -1155,$B$ -1156,"$X_n,X$" -1157,$(1-p)\gamma(dp)$ -1158,$X'=X$ -1159,$0.33$ -1160,$(1-p)/(p(\nu_p-l_p)^2)$ -1161,$\mu_U = 1-p = 0.995$ -1162,$j+1$ -1163,$q_{X+Y}=q_X+q_Y$ -1164,$\mathsf E[X_1\mid X=20]= 14$ -1165,$\mathsf Q_{X}$ -1166,"$u_{X,r}(p)=\psi_{X,r}^{-1}(p)$" -1167,$a_i=\mathsf E[X_i\mid X\ge \mathsf{VaR}_{p^**}(X)]$ -1168,$L_a^{a+da}=L_0^{a+da}-L_0^a$ -1169,$P\approx \mathsf E[A(1)] + k\mathsf{Var}(A(1))/2$ -1170,$c(X(\mathbf{v}))=c(\mathbf{v})$ -1171,$\mathsf{MRM}$ -1172,$^{*}$ -1173,"$s=0,1$" -1174,$\mathsf E[X] + \pi \mathsf{Var}(X)$ -1175,"$X(x,-x)\equiv 0$" -1176,$F(x):=\mathsf{P}(X\le x)$ -1177,$\max X$ -1178,$q=q(p)$ -1179,$1/m>0$ -1180,"$B\subset [0,1]$" -1181,$g(S(x))=1-p$ -1182,"$f:(0,1)\to (0,1)$" -1183,"$p_0,\dots, p_{n'}$" -1184,$X_1-X_0$ -1185,$\bar P = \bar S + \bar M$ -1186,$\rho(\tilde X)=\rho(X) + \rho(\tilde X-X)$ -1187,"$u\in D_n=\{ u \mid u^{(k)} \ge 0, k=1,\dots,n-1, u^{(n-1)}\text{ nondecreasing} \}$" -1188,$l(\mathbf X)=(\sum_i X_i^2)^{0.5}$ -1189,$\Pr(\cup_i E_i)=\sum_i \Pr(E_i)$ -1190,$s=S(x)$ -1191,$s_j < 1$ -1192,$\bar S(a+da)-\bar S(a)\approx \bar S'(a)da = S(a)da$ -1193,$\mathbf{\mathsf{VaR}_p(X_1+X_2)}$ -1194,$t-1$ -1195,$\mathcal D(X+c)=\mathcal D(X)$ -1196,$\tilde X_2 = X_2 -\mathsf E[X_2\mid X_1]$ -1197,$\mathsf E[X_i\mid X](x)$ -1198,"$s\in[0,1]$" -1199,$p=1-1/n$ -1200,$X(\omega)=X_1(\omega)+X_2(\omega)$ -1201,"$S(x) + d\,F(x) + (\delta^{\star}-d)\sqrt{S(x)F(x)}>1$" -1202,$S(x_#4)$ -1203,$\mathcal V$ -1204,$1-e^{-\lambda S(x)}$ -1205,$\beta>1$ -1206,$X_n=n1_A$ -1207,$d-1$ -1208,$g(S(x))\approx S(x)\approx 1$ -1209,$t_0$ -1210,$D_1$ -1211,$\mathcal E$ -1212,$\bar P=\mathsf E[W]+\lambda\sigma(W)$ -1213,$s\uparrow 1$ -1214,$Mg(0+)$ -1215,$S/L\ge A/L-1$ -1216,$\succeq$ -1217,$2\mathsf{VaR}_p(X_1) - \mathsf{VaR}_p(X)$ -1218,$Y = X + Z$ -1219,$)$ -1220,$1-(1-s)^m$ -1221,$p\to 1$ -1222,$\mathsf P(T^{-1}(A))=\mathsf P(A)$ -1223,$-zf(x)=(d/dx)g(S(x))$ -1224,$\rho_X(X_i)$ -1225,$n\Pr(Y > y_c)$ -1226,$P=\rho(X \wedge a)$ -1227,$s=0.02$ -1228,$F(q^-(p_0))=p_+>p_0$ -1229,$\Delta g(S)$ -1230,$\Delta$ -1231,"$\mu=10, \sigma=2$" -1232,$t=3$ -1233,$0\le q\le 1$ -1234,$\mathbb{Q}_k$ -1235,$L_a^y$ -1236,$X=30$ -1237,$l=\sum_i l_i$ -1238,$f:I\to\Omega$ -1239,"$f(x,y)=x^3/(x^2+y^2)$" -1240,$\Pr(X>\mathsf{VaR}_p(X))=1-p$ -1241,$g(0+)=\delta$ -1242,$S_i(x)$ -1243,$h=2$ -1244,$g'_\tau(s) = g'(s)/(1+\tau)\ge 0$ -1245,$\mathsf E_Q[X_i\mid X]=\mathsf E[X_i\mid X]$ -1246,$\mathbb{Q}(\{\omega_i\})=0$ -1247,$t \ne 0$ -1248,$\rho=\mathsf{TVaR}_p$ -1249,$\tilde M_i(a) = \bar M_i(a)-\tau_i a_i$ -1250,$a>10$ -1251,$x^+$ -1252,$A(-X)=-A(X)$ -1253,$g(s)=s^{1/3}$ -1254,$\{X = x\}$ -1255,"$p_1,p_1$" -1256,$0\le x \le 1000$ -1257,$U_s$ -1258,"$\{1,2,3\}$" -1259,$\kappa_i(x)\approx x -\sum_{j\not=i} \mathsf E[X_j]$ -1260,"$i=0,1$" -1261,$\mathsf{Var}(\Pi)$ -1262,$\mathsf E[Z \tilde X]$ -1263,$\mathsf{TVaR}_{0.75}(X_1)=10$ -1264,$g_k(s)=1-(1-s)^k$ -1265,$\mathsf E_\mathsf{P}[X_j]$ -1266,$g'(S_{X}(X))$ -1267,$(8t+10t)/2$ -1268,$\mathbf{\Sigma}$ -1269,$g(S(x_i-))=g(S(x_{i}))$ -1270,$\nu + \delta = 1$ -1271,$1-1/n$ -1272,$\Omega_1$ -1273,$\Delta g(S_j)$ -1274,$x\leftrightarrow u(x)$ -1275,$\eta=0.49$ -1276,$X=q(p)$ -1277,$\log(\mathit{EER}) = \gamma + \eta \log(\mathit{PFL}) + \beta \log(\mathit{LGD})$ -1278,$Y=-X_0$ -1279,$g'\circ S_{X\wedge a}$ -1280,$\mathsf E_{\mathsf{Q}}[X\wedge a] = \rho(X\wedge a)$ -1281,$s_2 - s_1$ -1282,$\mathbf{X_1(a)}$ -1283,$y < q_A(p)$ -1284,$\Delta\mathit{MV}$ -1285,$g'(s+)$ -1286,$w=E[w|s=0.1]=0.06405$ -1287,$f'_+$ -1288,$f_x=1/S_t$ -1289,$S(X(\omega))$ -1290,$\rho(X\wedge a)=\mathsf E[(X\wedge a)Z(X)]$ -1291,$\rho_2(X)$ -1292,$L$ -1293,$\partial a/\partial x_1=3x_1/a$ -1294,$g(s)\ge 0g(0) + sg(1)=s$ -1295,$T:\Omega\to\Omega$ -1296,$t>x$ -1297,$L^1$ -1298,$(a-X_{\mathsf{j}(a)})$ -1299,$\alpha=d_i$ -1300,"$A=\mathbb Q\cap [0,1]$" -1301,$Q_1\Delta X$ -1302,$f(L) \ge 0$ -1303,$\rho(X_1)=\rho(X_2)$ -1304,$\rho(\tilde X)$ -1305,$F_3$ -1306,$\mathsf{CTE}_p(X)$ -1307,$1_{U < s}$ -1308,$Q_2dX$ -1309,$p\to S\to gS \to \Delta gS$ -1310,$\Delta Q_{gc}(a)$ -1311,$g(s) = s^a$ -1312,$d^\ast = 1-(1-g^\ast)/(1-s^\ast)$ -1313,$g(s)=g(1-p)$ -1314,$\alpha_{Cat}$ -1315,"$\mathsf E[Y_{0,0}]+\lambda\sigma(Y_{0,0})=58.129$" -1316,"$D^f\rho_{X\wedge a,X}(X_i(a))$" -1317,$h=1+\lambda(f-\mathsf E f)$ -1318,$r_f$ -1319,$X = \sum_i X_i$ -1320,$x_3(S(x_2)-S(x_3))=x_3f(x_3)$ -1321,$\preceq_2$ -1322,$\Delta \bar Q$ -1323,$m_0$ -1324,$Q(a)=1-g(S(a))$ -1325,$\mathsf E[X\wedge a] = \dfrac{k}{\beta-1}F(a)-\dfrac{a}{\beta-1}S(a)$ -1326,$\bar P_i(x)$ -1327,$S\subset T$ -1328,$f(L)$ -1329,$D_n$ -1330,$R_M$ -1331,$Z_5$ -1332,$q^-=q^+$ -1333,$-\int xd(g\circ S)=\int g(S(x))dx$ -1334,$\tilde Z = \mathsf E[Z\mid X]$ -1335,$y\not=z$ -1336,$1-g_\tau(s)$ -1337,$\rho L = \iota Q$ -1338,$\rho(aX+bY) = a\rho(X) + b\rho(Y)$ -1339,$W \equiv T_{(1)}=min_k{T_k}$ -1340,$\lambda \rho(X)$ -1341,$Y=h(Z)$ -1342,$y^{\ast}-x^{\ast} < \epsilon$ -1343,$U/4$ -1344,$D\rho(X_0)=\{Z \}$ -1345,$X > A$ -1346,$1=\mathsf Q(\Omega)\not=\sum_n \mathsf Q(\{n\})=0$ -1347,$\sigma=0.25$ -1348,$\Delta \mathit{MV}_{gc}(a)$ -1349,$\Phi'(Z(s))Z'(s)=1$ -1350,$\bar q_{X_1+X_2}(s) \ge \bar q(s/2)$ -1351,$K = 5.029$ -1352,$1_{X>x_2}$ -1353,$S\Delta X$ -1354,$\bar{\mathbf M}$ -1355,$F_X(x):=\Pr(X\le x)$ -1356,"$G(X_1,\dots, X_n)'=(Y_1,\dots, Y_r)'$" -1357,$\mu_L=r_L +\pi$ -1358,$X=20$ -1359,$\mathsf P(X=\max(X))=0$ -1360,$r_a+r_l$ -1361,$D\rho_X(X_i) \ge \mathsf E[X_i]$ -1362,$S_1$ -1363,$\mathbf X / l(\mathbf X)$ -1364,"$w, 1-w$" -1365,$\mathcal D$ -1366,$-\rho(-X)\le \mathsf E[X] \le \rho(X)$ -1367,"$ (range.south)+(0, -1) $" -1368,$\mathsf{P}$ -1369,$X=\sum_{i=1}^n X_i$ -1370,$X_j=x$ -1371,$X_0=\mathsf E[X]$ -1372,$\Omega_a$ -1373,$\Pr(X > \mathsf{VaR}_p(X))$ -1374,$S_j$ -1375,$\beta>\alpha$ -1376,"$f(W_t,t)$" -1377,$\mathsf E[W\tilde X] \le \rho(\tilde X)$ -1378,$\mathsf E[X_ih(X)]=\mathsf E[\mathsf E[X_ih(X)\mid X]]=\mathsf E[\mathsf E[X_i\mid X]h(X)]=\mathsf E[\kappa_i(X)h(X)]$ -1379,$p\le S(x^*)$ -1380,$\phi(t)$ -1381,$S(x)=p$ -1382,$U/2$ -1383,$\int Zd\mathsf P=1$ -1384,$1+t$ -1385,$a_{1}'$ -1386,$r_h=-0.025$ -1387,"$(x_A,g(S(x_A)))$" -1388,$p(1-\nu(p))=p\delta(p)$ -1389,$\beta_i$ -1390,$1-S$ -1391,$p_{\mathit{pr}}$ -1392,$g(0+)=\lim_{t\downarrow 0} g(t)\ge 0$ -1393,$0\le \pi\le 1$ -1394,$Z=Z(X)$ -1395,$r_a$ -1396,"$\int_a^\infty g(S(x))\,dx$" -1397,$\prec X$ -1398,"$\{2, 3\}$" -1399,"$(0,1,2,3,4,8,8,8,8,9)$" -1400,$n\ge 3$ -1401,$=\mathrm{MV}(a-X)^+$ -1402,$g(s)/(1-g(s))$ -1403,$\Pr(X=y_j)$ -1404,"$E[Y\,dG/dF]$" -1405,$g(S_X(x))=1$ -1406,$q(p)=\inf\{x \mid F_X(x)\ge p \}$ -1407,$\mathit{NPV}_{\infty}$ -1408,$E[X_1 | X]$ -1409,$\beta_D$ -1410,$\sigma=0.1246$ -1411,$F(x;\alpha)$ -1412,$D_\infty$ -1413,"$(1,3)$" -1414,"$X, Y$" -1415,$q^-(p)=\mathsf{VaR}_p(X)$ -1416,"$i=1,\ldots,n$" -1417,$P/l-1 =\rho= \iota Q / l = \iota(C/l + g)$ -1418,$c(x)=\rho(\sum_i x_iX_i)$ -1419,$\omega_1=0$ -1420,$E_{\mathsf{Q_X}}$ -1421,$M_{2}\Delta X$ -1422,$S(x_#5)$ -1423,"$(\nu,\nu,\dots,\nu,\nu+10\delta)$" -1424,$\mathcal F'\subset \mathcal F$ -1425,$\Delta S_0$ -1426,$a_{d}$ -1427,$\tilde X(x) = x$ -1428,$A/L<1$ -1429,$X_n(\omega)$ -1430,$\bar P^a(\mathbf{v})$ -1431,$\int_0^1 f(s)ds = 1 - \alpha < 1$ -1432,$\mathcal{N}_{X}(X_i(a))$ -1433,$a-P$ -1434,$\mathsf{Q}(A)\le g(\mathsf{P})(A))$ -1435,$d=0$ -1436,$x\mapsto g(s)+g'(s)(x-s)$ -1437,$\mathsf{VaR}_{1-s}$ -1438,$\mathbf{Q_2\Delta X}$ -1439,$\rho_g(X\wedge a)=(\bar L + ra)/(1+r)$ -1440,$(a-X)$ -1441,$\omega'=1$ -1442,$1/6 + 2 /6 + 4/2 + 9/6$ -1443,$\rho_a(kX) = \rho(kX \wedge a(kX)) = \rho(kX \wedge ka(X)) = \rho(k(X\wedge a(X))) = k\rho(X\wedge a(X)) = k\rho_a(X)$ -1444,"$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 $" -1445,$(a_1'-a_1)^+$ -1446,$X\wedge a=\sum_i X_i(a)$ -1447,"$Q,\iota,M$" -1448,$\int_0^a g(S(x))dx$ -1449,$p>p^*$ -1450,$\{X\ge q(p)\}=\{X \ge 12\}$ -1451,$g(1)-g(0)=1$ -1452,$g(s)(1-q)$ -1453,$(g(S(x^-)-g(S(x)))/(S(x^-)-S(x))$ -1454,"$\sum_j X_{i,j}(a)\Delta g(S_j)$" -1455,"$\mathsf{P}(a,b]=b-a$" -1456,"$j=1,\dots,d$" -1457,$Z(\omega)=0$ -1458,"$\mathsf E[X_{t,d}\mid \mathcal F_0]=\mathsf E[X_{t_d}]$" -1459,$l(p)= \nu(p)-\sqrt{(1-p)/p}$ -1460,$\int_0^1 g(s)ds - 0.5$ -1461,$\rho_{g}$ -1462,$\prec_1$ -1463,$\mathsf E[X\wedge a] + d(a - \mathsf E[X\wedge a])$ -1464,$\epsilon v_1$ -1465,$\mathsf E X +\lambda {(X-\mathsf E X)^+}_1$ -1466,"$\phi(p) = (1-\alpha)^{-1}1_{[1-\alpha, 1)}(p)$" -1467,$S(M)=0$ -1468,$c\ge 0$ -1469,$\mathbf{\rho(X)}$ -1470,$p_1=1$ -1471,$\mathsf E[Z\mid X>a]=g(S(a))/S(a)$ -1472,"$x_{1,i}+x_{2,k(i)}$" -1473,"$(x_1, x_2)$" -1474,$\alpha_i'(x) \to 0$ -1475,"$\displaystyle\int_0^{F(a)} \kappa_i(q(p))\,dp + a\alpha_i(a)S(a)$" -1476,$\bar P(a)$ -1477,$q(U)$ -1478,$\iff\rho$ -1479,$F_g(x)$ -1480,$Q(a) = 1-P(a)= \nu F(a)$ -1481,$\mathsf P(\{x\})=0$ -1482,$1_V$ -1483,$R_Q$ -1484,$\mathcal D:=\{X\mid X\preceq_2 Y \}$ -1485,"$X_{j,i}$" -1486,$g(1-F(x))=1-\tilde p$ -1487,$p'$ -1488,$\beta_i(a)g(S(a))$ -1489,"$A\subset[0,\infty)$" -1490,$X_1/X$ -1491,$x$ -1492,$q_{\mathbf{v}}(p)$ -1493,$\rho(X) = \rho(X\wedge a) + \rho((X-a)^+)$ -1494,$1\not\in S$ -1495,$F(x):=\Pr(X\le x)$ -1496,$X_n=1/n$ -1497,$\rho_g(X)=\mu/b>\mu$ -1498,$\mathsf{VaR}_{0.99}(X)=1100$ -1499,$<1$ -1500,$S(X)$ -1501,$a=kP+Q$ -1502,$X\wedge a = \sum X_i(a)$ -1503,$A\subset \{ Z=0 \}$ -1504,$Z\circ T_i$ -1505,$a(X_i; X)\le \sup(X_i)$ -1506,"$Y_{1,2}$" -1507,$M_{2}$ -1508,$x \le 300$ -1509,$\implies c_i\ge 0$ -1510,$F(x)=1-s$ -1511,$h(0.9) = 1-\sqrt{0.1} = 0.684$ -1512,"$\alpha = 1, \kappa = 0.2$" -1513,$(8)(0.25)+(10)(0.25)=4.5$ -1514,$W_0=0$ -1515,$Q=S$ -1516,$X^{(d)}_i(a):=(X_i-d)^+$ -1517,${\mathcal{M}}$ -1518,$X = X_1 + X_2$ -1519,$V_t$ -1520,"$\mathsf P(\{ \omega\mid X(\omega)=X(\omega_0), \omega \le \omega_0 \})$" -1521,$\mathsf E[X_i\sum_j w_jZ_j]=\sum_iw_j\mathsf E[X_i Z_j]$ -1522,$m_3 := m_2$ -1523,$g(s)=(s+\iota)/(1+\iota)$ -1524,$\iota = \delta/\nu$ -1525,$r_X= r_f + \beta_X(r_m-r_f)$ -1526,$\mathsf E[X]+k\var(X)$ -1527,$Z\circ T\in \mathcal Q$ -1528,$\rho(X_1) \ge P_1$ -1529,$a-X$ -1530,$P(A)=1-p$ -1531,$10+0$ -1532,$\phi'(p)=-g''(1-p)>0$ -1533,"$\mathsf{TI,\ MON,\ SA,\ PH}$" -1534,$\Delta_1=a_1'-a_1$ -1535,$\mathit{RDS}_k$ -1536,$t=-ln(1-p)$ -1537,$C_i=c_i$ -1538,$\lim_{s\to 1} (g(s)-s)/(1-s) = \lim_{s\to 1} 1-g'(s)$ -1539,$\rho_i(X)$ -1540,$v(A\cap B) + v(A\cup B)\le v(A)+v(B)$ -1541,$\mathsf{TVaR}_{0.5}$ -1542,"$X_1, X_2$" -1543,$\rho=\sup$ -1544,$m_i$ -1545,$g'(s) = as^{a-1}$ -1546,$k\in\mathbb{R}$ -1547,$q(p)=F^{-1}(p)$ -1548,$E_4$ -1549,"$\psi_{X, m}(u)$" -1550,$f=(1-p)^{-1}1_A$ -1551,$<0$ -1552,$\mathbf{M}$ -1553,$X=X_1 + X_2$ -1554,$G=g$ -1555,$-q_{-Y}^-(1-p)$ -1556,"$\rho(\lambda P,\lambda R,\lambda a)=\lambda\rho(P,R,a)$" -1557,$1+bf$ -1558,$Y_j$ -1559,$\mathbf{\iota}$ -1560,$dP_g/dP_X$ -1561,$S(x)=d/dx(\mathsf E[X \wedge x])$ -1562,$M=g-S$ -1563,$FL$ -1564,$\int gS(x)dx=\int xg'(S(x))P_X(dx)$ -1565,$\mathit{MV}_{ro}(a) = a-\rho(X_{-1}\wedge a)$ -1566,$n+1$ -1567,$g'(s)=\phi(1-s)$ -1568,$X_i(a)\not= X_i\wedge a_i$ -1569,"$\mathbf{g(S)\,\Delta X}$" -1570,$\lim_{x\downarrow x_0} F(x)=F(x_0)$ -1571,$F(w) = 1-\exp(-w)$ -1572,$\mathbf{X_1/X}$ -1573,$\WCE_p(X) = \mathsf{TVaR}_p(X)$ -1574,$B_i^c$ -1575,$\Omega_a := \{\omega\in \Omega \mid (X\wedge a)=a \}$ -1576,$1/10$ -1577,$\mathsf E_{\mathbb{Q}}[(X-a)^+] \le \rho((X-a)^+)$ -1578,$Q_i(a)$ -1579,$Q>0$ -1580,$r_h-\mu_L$ -1581,$\mathbf{Z_8}$ -1582,$\mathsf E_{\mathbb{Q}}[X_i \mid X=x] = \mathsf E[X_ig'(S_X(X)) \mid X=x]/\mathsf E[g'(S_X(X)) \mid X=x] = \mathsf E[X_i \mid X=x]$ -1583,$s_j$ -1584,$\beta g(S)$ -1585,$\ge 0$ -1586,$E[u_j(W_j - X_j)]$ -1587,$\phi((x-\mu)/\sigma)/\sigma$ -1588,$X_{2}$ -1589,$E[X \wedge x+a]-E[X \wedge a]$ -1590,$\mathsf E[Z \mid X]$ -1591,$\mathsf{TVaR}_p(X)=25$ -1592,$X-(1+r)T$ -1593,"$\int_0^1 a'(tx)\,dt=\int_0^1 a(1)\,dt = a(1)=a'(x)$" -1594,$\mathsf E_{\mathsf Q}[X_i \mid X]$ -1595,$ (#1)+(#3) $ -1596,$g=F_G^{-1}(p_{\mathit{pr}})-1$ -1597,$X_{2}(a)$ -1598,$g(s)=s(1-s)$ -1599,$\mathsf{VaR}_{0.995}(U)-0.5=0.495$ -1600,$\kappa_2(10)$ -1601,$\lambda < 0$ -1602,$\mathit{ROE}(s) = fs/(1-f-s)$ -1603,$p_i$ -1604,$X_m$ -1605,$g(t) = r_0 + (1-r_0)t$ -1606,"$Y_{1,1}$" -1607,$s > s^*$ -1608,$\theta$ -1609,$g(s)=s^{1/2}$ -1610,$X\wedge a=a$ -1611,$\mathsf E[X_1Z]$ -1612,$\Pr(X\in A)=0$ -1613,$P=l + \iota Q$ -1614,$X-Y$ -1615,"$\mathbf{X\,\Delta S}$" -1616,$\log(\mathit{ROL}) = a + b \log(\mathit{EL}) + b X$ -1617,$q_{X_1+X_2}(p) \le q_{X_1}(p) + q_{X_2}(p)$ -1618,$k\ge 0$ -1619,$\Phi'(z)=\phi(z)$ -1620,$c^{-1}\log\mathsf E[e^{cX}]$ -1621,$q^-(p)=\inf \{ x \mid F(x) \ge p \}$ -1622,"$g'(s)=(1-p)^{-1}1_{[0,1-p]}$" -1623,$X(\mathbf{v})=\sum_i v_iX_i$ -1624,$s_0$ -1625,"$t=0,1$" -1626,$d^\ast = 2g^\ast-1$ -1627,"$(s_1,g(s_1))$" -1628,$g(s)=s$ -1629,$0\times\infty=0$ -1630,"$\bar Q_{0,t}:=a_{0,t}-\bar P_{0,t}$" -1631,$\mathbf{M_{1}}$ -1632,$q_X(p)$ -1633,$\rho_c$ -1634,$M(a)=g(S(a))-S(a)$ -1635,$\rho(X_n)=\rho(0)=0$ -1636,$c(S)=g(\Pr(S))$ -1637,"$\displaystyle\int_0^a \kappa_i(x) f(x)\,dx + a\alpha_i(a)S(a)$" -1638,$\mathsf E_\mathsf{Q}[X\mid A]$ -1639,$\mathbf{Z_\mathit{lin}}$ -1640,$\bar\iota = 0.12$ -1641,$\mathsf P(X=\sup(X))=0$ -1642,$\alpha_2(98)=0.9$ -1643,$p\delta(p)/p\nu(p)=\iota(p)$ -1644,$g_\tau(1)=1$ -1645,"$H(A, L, t)=LH(A/L, 1, t)$" -1646,$g_2F$ -1647,$X=X_0+X_1$ -1648,"$697.6 billion in 2016, $" -1649,$\bar Q=53.031$ -1650,$\mathsf E_{\mathsf{Q}}[\tilde X-X] \le \rho(\tilde X-X)$ -1651,$c(S\cup\{i\})=c(S\cup\{j\})$ -1652,$\mu_L=0.03$ -1653,$Q_0=\rho(V_0)=\rho(X_1)$ -1654,$g'(s-)=g'(s+)$ -1655,$\mathsf E[Xw(X)]/\mathsf E[w(X)]$ -1656,$U = X + Y$ -1657,$B=B(p)$ -1658,$\mathbf{gS}$ -1659,$9+1=10+0$ -1660,$n=67$ -1661,$a(X(\mathbf{v}))$ -1662,$v(\Omega)=1$ -1663,$p_Y=1-p_R$ -1664,"$p\,da$" -1665,$t\mapsto \rho(X+tY)$ -1666,$Y^S$ -1667,$g'(S(x)) = (1-p)^{-1}1_{x >\mathsf{VaR}_p(X)}$ -1668,$E_{\mathsf{Q_X}}[X_i(a)]$ -1669,$\rho(X)\le \rho(Y)$ -1670,$1-\tilde p=g(1-p)$ -1671,$\max_\mathsf{Q} \mathsf E_\mathsf{Q}[X] - \alpha(\mathsf Q)$ -1672,$R_f-R_L>0$ -1673,$\rho_c(X)$ -1674,$X^\star$ -1675,$X\wedge a'$ -1676,$a(W)=\mathsf E[W] + 4\sigma(W)$ -1677,$0.675=(6.258/7.613)^2$ -1678,$q<1$ -1679,$\alpha_1(90) = (0.0909 \times 0.0625 + 0.1 \times 0.0625)/(0.0625+0.0625)=0.0955$ -1680,$\mathsf E(X)=$ -1681,$g(Q)$ -1682,$\mathsf E[B]=p$ -1683,$\Pr(X< x)\le 0.75 \le \Pr(X\le x)$ -1684,"$X_2=0,0,0,0,1,1,1,4,24, 500$" -1685,$\bar P_i$ -1686,$\Pr(U\le \omega)=\omega$ -1687,$a(X)=3.769$ -1688,$\tilde X_2 = X_2 - \mathsf E[X_2]$ -1689,"$\rho(P,R,a)=\sqrt{(0.4P)^2+(0.25R)^2+(0.1a)^2}$" -1690,$\exp(x)$ -1691,$X_j$ -1692,$\mathsf E[X \mid X \ge q^+(p)]$ -1693,"$(anch.west |- lee.north)+(-0.125,0.25)$" -1694,$g(s)=20s\wedge 1$ -1695,$f(x_p)$ -1696,$\mathsf E_{\mathsf{Q}}[\cdot]$ -1697,$\Pr(X>0)$ -1698,$\{X=q_X(p) \}$ -1699,$EL(a)$ -1700,$30-11=19$ -1701,$x\in\mathbb{R}$ -1702,$p_R<0.5$ -1703,$\mathsf E[\Pi]$ -1704,$r=16$ -1705,$g(S(a))\ge S(a)$ -1706,$\beta_{1}$ -1707,$\beta_i(a)$ -1708,$N=71$ -1709,$\rho(X_1+X_2)\le \rho(X_1)+\rho(X_2)\le 0$ -1710,$a_{gc}$ -1711,"$1 between any of the layers, then $" -1712,$\mathcal{M}$ -1713,"$\sum_i \rho(X_i, p^*)=a$" -1714,$\int_0^\infty g(S(x))dx$ -1715,$t=1-p$ -1716,$\rho'(x)=U'(-x)$ -1717,"$\mathbf{D^f\rho_{X\wedge 30,X}(X_1)}$" -1718,$x=\mathsf{VaR}_{0.99}(X)$ -1719,$\alpha_i(x)-\kappa_i(x)/x=0$ -1720,$x\mapsto |x|$ -1721,$n\ge 2$ -1722,$D$ -1723,$\sigma(X)>\sigma(Y)=0$ -1724,$D\rho_X(X_2)$ -1725,$L_d^l(x)$ -1726,$\beta_1g(S)dX$ -1727,$\mathsf E[X_i]=14$ -1728,$p_j=\Delta S_j$ -1729,$x1$ -1732,$E[s|t]$ -1733,$\mathsf E[X_0]=80$ -1734,"$C(a)=\int_a^\infty S(x)\,dx + \tau a$" -1735,$\mathsf E[e^{hX}] = \exp(h\mu+\sigma^2h^2/2)$ -1736,$\beta=d^\ast-d$ -1737,$-0.00002$ -1738,$y=0$ -1739,$L_X$ -1740,$\lambda=0.5$ -1741,$g(s)=(1-p)^{-1}s\wedge 1$ -1742,$\rho(X) = \mathsf E[X] + \lambda \mathsf E[(X-\mathsf E[X])^+]$ -1743,$\sum M_i\Delta X$ -1744,$1\le x \le 2$ -1745,$f(x) \ge f(x_0) + f'(x_0)(x-x_0)$ -1746,$\mathsf E[Z_A]=1$ -1747,"$\Pr(A)\in [0,1]$" -1748,"$1,\dots,m$" -1749,$X\in L_p$ -1750,$x=1.5$ -1751,$u^{iv} \le 0$ -1752,$\mathbf{d}$ -1753,$1_{X > x}$ -1754,$S_{X_i}$ -1755,$xS(x)\to 0$ -1756,$(a-X)^+=a-(X\wedge a)$ -1757,"$j=0,1,\dots, n'$" -1758,$\mathsf{P}(\omega)$ -1759,$\beta_i(a)g(S(a))=\mathsf E_{\mathsf{Q}}[(X_i/X) \mid X>a]g(S(a))=\mathsf E_{\mathsf{Q}}[(X_i/X) 1_{X>a}]$ -1760,$\bar Q=a-\bar P$ -1761,$SdX$ -1762,$\sqrt{p}$ -1763,$L^p$ -1764,$\mu<0$ -1765,"$X_{i,i}(a)=X_{i,j}\dfrac{X_j\wedge a}{X_j}$" -1766,$\mathscr{M}$ -1767,$ so $ -1768,$1/4$ -1769,$\lambda\ge 0$ -1770,$d\bar S(a)/da=S(a)$ -1771,$(\alpha S)'(x)=-\kappa_i(x)f(x)/x$ -1772,$\sup f=1$ -1773,"$X_{t-2,3}$" -1774,$\beta_i(x)/\alpha_i(x) 0$ -1776,$\bar\nu a$ -1777,$\mathbf{\mathsf E[X_i\wedge a_i]}$ -1778,$a(1-f)$ -1779,$X\succeq Y$ -1780,$p_R$ -1781,$s_1 < s_2$ -1782,$1$ -1783,$\mathbb{Q}$ -1784,$a\le \dfrac{P-S}{\iota} + P\approx \dfrac{P-\mathsf E[X]}{\iota} + P$ -1785,$a_x=1/\lambda$ -1786,$\mathbf{\mathsf{VaR}_p(X_1)}$ -1787,$f:\mathbb{R}\to\mathbb{R}$ -1788,"$I=[0,1]$" -1789,$\rho(X)\le 0$ -1790,$B(0.5)$ -1791,$\mathsf E_G(X)$ -1792,"$i=1,2,\dots$" -1793,$r_D=1-D/L$ -1794,"$\min(X,a)$" -1795,$\Delta S$ -1796,$ is the total return on invested assets and $ -1797,$X(\psi)=X(\omega)$ -1798,$X_j\ge 0$ -1799,$\mathcal{S}$ -1800,"$i=1,\dots, n$" -1801,"$\rho_{a,\tau}(X)=v\rho(X\wedge a) + da$" -1802,"$(brR15 |- lee.south)+(-0.125,-0.25)$" -1803,$n\ge N$ -1804,$x_1 \wedge x_2$ -1805,$X_s = X_{s_1} + X_{s_2}$ -1806,$0$ -1859,$x_0 \in \{ x \mid F(x) \ge p \}$ -1860,"$\bar P(\mathbf{v}, a)$" -1861,$x_2(S(x_1)-S(x_2))=x_2f(x_2)$ -1862,$r_h=0$ -1863,"$S=[0,2\pi]$" -1864,$\mathcal E(X)=\mathsf E[(p X^+ + (1-p)X^-)/(1-p)]$ -1865,$gn$ -1866,$\mathbf{\Delta gS}$ -1867,$p=F(x)$ -1868,$\bar S_i(a) := \mathsf E[X_i(a)]$ -1869,$1/g'(s)$ -1870,$z(x)$ -1871,$-\sigma^2u''(w)\approx -cu'(w)$ -1872,$S(a+x)=d/dx(\mathsf E[X \wedge (a+x)-X \wedge a)$ -1873,$r=0.1$ -1874,$\beta_1$ -1875,"$i=1,\dots, M$" -1876,$S^{-1}(g_i)$ -1877,$X_t:=\mathsf E[X\mid \mathcal F_t]$ -1878,$\mathsf E_\mathsf{Q}[X\wedge a]$ -1879,$d =\iota/(1+\iota)$ -1880,$Z=g'(S_X(X))$ -1881,$i\not\in S$ -1882,$\mathsf E[v^T] \ge v^{\mathsf E[T]}$ -1883,$s+\delta p$ -1884,"$X_1=1+cos(X_3), X_2=1-cos(X_3)$" -1885,$(1+r)\lambda \mathsf E[X]$ -1886,$(1-p)^{-1}1_A$ -1887,$\rho=P/L-1=M/L$ -1888,$F(X)$ -1889,$\lambda=$ -1890,$\mathsf E_{\mathsf{Q}}[X]$ -1891,$\rho_g(X)=352$ -1892,$\rho(X)=\mathsf E_\mathsf{Q}[X]$ -1893,$x=0.5$ -1894,$A = -\log(p) = 5.298$ -1895,$\rho(X_{-1}\wedge a)$ -1896,$g'(S)dF(x)$ -1897,$-norm by integrating against a function with $ -1898,$(X-d)^+$ -1899,"$x=1000,2000,\ldots$" -1900,$\int_0^\infty S(x)dx$ -1901,$a=100$ -1902,$L(X)=k(X-\mathsf E X)$ -1903,"$\mathsf E[X_i] + \pi(X)\mathsf{cov}(X_i, X)/\mathsf{SD}(X)$" -1904,$+ \mathit{PV}_{r_f}(\text{Inv Inc tax})$ -1905,$S(x_1)(x_2-x_1)$ -1906,$m=q(p)$ -1907,$wx + (1-w)y\in C$ -1908,$m_X$ -1909,$A(\text{Bernoulli})$ -1910,$\mathcal{G}\subset\FF$ -1911,"$X,Y$" -1912,$\mathsf E_{QQ'}[X_i(a)] \ne \mathsf E_{QQ}[X_i(a)]$ -1913,$\tilde Q$ -1914,"$Y_{0,2}$" -1915,$E[T]=s$ -1916,$\max(X)<\infty$ -1917,$\rho(Z_2)$ -1918,$\alpha_2SdX$ -1919,$\mathsf E[\cdot\mid X]$ -1920,$c\ge 1/2$ -1921,$g(s)=\dfrac{s+\iota}{1+\iota}$ -1922,"$X_i(\mathbf{v}, a)$" -1923,$X \prec_n^* Y$ -1924,"$X\wedge a'=\min(X, a')$" -1925,$d=2$ -1926,$\mathcal D(X)=\rho(X)-\mathsf E[X]$ -1927,$s^\alpha$ -1928,$k(h):=\log\mathsf E[e^{hX}]$ -1929,$X(x)=\sum_i x_iX_i$ -1930,$\mathsf Q(\omega)=Z(\omega)\Pr(\omega)$ -1931,$1/6\le x < 2/6$ -1932,$p\ge r\ge 1$ -1933,$\rho(X_0)=\mathsf E[X_0Z]$ -1934,$\mathbf{B}(0)=\mathbf{P_0}$ -1935,$Q=(a-EL)/(1+\iota)$ -1936,$\mathsf E[Z]=\mathsf E[\mathsf E[Z\mid X]] = 0$ -1937,"$\rho(P,R,a)$" -1938,$t\mapsto v^t$ -1939,$\{ X=x\}$ -1940,$\omega \in \Omega$ -1941,"$j, p, S, \kappa_1, \Delta X, \Delta(X\wedge a)$" -1942,$0.375/1.5 = 0.25$ -1943,"$a(v_1(1+\epsilon),v_2)=a(v_1,v_2)+da$" -1944,$M_i$ -1945,$\alpha_i$ -1946,$p=1-\exp(-t)$ -1947,$\mathbf{\mu}$ -1948,$\rho(X - b)=\rho(X)-b\le 0$ -1949,$\rho(X) + c = \rho(X+c)\ge \rho(X) + \mathsf E[cZ]$ -1950,"$\boldsymbol{j, p, S, \kappa_1, \Delta X, \Delta(X\wedge a)}$" -1951,$x\ge 0$ -1952,$\rho(\lambda X) \le\lambda\rho(X)$ -1953,"$(1,1,\dots,1,1)$" -1954,$\rho(X_j)=\max_k \mathsf E_\mathsf{Q_k}[X_j]$ -1955,"$1-p, p$" -1956,$S(x)=(k/(k+x))^\beta$ -1957,$p = 0$ -1958,$\mathsf E[u(R - X)]=0$ -1959,$\var(Y_{d})=\sum_{s>d} \sigma_s^2$ -1960,$x_1$ -1961,$x=X(1-g^{-1}(1-\tilde p))$ -1962,$s < 1$ -1963,$\cdot$ -1964,$a'=a(1+r)$ -1965,$\phi(\cdot)$ -1966,"$i \in \{1,\dots,4\}$" -1967,$\gamma=r_f$ -1968,$\Delta A$ -1969,$P(X_{-1}(a))$ -1970,$0\le\lambda\le 1$ -1971,$\max$ -1972,$\Omega_0$ -1973,$\mathsf E[X^k]$ -1974,$0\le v\le 1$ -1975,$Y(\omega)=1$ -1976,$Q=A-P$ -1977,$0.75$ -1978,$a+y$ -1979,$\mathsf{Pr}$ -1980,$0.25$ -1981,$s=\mathit{EL}$ -1982,"$(1-g(S(x)),x)$" -1983,$\nu+10\delta$ -1984,$1=ps_g + (1-p)s_b$ -1985,$U(1)=2$ -1986,$\bar S_i(a)=\mathsf E[X_i(a)]$ -1987,$\Phi(-d^*)>0$ -1988,$\Pr(X\ge q(p))>1-p$ -1989,$x\to\infty$ -1990,$g(pq)=g(p)g(q)$ -1991,$P = \mathsf E[X] + \pi \mathsf E[|X-\mathsf E[X]|^p]^{1/p}$ -1992,$\frac{d}{dp}(1-p)^{-1}=(1-p)^{-2}=q^{-2}$ -1993,$\mathsf E X + c{(X-\tau)^+}_p$ -1994,$\rho(X)<\infty$ -1995,$\mu_L=r_L + \pi$ -1996,"$k=(0.04, 0.4)$" -1997,$\Delta S=p$ -1998,"$A,B$" -1999,$N(1-p)$ -2000,"$(\omega'=1, \omega'')\in B_k$" -2001,$P = \mathsf E[X] + \pi \mathsf E[((X-\mathsf E[X])^+)^p]^{1/p}$ -2002,$\mathbf{t}$ -2003,"$p_0,\dots, p_m$" -2004,$\tilde Z$ -2005,$\tilde X+X$ -2006,$dF(x) = dp$ -2007,$x_0 < \mathsf{TVaR}_{p_0}$ -2008,$\lambda\sigma$ -2009,$\mathsf E[(X-m)(1_{U_X\ge p}-B)] = 0$ -2010,$Z_j$ -2011,$m'(1) \to -1$ -2012,$\mathsf E[X\mid \mathcal F_{t+1}]$ -2013,$g(S_j)$ -2014,$g(s(t)) = m(t)+s(t)$ -2015,$A\subseteq \mathbb{R}^N$ -2016,$f(x)\ge f(x_0) + s(x-x_0)$ -2017,$p=0.9982$ -2018,$a=10$ -2019,$\mu + \lambda\sigma$ -2020,$\beta<\alpha$ -2021,$Z\ge 0$ -2022,$\bar\nu(x)$ -2023,$\mathsf E_\mathsf{Q}[X]\le \mathsf E_\mathsf{Q}[Y]$ -2024,$\mathbf{Z_3}$ -2025,$6.258$ -2026,$\rho(X)=-\rho(-X)$ -2027,$-\sigma^2/2$ -2028,$k>0$ -2029,$r = 0.12$ -2030,"$(3,4)$" -2031,$dG/dF=r(x)$ -2032,$F_0=2.5$ -2033,$F_g(b)-F_g(a)=g(S(a)) - g(S(b))$ -2034,$P_g$ -2035,$\kappa_i(x)=\mathsf E[X_i\mid X=x]$ -2036,$\bar S$ -2037,$p=F(a)=1-s$ -2038,$Z(\omega)<1$ -2039,$\alpha\equiv 0$ -2040,$Var(G)=c^2$ -2041,$a = a(X)$ -2042,"$x\in\Omega=[0,1]^N$" -2043,$1_{U_X\ge p}=1$ -2044,$r_h<0$ -2045,$g(S(x_i)-g(S(x_i-))$ -2046,$F(a)$ -2047,$L_d^{d+l}(x)=(x-d)^+ \wedge l$ -2048,$X_3$ -2049,$\bar P(a+y) - \bar P(a)$ -2050,$\bar P$ -2051,$x_{i+1}$ -2052,$-X_2$ -2053,$M_2\Delta X$ -2054,$(1+r)\mu$ -2055,$\bar P^a$ -2056,$\ge p$ -2057,$\mathsf E X + c\mathsf E[\vert X-\tau \vert^p]^{1/p}$ -2058,$\\mathbf{\1}$ -2059,$\displaystyle\int_0^\infty u(x) g'(S_X(x)) dF_X(x)$ -2060,"$\omega\in [k2^{-m}, (k+1)2^{-m}]$" -2061,$p=2$ -2062,$X=98$ -2063,"$0\le U, V\le 1$" -2064,$Y'$ -2065,$\displaystyle\int_0^\infty xf(x)dx$ -2066,$(1-g(s))/(1-s)$ -2067,$\mathsf E[X_i]/x$ -2068,$00$ -2079,$X_{1}(a)$ -2080,$\psi(u)=\Pr(Y > u)$ -2081,$\mathsf E[\cdot]$ -2082,$\Delta(X\wedge a)$ -2083,$P = S + M$ -2084,$(0.304-0.2)/(1-0.304) = 15$ -2085,$\mathsf E_{\mathbb{Q}}[X\wedge a] \le \rho(X\wedge a)$ -2086,$\omega_2$ -2087,$P/S-1$ -2088,$g(s)/s$ -2089,$\mathbf{S\Delta X}$ -2090,"$h(x)=\sup_{s\in[0,1]} g(s)-sx$" -2091,$C(t)$ -2092,$t=4$ -2093,$i^*$ -2094,$\rho(X)=\mathsf E[X] + c\sigma(X)$ -2095,$g(1)=1$ -2096,$C'_1+\cdots + C'_n$ -2097,$\mathsf E[X]=\sum_{\omega\in\Omega} X(\omega)\Pr(\omega)$ -2098,$E_i\cap E_j = \varnothing$ -2099,$s_1$ -2100,$BY \succ AR$ -2101,$0.8 \times 1.2 = 24/25$ -2102,$(g(s)-s)/(1-g(s))$ -2103,$a = 8.1484$ -2104,$Y\circ T_i$ -2105,$p=0.9999$ -2106,$Z_X$ -2107,$\beta_i(x) =\mathsf E_{\mathsf Q}[X_i/X\mid X>x]$ -2108,"$X_{0,1},X_{0,2},\dots, X_{0,N}$" -2109,$Z=0$ -2110,$\rho_g(X)=\mathsf E_{\mathbb{Q}}[X]$ -2111,$-k$ -2112,$\mathsf E_\mathsf{Q}[X]=\mathsf E[XZ]$ -2113,$v(A)=g(\mathsf{P}(A))$ -2114,"$\bar P_i(\mathbf{v}, a)$" -2115,$B_p$ -2116,$a_i=x_i(\partial a/\partial x_i)$ -2117,$N$ -2118,$\sup$ -2119,$\rho(\tilde X)=\mathsf E_{\mathsf{Q}}[\tilde X]$ -2120,$q_X(p)\le q_Y(p)$ -2121,$S(x)=s$ -2122,$X\preceq_n Y$ -2123,"$y,z\in X$" -2124,$\Omega_0 \times \Omega_1$ -2125,$df/dx=f$ -2126,$\mathsf{TVaR}_p(X)$ -2127,$X=8$ -2128,$Q\in\mathcal{Q}$ -2129,$0.125$ -2130,$P(X_{-1}\wedge a)$ -2131,$s < p$ -2132,"$n=1,2,\dots, m-1$" -2133,$S(x)\approx 1$ -2134,"$X_2=(0,1,2,3,4,8,6,4,0,9)$" -2135,$1.5$ -2136,$q_X(p) = X(T(p))$ -2137,$1-m\le 1$ -2138,$\mathsf E_\mathsf{Q}[X_1]$ -2139,"$k=1,\dots, n-1$" -2140,$X_{-1}+X_{0}$ -2141,$p<0.05$ -2142,$\delta$ -2143,"$\gamma([0,p])=C(p)$" -2144,$10$ -2145,$T(U)$ -2146,$\rho_a(X+c) = \rho((X+c)\wedge a(X+c)) = \rho((X+c)\wedge (a(X)+c)) = \rho((X\wedge a(X))+c) = \rho((X\wedge a(X))) + c=\rho_a(X)+c$ -2147,$\bar M_t$ -2148,"$x~\text{Unif}[0,1]$" -2149,$g'(S(X))$ -2150,$\tilde Z=\mathsf P(X=\sup(X))^{-1}1_{X=\sup(X)}$ -2151,$\bar P(a+da) -\bar P(a)$ -2152,$X(x)=1/x$ -2153,$x=\mathsf{VaR}$ -2154,$\beta_2g(S)dX$ -2155,$\sigma(X_d)$ -2156,$\mathsf Q(X>a)/\mathsf P(X>a)$ -2157,$\mu(dp)$ -2158,$c=(g-s)/(g(1-g))$ -2159,$\mathsf E[Y_d]$ -2160,$X\wedge a=a=90$ -2161,$\sigma(W)$ -2162,$1\le p\le \infty$ -2163,$X=4$ -2164,"$\sigma(L^\infty, L^1)$" -2165,$p_0\not= p_1$ -2166,$\mathsf E[X]+k\mathsf{Var}(X)=a(X)$ -2167,"$a_{0,0}'=a_{0,0}$" -2168,$\{\omega\mid X(\omega) > x\}$ -2169,$P_i$ -2170,$\lambda_2\not=1$ -2171,$p>0.9$ -2172,$E(X^k)=E(Y^k)$ -2173,$=v_f \mathsf E_\mathsf{Q}\left[\dfrac{X_i}{X}(X\wedge a)\right]$ -2174,$\bar P_t$ -2175,"$\Omega=\{ 1,2,3,4,5,6 \}$" -2176,$p<0.7$ -2177,"$a=10,20,40,50,60$" -2178,$-\infty+\lambda=-\infty$ -2179,$x=y$ -2180,$d=0.1/1.1$ -2181,$\beta_2>\alpha_2$ -2182,$\rho(X)=\mathsf E_{\mathsf Q}[X]$ -2183,$\Pr(E')+\Pr(E)=\Pr(\Omega)=1$ -2184,$v_f(\mathsf E_Q[X_i] - \mathsf E_Q[X_i/X(X-A)^+])$ -2185,$=\displaystyle\int_0^\infty x dF(x)$ -2186,$\mathcal Q=\{\mathsf Q_k\}$ -2187,$a(f + (1-f)/q)$ -2188,$\mathsf E_\mathsf{Q}[\cdot]$ -2189,$\lfloor x \rfloor$ -2190,$A\in\mathcal F$ -2191,$\mathsf E X + c\mathsf E[((X-\mathsf E X)^+)^p]^{1/p}$ -2192,$v(A)=\lambda(\pi_1(A))$ -2193,$\mathbf{M_{2}}$ -2194,$n\to\infty$ -2195,$\beta_i(x) =\mathsf E_{\mathsf{Q}}[X_i/X\mid X>x]=\mathsf E[(X_i/X)g'S(X))\mid X>x]$ -2196,$\Longleftarrow$ -2197,"$\eta_{p,\alpha}$" -2198,$\Omega$ -2199,$\mathsf{QCX}$ -2200,$\omega=\omega'$ -2201,$g(S_{\mathsf{j}(a)})(a-X_{\mathsf{j}(a)})=(0.5)(80-11)=34.5$ -2202,$z_p=\Phi^{-1}(p)$ -2203,$g_1(s)=s^{0.4}$ -2204,"$1-e^{-\lambda S(\mathsf{PML}_{n, \lambda})}=1/n$" -2205,$\mathbf{X_2/X}$ -2206,$\mathbf{\alpha_1S\Delta X}$ -2207,$q^-(U)$ -2208,$\mathbf{g_3(s)=s^{0.7}}$ -2209,$s=\exp(-a/b)$ -2210,$F(x)\ge p\iff q^-(p)\le x$ -2211,$\mathsf E_\mathsf{Q}[X]$ -2212,$(P-L)/L=P/L-1$ -2213,"$[p,1]$" -2214,$F_2$ -2215,"$\{H,T\}$" -2216,$\mathbf{g(S)}$ -2217,$a(1-p) + \mu p - \sigma\phi(z_p)$ -2218,"$(p, \mathsf E[X_i\mid X=q(1-g^{-1}(1-p))])$" -2219,$\rho(b-X)=b+\rho(-X)$ -2220,$s<1$ -2221,$g''(s)=-s^{3/2}/4$ -2222,$D^n\rho_X(X_1)=6.2048$ -2223,$\Delta X\wedge a$ -2224,$v=1/(1+r)$ -2225,$v_f\mathsf E_Q[X_i]$ -2226,$(1-p)^{-1/2}/4$ -2227,$T(X):=y\wedge (X-r)^+$ -2228,$x=S^{-1}(g^{-1}(u))$ -2229,$A(X+c)=A(X)+c$ -2230,$\mathit{EGL}_{gc}(a)$ -2231,"$c\in[0,1/2]$" -2232,$\sigma=2.58$ -2233,$a_x=4$ -2234,$dp=\exp(-t)dt$ -2235,"$\beta_i(a) = \dfrac{\sum_{j:X_j>a} (X_{i,j}/X_j) \Delta g(S_j)}{\sum_{j:X_j>a} \Delta g(S_j)}$" -2236,$X = X\wedge a + (X - a)^+$ -2237,$(1-p)/(p\nu_p^2)$ -2238,$u$ -2239,$\omega$ -2240,$\mathsf{TVaR}_{0.8}(X+tX_1)$ -2241,$\Pr(X > x)$ -2242,"$\rho_g(X)= \sum_j X_j\,\Delta g(S_j)$" -2243,"$X_1,\dots,X_n$" -2244,$D\rho_{X}(Y) \subset D\rho_{X\wedge a}(Y)$ -2245,$\lambda=\dfrac{1}{1+\rho}$ -2246,$q^-(s)=\mathsf{VaR}_s(X)$ -2247,$=v_f \mathsf E_Q\left[\dfrac{X_i}{X}(X\wedge A)\right]$ -2248,$v_i$ -2249,"$p=0.01, 0.02, \dots, 0.99$" -2250,$\mathsf{VaR}\_p(X)$ -2251,$a_0$ -2252,$0\le b\le 1$ -2253,"$A=(a,b]$" -2254,$\rho(X)=\max_\mathsf{Q} \mathsf E_\mathsf{Q}[X]$ -2255,$a(\mathbf{v}) =\mathsf{TVaR}_p(X(\mathbf{v}))= (1-p)^{-1}\int_p^1 q_{\mathbf{v}}(s)ds$ -2256,$-g$ -2257,$q^-(p) := \sup\ \{x \mid F(x) < p \} = \inf\ \{ x \mid F(x) \ge p \}$ -2258,$p(\omega)\ge 0$ -2259,$D/L>1$ -2260,$\rho(X)=\mathsf E[f_X X]$ -2261,$-m_2/(1-s_2)$ -2262,$g(1-F(x))=1-p$ -2263,$h(1_{X\le a})$ -2264,$E(\pi)$ -2265,$\mathsf{TVaR}_{0.95}(X)$ -2266,$b-X\ge 0$ -2267,$Z = \sum_j X_j$ -2268,$X+Z$ -2269,$\mathsf{VaR}_{0.75}(X)=90$ -2270,$QR_Q = aR_A + PR_L$ -2271,$x=\lambda y + (1-\lambda)z$ -2272,$dS=-dF$ -2273,$\mathsf E[X_i \mid X=q(p)]$ -2274,$s \to 1$ -2275,$\mathsf E[X]\le \mathsf E[Y]$ -2276,$\tilde M(a)=\bar M(a)-\tau a$ -2277,$P = \log(\mathsf E[e^{\pi X}])/\pi$ -2278,"$(-\mathsf x*.8, 2*2)$" -2279,"$(ccc.south |- mcc.south)+(0,-0.5)$" -2280,"$[0,1]\to[0,1]$" -2281,$p=\infty$ -2282,$\bar P(a) = \rho_g(X\wedge a)$ -2283,$0\rho_2(X)$ -2285,$s(t)$ -2286,$\rho(W_1\wedge a_0)$ -2287,$0.8 \le p < 0.9$ -2288,$\epsilon_2$ -2289,$k=0$ -2290,$\Delta X_j=X_{j+1} - X_j$ -2291,$\iota:1$ -2292,"$x_{2,1}$" -2293,$Y_{d}=\sum_{s>d} X_{s}$ -2294,"$\phi(x_1,...,x_n)$" -2295,$Z\in\mathcal Q$ -2296,$\mathbf{Z_7}$ -2297,$\iota^\ast$ -2298,$X-P$ -2299,$g(s)q=0.1839$ -2300,$X_2=x-t$ -2301,"$X_{t+2,1}$" -2302,$\mathsf{MON}$ -2303,$G(x)= 1-g(1-F(x))$ -2304,$g'(s)\to\infty$ -2305,"$j \in \{5,\dots,8\}$" -2306,$e^{-r_Dt}$ -2307,"$\mathbb{R}=(-\infty, \infty)$" -2308,$\rho((X-a)^+)$ -2309,$Q_t$ -2310,$\Pr(B)=0$ -2311,$X_0 < \dots < X_{N-1}$ -2312,$\Pr(X=x_i)=\lambda_i/\lambda$ -2313,"$B_4 = [\epsilon_1, \epsilon_2]$" -2314,$a(w_1X_1+w_2X_2;X)=w_1a(X_1;X)+w_2a(X_2;X)$ -2315,$(P-L) / (A-P)=$ -2316,$AR\succ BR$ -2317,$\mathsf E[X\wedge a]= 2.4982$ -2318,$a(x)=xa(1)$ -2319,$X(\mathbf{v})$ -2320,"$x_{1,1}$" -2321,"$d, r>0$" -2322,"$\phi(s)= g'(1-s) = \frac{1-w}{1-p_0}1_{[p_0, 1)}(s) + \frac{w}{1-p_1}1_{[p_1, 1)}(s)$" -2323,"$S\subset \Omega=\{1,\dots,N\}$" -2324,$\rho(\mathsf E[X_2\mid X_1])\le \rho(X_2)$ -2325,$x\le 0$ -2326,$\mathbf{d=1}$ -2327,$S_0=1$ -2328,$f(x)=|x|$ -2329,$S_t \ge 0$ -2330,$p=F(a)$ -2331,$\Psi^{-1}(t)=\log(-\log(t))$ -2332,$\mathsf E[X\mid X>2000]-2000=\mathsf{TVaR}_{F(2000)}(X)-2000=624$ -2333,$q(U_X) > m$ -2334,$Y_s=(Y\mid Y\le y_c)$ -2335,$\mathsf{P}(d\omega)$ -2336,$h(0)$ -2337,$\mathbf{Z_\mathit{lift}}$ -2338,$P_i/v_i$ -2339,$\lambda > 0$ -2340,"$c(1,2) - c(2)$" -2341,"$(0,1]$" -2342,$t<0$ -2343,$\mathsf{COMON}$ -2344,$\beta_i(x)/\alpha_i(x)> 1 > g(S(x)) / S(x)$ -2345,$\mathsf E[XM]$ -2346,$\int_0^\infty (1-F(x))dx=\int_0^\infty xdF(x)$ -2347,$(dW_t)^2=dt$ -2348,$\mathbf{a=0.93}$ -2349,$\mathsf{TVaR}_{0.95}(X)=3699$ -2350,$g(0^+) = r/(1+r)$ -2351,$x\mapsto 1/x$ -2352,$m\in\mathbb{R}$ -2353,$-S(a)+\tau=0$ -2354,$\mathsf{VaR}_{0.7}(X_i)=-\log(0.3)=1.204$ -2355,$\rho(c)\ge c$ -2356,$\beta_i(X)$ -2357,$0.8\le p<0.9$ -2358,$\mathsf P(X \le q_X(p)) > p$ -2359,$1/X$ -2360,$\displaystyle\int_0^1 X(p)dp$ -2361,$\kappa_1(x)=\mathsf E[N_1/(N_1+N_2)]x$ -2362,$\rho_c\leftrightarrow\mathcal Q$ -2363,$U(X)\ge U(Y)$ -2364,$ = \mathsf E_{\mathsf{Q}}[X_i\mid X= x]$ -2365,$\lambda X_1 +(1-\lambda) X_2$ -2366,$MV = \bar Q + \mathit{NPV}_{\infty}$ -2367,$g(s)=1-(1-s)^m$ -2368,$g(0.05)=0.05\nu + \delta=0.1364$ -2369,"$\mathcal F_0=\{\varnothing, \Omega\}$" -2370,$p(x) = \Pr(\{\omega\mid X(\omega) = x\})=\Pr(X=x)$ -2371,$g(S(x)) = 1 - h(F(x))$ -2372,$g(s)\le s$ -2373,$L_1$ -2374,$X_1=1000$ -2375,$S$ -2376,$x < y$ -2377,$p>0.5$ -2378,$x=(y-\mu)/\sigma$ -2379,$a\to\infty$ -2380,$X+tX_1$ -2381,$M = \beta g(S)-\alpha S$ -2382,$0 < \nu = 1-\delta < 1$ -2383,$d=(\log(a/S_0)-(r-\sigma^2/2)t)/\sigma\sqrt{t}$ -2384,$X(\omega)=1/\omega$ -2385,$1/n$ -2386,$\mathsf E[X] + \pi\mathsf E[X]$ -2387,$H(X)>-H(-Y)$ -2388,$s/(1-p) \wedge 1$ -2389,$\mathsf E[X] + \pi\var(X)$ -2390,$\Phi$ -2391,$\lambda y=x$ -2392,$\mathsf{MON'}$ -2393,$g'(S_X(X))$ -2394,$b<1$ -2395,$w < s$ -2396,$m_2$ -2397,$\le c$ -2398,$n-1$ -2399,$qX$ -2400,$\bar P_2$ -2401,"$(4,3)$" -2402,$(X_i)_i$ -2403,$20+10t$ -2404,$s=1-\alpha$ -2405,$Z=d\mathsf Q / d\mathsf P\ge 0$ -2406,$X_i(a) = aX_i/X$ -2407,"$c(1,2,3)-c(2,3)$" -2408,$\sum_i q_iX_i$ -2409,$\mathbf{Q_{2}\Delta X}$ -2410,"$H_k(X):=\mathsf E[\max(X_1\dots, X_k)]$" -2411,$\kappa_j(x)/x > \alpha_j(x)$ -2412,$a_i'$ -2413,$-\int xdS=\int Sdx$ -2414,$c\ge 1$ -2415,$\mathbf{B}(1)=\mathbf{P_3}$ -2416,"$\bar Q_{0,0}:=a_{0,0}-\bar P_{0,0}$" -2417,$p_- < p_0 < p_+$ -2418,$g'(t)=1-r_0$ -2419,$q(p)=\mathsf{VaR}_p(X)$ -2420,$g(0+):=\lim_{s\downarrow 0}g(s)$ -2421,$z\ge 0$ -2422,$ \& $ -2423,$A\setminus B$ -2424,$(k_1!)(k_2!)\dots$ -2425,$Q(x)=1-P(x)$ -2426,$\sup(X)$ -2427,$1=\delta+\nu$ -2428,$=1/\lambda-1=(1-\lambda)/\lambda$ -2429,$U_X$ -2430,"$\mathbf{X\,\Delta g(S)}$" -2431,$\mathit{EGL}_{ro}(a)$ -2432,$q_X$ -2433,"$i=1,2,\dots,10000$" -2434,$Z=z(X)$ -2435,$\bar{\mathbf P}$ -2436,$\{X > x \}$ -2437,$X_{\mathsf j(a)+1}>a$ -2438,$g_j<1$ -2439,$\rho(X)=0$ -2440,$\sum_i x_iX_i$ -2441,$Xq$ -2442,$\phi(p)=g'(1-p)=b(1-p)^{b-1}$ -2443,$N=1000$ -2444,$\mathsf E_{\mathsf{Q}}[X]=\infty$ -2445,$A\subseteq \mathbb{R}^n$ -2446,$a=90$ -2447,"$g:[0,1]\to [0,1]$" -2448,$q(p)$ -2449,$g(s)=\nu s+\delta$ -2450,$m=$ -2451,$\mathbb{Q}\in\mathcal Q$ -2452,"$q(p)\phi(p)\,dp$" -2453,$x>\mathsf{VaR}_p(X)$ -2454,$\hat x > x$ -2455,$\text{VaR}_{0.99}$ -2456,$P_X\{X=M\}=0$ -2457,$X=X_0+X_{-1}+X_{-2}+X_{-3}$ -2458,$x>0$ -2459,"$X_{i,j}$" -2460,$a_1=\int_0^1 (\partial a/\partial x_1)dt=\partial a/\partial x_1$ -2461,$\mathsf E[X(1_{U_X\ge p}-B)]=\mathsf E[(X-m)(1_{U_X\ge p}-B)]$ -2462,$1=\bar\nu+\bar\delta$ -2463,$(1-p)/p=1$ -2464,$\mathsf E[X_i(v_i)]=v_i\mathsf E[X(1)]$ -2465,$s=S(a)$ -2466,$\partial\rho(Z)$ -2467,$\mathbf X$ -2468,$\rho(W_1\wedge a_1 \wedge (a_0-X_1))=\rho(W_1\wedge a_1)$ -2469,$\sum_i \kappa_i(x)=x$ -2470,$(g(s_0)-g_0)/s_0 \ge g'(s_0)$ -2471,$g(s)=s^{0.4}$ -2472,$X_n(0)=1$ -2473,"$X_{t,2}$" -2474,$W=Z$ -2475,$\phi(x):=(2\pi)^{-1/2}\exp(-x^2/2)$ -2476,$g(s)=\sqrt{s}$ -2477,$1-p=S(x)$ -2478,$\mathsf E_{\mathsf{Q}}[Y \mid X] = \mathsf E[Y \mid X]$ -2479,$p(\delta_p-il_p)$ -2480,$\alpha(X)$ -2481,$=1$ -2482,$g''$ -2483,$f=f_X$ -2484,$dW_t\approx W_{t+dt}-W_t$ -2485,$X(\omega_1) > Y(\omega_1)$ -2486,$H_g(X) \le H_g(Y)$ -2487,$M:=\max(X)$ -2488,"$0,10,20$" -2489,$1/9=0.11\dot 1$ -2490,$a=80$ -2491,$n-2$ -2492,"$((0, x), (1-p, p))$" -2493,$P=D=L/(1+R_L)$ -2494,$w(A)\le v(A)$ -2495,$\Pr(X\ge x)\ge 1-p\ge \Pr(X> x)$ -2496,$2^{20}\approx 1$ -2497,$^{**}$ -2498,$\mathbf{X_{g}}$ -2499,$\mathsf{LI}\iff\mathsf{SSD}$ -2500,$p_j$ -2501,$P$ -2502,$s_0\mathsf{VaR}_p(X)]$ -2530,$\mathcal M_\rho=\{ m \}$ -2531,$\mathsf E[kX]$ -2532,$f(p)=\alpha(1-\alpha)(1-p)^{\alpha-1}$ -2533,"$L_a^{a+y}(x)=\min(y, \max(x-a,0))$" -2534,$(X-a)^+$ -2535,$\omega''$ -2536,$0.20$ -2537,$g(X_n)=1$ -2538,$M_i\Delta X$ -2539,$p=0.01$ -2540,$w=0$ -2541,$f'_-(x)=\lim_{h\uparrow 0} (f(x+h)-f(x))/h$ -2542,$B_1 \succ A_1$ -2543,$1-f$ -2544,$X_0 < X_1 < \dots < X_{n'}$ -2545,$a=$ -2546,$Q_{2}\Delta X$ -2547,$\kappa_i'(x)=1$ -2548,$X'\Delta S$ -2549,$a=\alpha(X)$ -2550,$X_2=1000$ -2551,$\alpha(\mathsf P)=0$ -2552,"$h(p)=p/(1+\iota(p))=\nu(p)\, p$" -2553,$\mathsf E[X_2\mid X=20]=6$ -2554,"$s,p$" -2555,$F(x)=u$ -2556,$a_i = \mathsf{VaR}_p(X) - \mathsf{VaR}_p(\sum_{j\not=i} X_j))$ -2557,$z(X)$ -2558,$n_s\ge 0$ -2559,$x_6^1+x_6^2=10+1=11=x_6$ -2560,$g(S(x)$ -2561,$0\le \lambda \le 1$ -2562,$(\mu-\sigma^2/2)t$ -2563,$(\delta^{\star}-d)\sqrt{S(x)F(x)}$ -2564,"$\alpha_p = 1- (\| (X-\eta_{p,\alpha})^+\|_{p-1} / \| (X-\eta_{p,\alpha})_- \|_{p})^{p-1}$" -2565,$\alpha (1-s)^\alpha/(1-s)$ -2566,$d\to\infty$ -2567,$p(\nu_p-l_p)$ -2568,$g(s)=s^2$ -2569,$a=a(\mathbf{v})$ -2570,$\sup_\mathsf{Q} \mathsf E_\mathsf{Q}[X]$ -2571,$\int_0^1 1-g(s)ds=1-\int_0^1 g(s)ds < 0.5$ -2572,$X_i>0$ -2573,$i= \alpha/(1-\alpha)$ -2574,$X\ge x$ -2575,$Z(x)=g'(S(x))$ -2576,"$c = 1.0, 1.5$" -2577,$a_{d}=a(Y_{d})$ -2578,$\mathsf{SD}(X)$ -2579,$-A(-X)$ -2580,$t\ge 0$ -2581,"$\Omega=\{0,\dots,99\}$" -2582,$g'(S(x))\ge 0$ -2583,"$p~\text{Unif}[0,1]$" -2584,$R_A=R_f$ -2585,$\mathsf{VaR}_p(X)=q^-(p)$ -2586,$E(u(X)) \le E(u(Y))$ -2587,$(\beta_i g(S))'(x)=-\mathsf E[X_i\mid X=x]g'(S(x))f(x)/x=-\kappa_i(x)g'(S(x))f(x) / x$ -2588,$a \ge 1$ -2589,"$\mathsf{biTVaR}_{0,p}^w(X)$" -2590,$\iota^\ast = (g(s^\ast)-s^\ast) / (1 - g(s^\ast))$ -2591,$g'(s)\ge 0$ -2592,"$X:\Omega\to [0,\infty]$" -2593,"$\mathsf{TVaR}_{0.95}(X)=\int_0^{1000}g(S(x))\,dx$" -2594,$\rho(X_n)\to \rho(X)$ -2595,$\lambda_{obj}$ -2596,$W_0$ -2597,$0=q(0)=q(Y+(-Y))\le q(Y) + q(-Y)$ -2598,$cv=0.287$ -2599,$g_\tau(0)=0$ -2600,$0.41$ -2601,$\mathsf P(X=q_X(p))>0$ -2602,$p=0.8$ -2603,$\kappa_1(10)$ -2604,$\mathsf E_\mathsf{Q}[X_i(a)]$ -2605,$\mathsf E[(X-\mathsf E X)^+]$ -2606,"$[0,p)$" -2607,$a_1'$ -2608,$S(x)=1-\Phi((x-\mu)/\sigma)=\Phi(-(x-\mu)/\sigma)$ -2609,$X_{t+1}$ -2610,$X=0$ -2611,$p\mapsto g(1-p)$ -2612,$\downarrow$ -2613,$X\wedge 20$ -2614,$\mathsf{TVaR}_1( X )$ -2615,"$x_1, x_2$" -2616,$\bar P_{2}$ -2617,$\Sigma$ -2618,$B\subset A$ -2619,$\bar P=\mathsf{TVaR}_{p^\ast}(X)$ -2620,$\bar P^a_g(X_i\subseteq X)$ -2621,"$\mathcal{M} = \{ f \mid \|f\|_q\le c, f\ge 0 \}$" -2622,$X \preceq_m Y$ -2623,"$\mathsf E[X_i(a)\,g'(S_{X\wedge a}(X\wedge a))]$" -2624,$qX_i$ -2625,$X \prec_n Y$ -2626,$X(\omega)\Pr(\omega)$ -2627,$\bar\iota=0.10$ -2628,$a=18000.0$ -2629,$\mathsf{TVaR}_{0.95}(X)=1000$ -2630,$s_0/2^{n+1}$ -2631,$\delta(x)$ -2632,$H[X]$ -2633,$\rho(X_0+Y) \ge \rho(X_0) + \mathsf E[YZ]$ -2634,"$x_1,x_2$" -2635,$>100$ -2636,$dh - h_x dx = (r_h-\mu_L)(h-h_x x)dt$ -2637,$\alpha<1$ -2638,$2.576\times 6.258$ -2639,$\mathsf{TVaR}_{p*}(X)=a$ -2640,$\kappa_i(X) = X_i$ -2641,$a_i = a(X_i; X)$ -2642,$\rho(X_n(t))+t\pi$ -2643,$\mathsf{TVaR}_{p}$ -2644,$g(A)/p=59.142$ -2645,$Z(S_X(x))=-(x-\mu)/\sigma$ -2646,$\nu=1-\delta$ -2647,$\{\omega\in\Omega\mid X(\omega)\le x\}$ -2648,$X_n= X_g-X_c$ -2649,$s^*$ -2650,$\bar P_{0}$ -2651,$X\wedge 30$ -2652,$k+1/2$ -2653,$\mathsf{CTE}_{p_0}=\mathsf E[X \mid X \ge x_0]$ -2654,$\lambda=5$ -2655,$D_3$ -2656,$\ge c$ -2657,$\kappa_i(X)$ -2658,$L(X)=w(X)/\mathsf E[w(X)]$ -2659,"$\mathsf{PH,SA,CX}$" -2660,$\phi:=\rho\circ F$ -2661,$u_j(x) = 1 - exp(-\lambda_j x)$ -2662,$\mathsf E[X]+\var(X)/\mathsf E[X]$ -2663,$M_{1}$ -2664,$\mathsf E[YZ_\epsilon]\to\mathsf E[YZ]$ -2665,$X-V$ -2666,"$\bar P_i = \sum_{j} X_{i,j}\Delta g(S_j)$" -2667,$f(t)$ -2668,"$[0, \epsilon_1]$" -2669,$\pi=1$ -2670,$\psi(0)=1-\Pr(Y=0)=1-\Pr(M=0)=\frac{1}{1+r}$ -2671,$a_l \le 1$ -2672,$P_g\not\ll P_X$ -2673,$\delta_i=\delta$ -2674,$p_Y>0.5$ -2675,$-g'(1-p)<0$ -2676,$\rho(X+Y) = \rho(X) + \rho(Y)$ -2677,"$(0.5,1]$" -2678,$F(x_0)=p_+$ -2679,"$(X_i, X)$" -2680,$\mathsf E[X]=\mathsf{TVaR}_0(X)$ -2681,$l(kX)=k\rho(X)$ -2682,$\int udv = uv - \int vdu$ -2683,$r_P-\mu_L$ -2684,$\mathsf E[X_i\mid X\le a]F(a) + a\mathsf E[X_i/X\mid X >a]S(a)$ -2685,$\bar P(a)=\displaystyle\int_0^a g(S(x))dx$ -2686,$g(x) = (x-\mu)^2$ -2687,$\mathsf{biTVaR}(Y)=\mathsf{TVaR}_{p^\ast}(Y)$ -2688,$0$ -2689,$p=1-g(1-F(x))$ -2690,$\bar S_i(3463)$ -2691,$X=X(\omega)$ -2692,$\int_0^1$ -2693,$\esssup(X)g(0-)$ -2694,$\mathsf E[X \mid U]$ -2695,$\tilde p$ -2696,$\bar P'$ -2697,$\sum_i E[X_i|anything]\le _{cx} \sum X_i \le_{cx} F_{X_i}^{-1}(U)$ -2698,$\lambda = \lambda_0+\lambda_1$ -2699,$X_{-2}=C_1 + \cdots + C_n$ -2700,$X_{0}$ -2701,$\rho_g = \int g(S)$ -2702,$a={{break_even}}$ -2703,$x=q(1-g^{-1}(1-\tilde p))$ -2704,$0.5\le p^* \le 0.75$ -2705,$X(\omega)=1$ -2706,$P(a)=S(a)+\delta F(a)$ -2707,$\mathsf{TVaR}_{1-c\epsilon}(X) = \mathsf{VaR}_{1-\epsilon}(X)$ -2708,$q(p)=S^{-1}(1-p)$ -2709,$d_i= i/(1+i)$ -2710,$P=\sum_i P_i$ -2711,$B_i$ -2712,$a_1=a(W_1)$ -2713,$\rho=\esssup=\mathsf{TVaR}_1$ -2714,$\sigma(X)^2$ -2715,$g'(s)=1/(1-p)$ -2716,$0.4$ -2717,$f'_+(x)=\lim_{h\downarrow 0} (f(x+h)-f(x))/h$ -2718,$g'(1)=1$ -2719,$\mathcal Q_1$ -2720,$X\le m$ -2721,$\mathsf E_\mathsf{Q_k}[X_j]$ -2722,$\mathbf{\min a}$ -2723,$dt^2$ -2724,$q=p$ -2725,$\sigma_d = \mu_d/5$ -2726,$\mathsf E[cZ]=c\mathsf E[Z]=c$ -2727,$\mathsf E[(X-\mu)^2]$ -2728,$Q_0$ -2729,$X_t=X_{t+1}$ -2730,$g(s)=0.1995$ -2731,$\log(0)=-\infty$ -2732,$\mathsf{VaR}_p(X_1)$ -2733,$W_t$ -2734,$Z_{\tilde X}$ -2735,$U0$ -2748,"$\{4,5\}$" -2749,$h(x)=f(x)/S(x)$ -2750,$S\Delta X\wedge a$ -2751,$r_U$ -2752,$\mathsf E_\mathbb{Q}[X]$ -2753,$\mathsf E[X_d]$ -2754,$(c(S\cup \{i\})-c(S))$ -2755,$\mu(dp)=f(p)dp$ -2756,$X\preceq_2 Y$ -2757,$P \le \dfrac{S}{\lambda} \approx \dfrac{\mathsf E[X]}{\lambda}$ -2758,$v(E)$ -2759,$\{Z\circ T\mid T:\Omega\to\Omega\text{\ PPT}\}$ -2760,$\mathsf E[X_ih(X)]$ -2761,$S(x_i-)-S(x_i) =\Pr(X=x_i)$ -2762,$6/6$ -2763,$\Phi(\Phi^{-1}(s) + \lambda)$ -2764,"$[0, -k]$" -2765,$\mathcal E(X)=c\mathsf E[X^2]$ -2766,$\mathsf P(X=\mathsf{VaR}_p(X))>0$ -2767,$\mathsf E[X] = \displaystyle\int_\Omega X(\omega)\Pr(d\omega)$ -2768,$\rho(W_0\wedge a_0)=\bar P_0 +\bar P'$ -2769,$U(t)$ -2770,$p=1-s$ -2771,"$\sum_i a(X_i, p^*)=a$" -2772,$q_{X_1}(p)+q_{X_2}(p)=q_{X_1+X_2}(p)$ -2773,$8.5$ -2774,$\mathbf{\Omega}$ -2775,$M_{1}\Delta X$ -2776,$\bar P_i(a)$ -2777,$T_i$ -2778,$L_0^y$ -2779,$2$ -2780,$\rho(c)=\rho(0+c)=\rho(0)+c$ -2781,$U_X(\omega)=F(X(\omega)-) + V(\omega)(F(X(\omega)) - F(X(\omega)-))$ -2782,$s = 1-10^{-15}$ -2783,$s/g(s)$ -2784,$\alpha f$ -2785,$\{\omega\in\Omega \mid X(\omega)=x\}$ -2786,"$[0,1]$" -2787,$a(\mathbf{v})=\mathsf{TVaR}_p(\mathbf{v})=\mathsf E[X\mid X > q_{\mathbf{v}}(p)]$ -2788,$M=\varnothing$ -2789,$1/(1-p)$ -2790,"$(0,0),\ (1,0),\ (1,1)$" -2791,$t\mapsto \rho(X) + t\mathsf E_{\mathsf Q_X}[Y]$ -2792,$\omega_i\in B$ -2793,$g'(1)=\alpha$ -2794,$\le 1$ -2795,$-\rho(-X)$ -2796,$g(S(x_i-))-g(S(x_{i-1}))$ -2797,$f_{opt} = 1-s/g$ -2798,$\Delta \mathit{MV}_{ro}(a)$ -2799,$Q_i=a_i-P_i$ -2800,$0.0476/(1-0.0476)=0.05$ -2801,$q=0.9215$ -2802,$f(x)dx$ -2803,$\mathcal F'$ -2804,$\nu (1-s)$ -2805,$p=\Phi((a-\mu)/\sigma)$ -2806,$g'(1-s)$ -2807,$\Pr({\omega})=1/6$ -2808,$P(X_{-1}\wedge a_{ro})=9196.39$ -2809,$\omega_0$ -2810,$g_2(s) = 2s/3 + 1/3$ -2811,$\bar S(a):= \mathsf E[L_0^a(X)]=\mathsf E[X\wedge a]$ -2812,$x^\ast$ -2813,$2/3$ -2814,$\iota(s)=w/(1-w)$ -2815,$\phi(0)=0$ -2816,$\log(S) =\mu t$ -2817,$a\le (P(1+\iota)-S)/\iota$ -2818,$g'(s_1) \ge (1-g(s_1))/(1-s_1)$ -2819,"$U, V$" -2820,$s^{0.642}$ -2821,$\kappa_i(x)=mx/(m+n)$ -2822,$C\mathsf X$ -2823,$s_0=1$ -2824,"$\Omega=\{1,2\}$" -2825,$\min_{\eta\in \mathbb{R}} \eta + \alpha \mathsf E[(X-\eta)^+] -\beta\mathsf E](X-\eta)^-]$ -2826,$X\mapsto\int X(\omega)Z(\omega)\mathsf(d\omega)$ -2827,$\rho(X_0+\epsilon Y)-\rho(X_0)$ -2828,"$\sigma_A,\sigma_L$" -2829,$P(a)=g(S_X(a))$ -2830,$Z_\epsilon\to Z$ -2831,$\alpha_i(x) =\mathsf E[X_i/X\mid X>x]$ -2832,$a\beta_1g(S)$ -2833,"$\mathsf{biTVaR}_{p,1}^w$" -2834,"$(s^\ast, g(s^\ast))$" -2835,$a^\star$ -2836,$\mathsf E_{\mathsf Q}[\cdot]$ -2837,$\mathsf E[X_i\mid \{X=X(\omega)\}]$ -2838,$\beta_i(x)$ -2839,"$\mathbf{S\,\Delta X}$" -2840,$\rho(X+tY)\ge \mathsf E_{\mathsf Q_X}[X+tY]=\mathsf E_{\mathsf Q_X}[X]+\mathsf E_{\mathsf Q_X}[tY]=\rho(X)+t\mathsf E_{\mathsf Q_X}[Y]$ -2841,$\mathsf EPD_s(X)$ -2842,$\rho(X)=1.169$ -2843,$S(a)=\mathsf E[1_{X>a}]$ -2844,$U(\omega)=\omega$ -2845,${X}$ -2846,$D^f\rho_{X;\tilde X}(X_i)$ -2847,"$d(g(S(x)))/dx=-g'(S(x))\,dF/dx$" -2848,$S(x+a)$ -2849,$\rho(0)=0$ -2850,$\succeq^2$ -2851,$\mathsf E_{\mathsf Q}[Y] = \mathsf E[Yg'(S_X(X))]$ -2852,$a=\mathsf E[X \mid X > q(p)]$ -2853,$\mathbf{Q_1\Delta X}$ -2854,$\rho(\lambda X)=\lambda \rho(X)$ -2855,$q(p) \times \phi(p)dp$ -2856,$E(X_{-1}(a))=\bar S_0(a)$ -2857,$X\mapsto \mathsf E[XZ]$ -2858,$q_2(t)=t^2$ -2859,$\sigma^2=\sigma_A^2 + \sigma_L^2 - 2\rho\sigma_A\sigma_L$ -2860,"$(0.5, 0.5)$" -2861,$a_lp}$ -2881,$\int X=0$ -2882,$\mathsf{j}(0)=0$ -2883,$g'(S(x))>1$ -2884,$0<\alpha\le 1$ -2885,$-q(-Y)$ -2886,$X_c$ -2887,$r_f /(1+ r_f)$ -2888,"$[1,\infty)$" -2889,$4.75$ -2890,$D_c$ -2891,"$X_{t-2,1}$" -2892,$L\mathsf{VaR}_p(X)}$ -2904,$\gamma(ds)$ -2905,$Z=20\cdot1_A$ -2906,$X_n(\omega)= 1$ -2907,$\mathsf{Var}(X)$ -2908,$\bar M_t = \bar P_t - \mathsf E[Y_{t}]$ -2909,$f=(1-p)^{-1}1_{W}$ -2910,$\rho(X_n)=0$ -2911,$1_{X\le a}$ -2912,$af\le 1$ -2913,$ for estimates $ -2914,$X+W$ -2915,$X=\mathsf E[Y \mid \mathcal F']$ -2916,$\mathsf E[(-Y)Z]\ge 0$ -2917,$gS$ -2918,$\Delta X_7$ -2919,$Z=\tilde X_2$ -2920,$a\alpha_i(a)=\kappa_i(a)$ -2921,$B-p(\nu(p) + il(p))$ -2922,"$(0,0,0,0,0,0,5,0,0,5)$" -2923,$\mathsf{VaR}_p(X)=q_X^{-}(p) = \sup \{ x\mid F_X(x) < p \}$ -2924,$\lambda\mathsf E[X]$ -2925,$\omega\in\Omega$ -2926,$g=0$ -2927,$L_0^a$ -2928,$-5.91$ -2929,$X_i=\mathsf E[X_i\mid X]$ -2930,$\bar q_{X_1+X_2}(s)=q_{X_1+X_2}(1-s)$ -2931,$\Pi=B-p\nu(p)$ -2932,"$Y_{2,1}$" -2933,$\rho(X)=\mathsf E_{\mathsf Q_X}[X]$ -2934,$U(a)=-s$ -2935,$\rho(X+Y) = \rho(\lambda(X/\lambda) + (1-\lambda)(Y/(1-\lambda))))$ -2936,$P(x)$ -2937,$r=(1+\bar\iota)/(1+\tau)-1$ -2938,$\mathbf{d=2}$ -2939,$\mathbf{x_2}$ -2940,$\rho(-X)=-\rho(X)$ -2941,$R_L=-k R_f + \beta_L(R_M-R_f)$ -2942,$g(t)$ -2943,$N := \lceil (1-p)M \rceil$ -2944,$\{2\}$ -2945,"$(\nu,\delta)$" -2946,$p\to\infty$ -2947,$1\le\lambda$ -2948,$\rho E/(1-\tau) - rA$ -2949,$\mathbf{\beta_{2}g(S)\Delta X}$ -2950,$x\mapsto x^{1/2}$ -2951,"$j=0,\dots,m=8$" -2952,$\beta_i(a)/\alpha_i(a) > 1$ -2953,$=\displaystyle\int_0^\infty x f(x)dx$ -2954,$a_l-1<0$ -2955,"$\mathcal F'=\{\varnothing, \Omega \}$" -2956,$F_Y^{-1}(V)=q_Y(V)$ -2957,$Z_{\mathit{lin}}$ -2958,$0\le p_0\le p^*\le p_1\le 1$ -2959,$log(x)$ -2960,$\mathsf{TVaR}_0( X )=\mathsf E[X]$ -2961,$\nu=\nu(a)<1$ -2962,$X_1$ -2963,$X(\cdot)$ -2964,"$Z=(0,0,0,0,0,0,0,0,5,5)$" -2965,$Z_{\mathit{lift}}$ -2966,"$\mathbf{B}:\left [0,1 \right ] \ni t \mapsto (x(t),y(t)) \in \mathbb{R}^2$" -2967,"$(\Omega, P)$" -2968,$0 \le p<1$ -2969,$\mathbf{\Delta X}$ -2970,$\mathsf Q_X$ -2971,$1_A/\Pr(A)$ -2972,$n < N-1$ -2973,$\bar P(x)=\int_0^x P(t)dt$ -2974,$F(x_0)\ge p$ -2975,$X-a$ -2976,$\alpha_i(a)S(a)=\mathsf E[(X_i/X)1_{X>a}]$ -2977,$Z\not=0$ -2978,$\mathsf E X +\lambda_1 {(X-\lambda_2 \mathsf E X)^+}_1$ -2979,$\sigma(Z)=\sqrt{\var(Z)}$ -2980,$\rho(\cdot)$ -2981,$p = (1-s)$ -2982,$p=0.417$ -2983,$(j)$ -2984,"$\int_{[0,1]}$" -2985,$y^{\ast}:=\min(y)$ -2986,$\mathsf E[X]=1/\beta$ -2987,$\mathsf E[X_i (X\wedge a)/X]$ -2988,"$ (MA.south)+(0, -1) $" -2989,$q^-(U(\omega))$ -2990,$Q_2\Delta X$ -2991,"$\mu=0.1, \sigma=0.15$" -2992,$\mathsf E[X_2Z]$ -2993,$P_X(dx)$ -2994,$Y_{2}$ -2995,$Q_1dX$ -2996,${}^nS_X(t)\le {}^nS_Y(t)$ -2997,$(0.5)(20)+(0.5)(30)=25$ -2998,$\rho(X)\not=\sum_i\rho(X_i)$ -2999,"$M\subset \{1,\dots, n\}\setminus \{i, j\}$" -3000,$u(x)=(1-e^{-\pi x})/\pi$ -3001,$\Pr(p(\omega)=0)=0$ -3002,$55+0.675\times 3.807=57.572$ -3003,$D \rho(X_0)$ -3004,$\alpha(1-f)$ -3005,$90$ -3006,$L_{250}^{\infty}(x)$ -3007,$q(1)=\infty$ -3008,"$(p,q(p))$" -3009,$\prec_2^*$ -3010,$1/r$ -3011,"$\mathbf{v}=(v_1,\ldots,v_n)$" -3012,$\rho(X+X_i)=\rho(X)+\rho(X_i)$ -3013,$\displaystyle\int$ -3014,$\alpha_i(x)<\kappa_i(x)/x$ -3015,$\Pr(X_n=1)=1/n$ -3016,"$\{1,2,3,4,5,6\}$" -3017,"$(X_1,\dots, X_n)'$" -3018,$1_A(x)=0$ -3019,$X_{-4}=x$ -3020,$\mu-\sigma^2/2$ -3021,$s(1)=s_3=1$ -3022,$g(x)=0$ -3023,$L_0^{a+y}=L_0^a+L_a^{a+y}$ -3024,$x_{#4}$ -3025,$n\ge 1$ -3026,$2.576$ -3027,$Q_j=1-g(S_j)$ -3028,"$B_2=[0,0]$" -3029,$\sum c_i^2$ -3030,$\mathsf E[X_1\mid X=x]$ -3031,$X_i(v_i)$ -3032,$\alpha_i(a)S(a)$ -3033,$X(\omega)=0$ -3034,$U \ge U_s$ -3035,$g_i$ -3036,$\lambda=(1-\alpha_p)^{-1}$ -3037,$\bar P = a - \bar Q$ -3038,$p(1-\nu(p)-il(p))$ -3039,$Z_{a}(x)=g(S_X(a))/S_X(a))$ -3040,$X(\omega_1)a'$ -3042,"$0.1, 0.4, 0.5,\dots, 0.9$" -3043,"$I(q,p) \ne I(p,q)$" -3044,$k=-\log(p)/u$ -3045,$S(x_{i-1})-S(x_{i})=S(x_i-(x_i-x_{i-1}))-S(x_i)=-S'(x'_i)(x_i-x_{i-1})=f(x'_i)(x_{i}-x_{i-1})$ -3046,"$x,y\in C$" -3047,$g^{-1}(x)\le s$ -3048,$f(x) < f(y)$ -3049,$\iota^{\star}$ -3050,$(1+\rho)\mathsf E[C]$ -3051,$Z(\omega)=\dfrac{1}{1+r}\dfrac{\mathsf Q(\omega)}{\mathsf{P}(\omega)}$ -3052,$\zeta_{s} = \Phi^{- 1}(s)$ -3053,$\displaystyle\int_\Omega X(\omega)p(\omega)\Pr(d\omega)$ -3054,$\iota$ -3055,$\rho \ge \mathsf E[X]$ -3056,$d^* = D/L^*$ -3057,$\mathbf{K}$ -3058,"$\rho(X) = \max\{\rho_c(X), \mathsf{TVaR}_{0.8}(X) \}$" -3059,$S(x)=1$ -3060,$g(s)=\Phi(\Phi^{-1}(s)+\lambda)$ -3061,$\mathbf r$ -3062,$\rho_g(X)<\infty$ -3063,$M=\iota Q$ -3064,$\mathcal D(X)+\mathsf E[X]$ -3065,$q + 2pq + 3p^2q+\cdots=q(1+2p+3p^2+\cdots)=1/q$ -3066,$g'(1-p^* )=1$ -3067,$-1$ -3068,$\Pr(X< q(p))\le p \le \Pr(X\le q(p))$ -3069,"$ In general, define $" -3070,"$(4,2)$" -3071,$\alpha=1$ -3072,$\alpha_{Cat} \le \beta_{Cat}$ -3073,$R_S$ -3074,$dt$ -3075,$E_i\in\mathcal F$ -3076,"$\bar P_{0,0}:=\rho(Y_{0,0})$" -3077,$\mathcal V(X)=\mathsf E[X]+c\mathsf E[X^2]$ -3078,$a_1 < a_0-X_1$ -3079,$\mathsf E[X]=28$ -3080,$ is different from the contact function $ -3081,$t < 2/3$ -3082,"$\omega\in[0,1]$" -3083,"$h(x):=H(x, 1, t)$" -3084,$g(s)=3s$ -3085,$p=0.5$ -3086,"$\lambda\rho(X) + (1-\lambda)\rho(Y) \le \max(\rho(X),\rho(Y))$" -3087,$\{n_s\}$ -3088,$S(X_0)$ -3089,$r_m$ -3090,$X_i=X_i(a)$ -3091,$\mathsf{CTE}_p(X) := \mathsf E[X \mid X \ge \mathsf{VaR}_p(X)]$ -3092,$\bar Q_{act} = \bar Q - F_0$ -3093,$\beta_i/\alpha_i$ -3094,$\bar P(a)= (1-e^{-a\alpha\beta})/(\alpha\beta)$ -3095,$S(x)=e^{-x/\mu}$ -3096,$\mathsf E[Xe^{\pi Z}]/\mathsf E[e^{\pi Z}]$ -3097,$M(x)/(1-S(x))$ -3098,$\Pr(Xq_X(p)}$ -3105,$d(g(S(x))/dx=g'(S(x))f(x)$ -3106,$p={{p}}$ -3107,$\rho(X) = \mathsf E[X] + c\mathsf E[X-\mathsf E[X]]^+$ -3108,$v\in V$ -3109,$\iota=0.10$ -3110,$\hat p > p$ -3111,"$C(S_0, a, t)$" -3112,$c = 0.5(0.5)2.5$ -3113,$M = 0.603$ -3114,$\Pr(A\cup B)=\Pr(A)+\Pr(B)$ -3115,$A/(A-P)$ -3116,"$A,B,C,D$" -3117,$h=\sin(77 s)$ -3118,$\sup \{ \mathsf E[X\mid A] \mid \Pr(A) > 1-p) \}$ -3119,"$\mathbf{X\,p}$" -3120,$g(s)=1-(1-s)^3$ -3121,$\phi\in \mathcal E$ -3122,$F_1 \prec_1 F_0$ -3123,$\lim_{s \downarrow 0}1/g'(s)$ -3124,$\Delta_j =g'(s_j-)-g'(s_j+)=\phi((1-s_j)+)-\phi((1-s_j)-)$ -3125,$\Omega_0:=\{\omega\in \Omega\mid X(\omega)=\max(X)\}$ -3126,$f(s)\le s$ -3127,$\bar\iota(a)$ -3128,$j>0$ -3129,$n=1$ -3130,$S_0$ -3131,$g(S(x))=g(S(x-))=1$ -3132,$A\subset\mathbb{R}$ -3133,$f(p)=(1-p)\phi'(p)=-(1-p)g''(1-p)$ -3134,$ then $ -3135,$\epsilon_1$ -3136,$i>0$ -3137,"$0, 1, 90$" -3138,$\beta_1<\alpha_1$ -3139,$\nu p$ -3140,"$n=1, p=1/{{p}}={{pf}}$" -3141,$q(U)=F^{-1}(U)$ -3142,$\sqrt{0.1}=0.316$ -3143,$\ge 0.95$ -3144,$vL + da$ -3145,$g'(s)=\nu$ -3146,$b=0.5$ -3147,$\mathbf{x_1}$ -3148,$a < b_h$ -3149,$L>d$ -3150,"$a_{0,2}$" -3151,$a_i=a(X_i; X)$ -3152,$\mathsf E_F(h(X))$ -3153,$\dot f(t)=a(x)$ -3154,$A^c$ -3155,"$\mathsf P((a,b])=b-a$" -3156,$1-p$ -3157,$\lim_{s \downarrow 0} s/g(s) = \lim_{s \downarrow 0}1/g'(s)$ -3158,$\rho_\mu$ -3159,$\bar F(a)$ -3160,$P(X_{0}(a_{gc}))$ -3161,$\mathbf{\alpha_2}$ -3162,$20+8t>20+10t$ -3163,$b=1$ -3164,$p_0 = p^\ast = p_1$ -3165,$Z\in L^1$ -3166,$Y$ -3167,$g(S(x))=u$ -3168,$\phi'(s)\ge 0$ -3169,$x\mapsto (x-d)_+^{n}$ -3170,$\{X \le x^*\}$ -3171,$\mathbf{M_1\Delta X}$ -3172,"$X_1=0,0,0,0,1,1,2,3,20, 400$" -3173,$\mathsf E[XZ]$ -3174,$m_1=m_2$ -3175,"$\dfrac{\partial\rho}{\partial P} = \dfrac{0.4^2 P}{\rho(P,R,a)}$" -3176,$u = g(S(x))$ -3177,$\mathsf E[\mathsf E[Z\mid X]]=\mathsf E[Z]$ -3178,$\rho(X)=g(q)$ -3179,$\bar M=\bar P-\bar S$ -3180,$\mathsf E[Z\mid X]=0$ -3181,$\mathbf{s_1}$ -3182,$q_Y(1-U)$ -3183,$h(s)$ -3184,$f^{-1}(A)\in\mathcal B$ -3185,$\beta_1g-\alpha_1S$ -3186,$X_2(a)$ -3187,$g'(s)=bs^{b-1}$ -3188,$\mathsf P(A)=1-p$ -3189,$dF(x)$ -3190,"$(0,g_0)$" -3191,$\kappa_1(X)$ -3192,$x \mapsto -x$ -3193,$A(1_{X>x_1} + 1_{X>x_2})= A(1_{X>x_1}) + A(1_{X>x_2})$ -3194,${Z}_p \le c$ -3195,$X:\Omega\to\mathbb{R}$ -3196,$C_1+\cdots + C_n$ -3197,$0=\Pr(X<1)<\Pr(X\le 1)=1/6$ -3198,$\rho(X)=\mathsf E[XZ]$ -3199,$\tilde X\wedge a$ -3200,$d+v=1$ -3201,"$\Omega=[0,1]$" -3202,$q_Y$ -3203,$D\rho_X(\cdot)$ -3204,$g^{-1}(u)$ -3205,$\sum_{i}X_{i} = X$ -3206,$g_{ROE}$ -3207,$>1-p$ -3208,$a=\mathsf{VaR}_{1-\tau}(X)$ -3209,$h(x):=f(x)/S(x)$ -3210,$X_n(\omega)=1$ -3211,$\mathbb{R}$ -3212,$S_Y$ -3213,$\chi^2$ -3214,$X=X' + X''$ -3215,$(X\wedge a)\Delta g$ -3216,$f(x)\approx 0$ -3217,$ but if $ -3218,$Q=(a-EL)/(1+r)$ -3219,$\max_{\mathsf{Q}} \mathsf E_\mathsf{Q}[0] -\alpha(\mathsf Q) =\max_{\mathsf{Q}} -\alpha(\mathsf Q)= -\min_{\mathsf{Q}} \alpha(\mathsf Q) = 0$ -3220,$a\ge 0$ -3221,$\mathbf{F}$ -3222,$N=5$ -3223,$\{ X=x \}$ -3224,"$D_n,D_n^*$" -3225,$X_d$ -3226,$Z=\mathsf E Z$ -3227,$\rho_g(X)=\mathsf E_{\mathsf{Q}}[X]$ -3228,$P=g(s)$ -3229,$\int xdF(x)=\int xf(x)dx$ -3230,$\mathsf E[X] = \mathsf E[\mathsf E[X\mid Y]]$ -3231,$X({\mathbf{v}})$ -3232,$g(s)=s^{0.9}$ -3233,"$X_{t,1}$" -3234,$x=X(p)$ -3235,$\mathsf E[X_i(a)]$ -3236,$\mathsf E[1_{U_X\ge p}]=\mathsf E[B]$ -3237,$\hat s$ -3238,$\mathsf E X + c{ X-MX }$ -3239,$\sigma_i^2$ -3240,"$(1-s, 1-g(s))$" -3241,$\ge$ -3242,$h(p)$ -3243,$\max(X)=1$ -3244,$R_f$ -3245,$\mathbf{X_1}$ -3246,$\phi(s)=0$ -3247,$P = \mathsf E[X] + \pi \mathsf{Var}(X)$ -3248,$\mathsf{Var}(X+c)=\mathsf{Var}(X)$ -3249,$\mathsf{TVaR}_{0.975}$ -3250,$l^\infty$ -3251,$\mathsf E[Yg'(S(X))]$ -3252,$x_p=\mathsf{VaR}_p(X)$ -3253,$\sum v_iX_i$ -3254,$R$ -3255,$s=0.5$ -3256,"$(1-S(x),x)=(p,q(p))$" -3257,$0!=1$ -3258,$\rho(U)=1$ -3259,$x=1000$ -3260,$m(1)=0$ -3261,$a_{t} = a_{t-1}$ -3262,$A=\{X(\omega) > x\}$ -3263,$\beta_i(a)g(S(a))=\mathsf E_{\mathsf{Q}}[(X_i/X) 1_{X>a}]$ -3264,$\mathsf E[X\wedge a(X)]$ -3265,$0.1 < s < 0.2$ -3266,$p < 1$ -3267,$g(0+)\ge 0$ -3268,"$3.129=\lambda \sigma(Y_{0,0})$" -3269,$\mathsf E[X\mid \mathcal F_t](\omega)$ -3270,$\beta_i(x)/\alpha_i(x)> 1 > S(x) / g(S(x))$ -3271,$S(x)\approx k x^\alpha$ -3272,"$\mathit{EGL}_{gc}(a)>\max(0, \mathit{EGL}_{ro}(a))$" -3273,$\alpha_1(99)=0.1$ -3274,$\mathsf{TVaR}_p(X)=80$ -3275,$m\ge n$ -3276,$(a-X)^+$ -3277,$M_1dX$ -3278,$(X\wedge l)(\omega)=X(\omega)\wedge l$ -3279,$a=a[X]$ -3280,$\mathbf{a_{2}}'$ -3281,$a^{\star}(X)-a(X)$ -3282,$\mathit{PV}_{r_X}(X) + \mathit{PV}_{r_f}(\text{UW profit tax})$ -3283,$A-A\Phi(d^*)=A\Phi(-d^*)$ -3284,$\Pr(\varnothing) =0$ -3285,"$j=0,\dots, m-1$" -3286,$n\Pr(Y\le y_c)$ -3287,$(P-S)/(a-P)\ge \iota$ -3288,$(1-p)^{-1} \min_x x(1-p) + \mathsf E[(X-x)^+]$ -3289,$r^*$ -3290,$F(x)=\P(X\le x)$ -3291,$\bar Q(x)$ -3292,$q^-(p)=\sup\ \{ x\mid \Pr(X < x) < p \}$ -3293,"$(a,b] \subset [0,1]$" -3294,$\mathsf E[X_iX]$ -3295,$\mathbf{s}$ -3296,$\rho(X_0) = \mathsf E[X_0Z]$ -3297,$\iff$ -3298,$\exp$ -3299,$\mathbf{1_{X>x}}$ -3300,$D>L$ -3301,"$\mathsf{biTVaR}_{0,1}^{0.0476}$" -3302,$\iota(0.5)=\iota^{\star}$ -3303,$\kappa_{2}$ -3304,$n \ge 1$ -3305,$Y\in L^\infty$ -3306,$\lim_{s \to 1}{\mathsf E[ r_{s} ] = - 1}$ -3307,$0 \ge \rho(-X+a)=\rho(-X) + a \ge -\rho(X) +a$ -3308,$\mathsf E[XZ]=\mathsf E[X\mathsf E[Z\mid X]]=0$ -3309,$a>0$ -3310,$\mathbf{X_2}$ -3311,$1\le p \le \infty$ -3312,$\mathit{PFL}$ -3313,$X_i(a)=X_i\dfrac{X\wedge a}{X}$ -3314,$\mathbf\Omega$ -3315,$g'(1)$ -3316,$0\le \alpha\le 1$ -3317,$g(S(x))=0$ -3318,$\rho\ge 0$ -3319,$\nu(p)=1/(1+\iota(p))$ -3320,"$[0,\infty)$" -3321,$\uparrow$ -3322,$a_i + b_i\ \mathit{EL}$ -3323,$\mu t + \sigma dW_t -\sigma^2 dt /2 +o(dt)$ -3324,$h$ -3325,$4/6$ -3326,$X_2=c_2+2Y$ -3327,$-Y\ge 0$ -3328,$S(x_2)(x_3-x_2)$ -3329,$0\le\lambda \le 1$ -3330,$\mathsf E[X_iZ]=\rho_g(X)/2$ -3331,$x \ge x^\ast$ -3332,$1/4 < s\le 1$ -3333,$A_X = 5.976$ -3334,$\rho(X+Y)\ge$ -3335,$\mathbb{Q}(\Omega_a) >0$ -3336,$\mathbf{t+2}$ -3337,$M = r K$ -3338,$X_n(\omega)=n$ -3339,$r = 0.6565$ -3340,$\nu^{\star}$ -3341,$-\rho(-X) =b-\rho(b-X)$ -3342,$\mathsf E_{\mathsf{Q}}[Y\mid X]\mathsf E[Z\mid X] = \mathsf E[YZ \mid X]$ -3343,$\alpha_1SdX$ -3344,"$a(\cdot, p)$" -3345,$\tau \ge t+d$ -3346,$\mathsf E[u(P-X)]=0$ -3347,$\mathsf E[X_1]=4.75$ -3348,$\mathbf{Q}$ -3349,$\mu(\{p\})=1$ -3350,$c\approx -\sigma^2u''(w)/u'(w)$ -3351,$X\wedge a=\sum X_i(a)$ -3352,$\rho(X)\ge\rho(X+Y)\ge \rho(X)+\mathsf E[YZ]$ -3353,$\mathbf{M\Delta X}$ -3354,$\mathsf E[XB]$ -3355,$\kappa_i(x)=E[X_i \mid X=x]$ -3356,$\lambda_0$ -3357,$\epsilon /2^{n+1}$ -3358,$\nu(x)$ -3359,$S(x)=\exp(-\int_x^\infty h(t)dt)$ -3360,$g(P)$ -3361,$2x$ -3362,$P(a) = g(S(a))$ -3363,$[F(x)](\cdot)$ -3364,"$\Omega=\{\omega_1, \ldots, \omega_6\}$" -3365,$\mu-\sigma^2/2=0.0992$ -3366,$F(p)=0.6$ -3367,$\rho(X_j)$ -3368,$\mathbf{M_2\Delta X}$ -3369,$y=a$ -3370,"$\mu,\sigma$" -3371,$g_i=g^{-1}(u_i)$ -3372,$u=0.1$ -3373,$1_{U>s}$ -3374,"$\rho(X)=\int g(S(t))\,dt$" -3375,$S=\mathsf E[X\wedge a]$ -3376,$\{ x \mid F(x) \ge p \}$ -3377,"$\mathsf E_{\mathbb{Q}}[Y]=\mathsf E[Y\,g'(S(X))]$" -3378,$g(s)q$ -3379,$\mathsf{VaR}_1(X)$ -3380,$\sigma_L$ -3381,$\mathsf E[(X-a)^+]/\mathsf E[X]$ -3382,$Q=1-g$ -3383,$L_a^{a+y}(X)$ -3384,$\rho(X)=\mathsf{SD}(X)$ -3385,"$\int_{[a,b]} h(x)dF(x)$" -3386,$\bar\nu(a)=1/(1+\bar\iota(a))$ -3387,$-g''(1-p) = \phi'(p) = (1-p)^{-1}f(p)$ -3388,$g(S_X(X))$ -3389,"$(\Omega, \mathcal F, \mathsf P)$" -3390,$0\le p\le 1$ -3391,"$D^f\rho_{X\wedge a,X}(\cdot)$" -3392,$P = \mathsf E[X] + \pi \max(X)$ -3393,$\mathbf{g_1(s)=s^{0.4}}$ -3394,$V^{\ast}(1)=p/(1+r-p)$ -3395,$H_k(X)=H_{g_k}(X)$ -3396,$\partial\bar P/ \partial a$ -3397,$f(x)/S(x)$ -3398,"$X_{t,d}$" -3399,$a_{d} = \mathsf E[Y_{d}]+4\sigma(Y_{d})$ -3400,$a=\mathsf E_\mathsf{Q}[X]$ -3401,$\Delta S=0$ -3402,$\mathcal V(X)=\frac{1}{1-p}\mathsf E[X^+]$ -3403,$s_3=1$ -3404,$0< a\le 1$ -3405,$B(1_{X\le x})$ -3406,$2^{-t+1}$ -3407,$\beta < \alpha$ -3408,"$\bar P_i(\mathbf{v},a)$" -3409,$\sum \Delta g(S)_jX_j$ -3410,$\mathsf E X+\lambda\sigma(X)$ -3411,$\rho(0) = \rho(0+0)\le \rho(0)+\rho(0)$ -3412,$a<\infty$ -3413,$X=Y/\lambda$ -3414,$a\alpha_i(a)$ -3415,$q(1-s)$ -3416,$\mathbf{X_{1}}$ -3417,$a_2 = 2.157$ -3418,$\mathsf{TVaR}_p = 20(0.55x_{67}+x_{68}+x_{69}+x_{70})/71$ -3419,$\mathsf{TVaR}$ -3420,$q(\psi)$ -3421,$\mathsf E_\mathsf{Q}[X1_A] / \mathsf E_\mathsf{Q}[1_A]$ -3422,$a_{ro}:=\mathit{VaR}_{p}(X_{-1})={{a_x0}}$ -3423,$( x_{(j)}-x_{(j-1)} )$ -3424,$l(\mathbf X)$ -3425,$p\nu(p)$ -3426,$w_{0.75}$ -3427,$0.7 \ge p < 0.8$ -3428,$\omega_1=1$ -3429,"$(1-g(S(x)),x)=(p,q(1-g^{-1}(1-p))$" -3430,$v=1/(1+\iota)$ -3431,$f$ -3432,$\rho(X)=\mathsf E[h(X)L(X)]$ -3433,$a(X_i)=2.665$ -3434,$\mathsf E[e^{kX}]$ -3435,$\mathbf{B}'(0) = -3\mathbf{P_0}+3\mathbf{P_1}$ -3436,$g_3(s)=s^{0.7}$ -3437,$1-\hat p$ -3438,$P(A\cup B)\le P(A)+P(B)$ -3439,"$(2,-\mathsf x*0.75)$" -3440,$\iff \rho$ -3441,$0\le s\le \epsilon$ -3442,$\rho(X)\le c$ -3443,$X_n(\omega)\to 0$ -3444,$q(p)=25$ -3445,"$(0,3)$" -3446,$g(s)=sv+d$ -3447,$\mathbf{2\mathsf{VaR}_p(X_1)}$ -3448,$a=P+S$ -3449,$\mathsf E[(X-a)^+]$ -3450,"$(x,y)\not=(0,0)$" -3451,$\bar P_0$ -3452,$S=1-F$ -3453,$-t$ -3454,$f(x) = \dfrac{dF}{dx}$ -3455,$-g''(s)=\alpha(1-\alpha)s^{\alpha-2}$ -3456,$\sigma=1$ -3457,$P(a)=1-Q(a)=1-h(F(a))$ -3458,$\delta=\dfrac{\iota}{1+\iota}=\dfrac{M}{a}$ -3459,$s\le s^*$ -3460,"$\mathbf{j, p, S, \kappa_1, \Delta X, \Delta(X\wedge a)}$" -3461,$a' := (1-S)\Delta X$ -3462,$w/s = g'(s-) - g'(s+)$ -3463,$e^{\mu_L}-1$ -3464,$X=m$ -3465,$k(s)$ -3466,$\mathsf Q(A)=\int_A f(\omega)\mathsf P(d\omega)$ -3467,$\Pr(X_n>\epsilon)\to 0$ -3468,$(g-S)dX$ -3469,"$k, b$" -3470,$p^*$ -3471,$\int_0^\infty xf(x)dx$ -3472,$\Delta P$ -3473,$\mathbf{g(S)\Delta X}$ -3474,$r$ -3475,$s+\delta p = 1-\nu p$ -3476,$\mathsf{Var}(\lambda X)=\lambda^2\mathsf{Var}(X)$ -3477,"$m_0, s_1, m_1, s_2, m_2$" -3478,$=\displaystyle\int_0^\infty x \P_X(dx)$ -3479,$\mathit{NPV}_1 = \bar Q - \bar Q_{act} = F_0$ -3480,$\rho_g(X)=35.2$ -3481,$Z=Y-X$ -3482,$\mathsf{TVaR}_{0.642}$ -3483,$g(S(x_B))-g(S(x_B-))$ -3484,$u = \alpha_i(x)S(x)$ -3485,$\alpha_1 < \alpha_2$ -3486,$Z(g(s))=Z(s)+\lambda$ -3487,$\mathit{NPV}_{\infty}=a_xF_0$ -3488,$e^{-kX}/\mathsf E[e^{-kX}]$ -3489,"$X_{1,0}=\cdots=X_{m,0}=X_0=0$" -3490,"$\Omega=\{\omega_1, \omega_2 \}$" -3491,$a(X_i+X_j) < a(X_i)+a(X_j)$ -3492,$m=0.25$ -3493,$\mathbf{M=g(S)-S}$ -3494,$\{Y\mid Y\preceq_2 Z\}$ -3495,"$(de.east |- lee.north)+(0.375,0.25)$" -3496,$c(\{i\})=c(i)$ -3497,$\hat g(s)=1-g(1-s)$ -3498,$W_{s+t}-W_s$ -3499,"$(1,2)$" -3500,$1-s$ -3501,$D_2$ -3502,$x=200$ -3503,$\mathbf{v}$ -3504,"$(0,0,0,0,0,5,0,0,0,5)$" -3505,$P=l + \delta(a-l)$ -3506,$S/L$ -3507,"$\int_0^a F(t)\,dt$" -3508,$\mathbf{X_3}$ -3509,$\int_0^s \mu(dt)/(1-t)$ -3510,$Z\circ T$ -3511,$g(S(a))$ -3512,$\mathsf E[\iota Q] = \mathsf E[\iota]\mathsf E[Q]$ -3513,$\mathcal M_\rho$ -3514,$F(a+)=\lim_{x\downarrow a} F(x)$ -3515,$f<1$ -3516,$\mathcal F_0\times \mathcal F_1$ -3517,$\alpha>1$ -3518,$\rho(Y)$ -3519,$\mathsf E_{\mathsf{Q}}[(X - a)^+] = \rho((X - a)^+)$ -3520,$\mathsf Q(\omega)\ge 0$ -3521,$\lim_{s \uparrow 1}g'(s)$ -3522,$k>2$ -3523,$S\to Y$ -3524,$\Pr(\Omega)=1$ -3525,$s'(t)$ -3526,$g'\circ S_X$ -3527,$s=0.1$ -3528,$g = s/(1-f)$ -3529,$\Delta g(S_j)=g(S_{j-1})-g(S_j)$ -3530,$g(s)=A(1_{U < s})$ -3531,$A\wedge L$ -3532,"$5^{-1},5^{-2},5^{-3},\dots$" -3533,$g'(1-s)+g(0+)\delta_1$ -3534,$+$ -3535,$c(\alpha)x^\alpha g(x)$ -3536,$\mathit{NPV}_{\infty} = a_xF_0$ -3537,$v/\sqrt{n}$ -3538,$X_h$ -3539,"$\mathsf{cov}(X_i,X)$" -3540,"$(p,t)$" -3541,$e^{-rt}S_t$ -3542,$9+1$ -3543,$(x-d)^+ \wedge l$ -3544,$\mathsf Q(B) = \mathsf P(A\cap B)/\mathsf P(A)=\mathsf P(A\cap B)/(1-p_0)$ -3545,$Y_i$ -3546,$\sqrt{x}$ -3547,$\rho(X-X)=\rho(X)+\rho(-X)=0$ -3548,$dG/dF=g'(S(x))$ -3549,$D_m\subset D_n$ -3550,$\mathsf E[X_m\mid X_{m+n}=x]=mx/(m+n)$ -3551,"$[0,1]\subset\mathbb R$" -3552,$r-1$ -3553,$d_f = r_f / (1+r_f)$ -3554,$\hat q(p)=q(1-g(1-p))$ -3555,$X=Y$ -3556,$U^{1/b}$ -3557,$X\preceq_1 Y$ -3558,$E(X-q(X))^+$ -3559,$X_{-2}$ -3560,$t=U_X(s)$ -3561,$3^{30}=2.06\cdot 10^{14}$ -3562,$\rho(kX)\ge k\rho(X)$ -3563,$M(x)=P(x)-S(x)$ -3564,$H$ -3565,$a=\mathsf{VaR}$ -3566,$\int X_n=1$ -3567,"$\displaystyle\int_0^a \kappa_i(x)f(x)\,dx + a\alpha_i(a)S(a)$" -3568,$\alpha_1(90) = (0.0816 \cdot 0.0625 + 0.1 \cdot 0.0625)/(0.0625+0.0625)=0.01135/0.125=0.0908$ -3569,$\mathsf E[X_i\mid X=x]$ -3570,$c_i$ -3571,$0 \le X_i(a) \le X_i$ -3572,$\sup_i f_i$ -3573,$D\rho_X(X_1)=6.2085$ -3574,$+\mathsf{NORIPOFF}$ -3575,"$(a,b)$" -3576,$\mathsf E[g(X_n)]\to \mathsf E[g(x)]$ -3577,$\tilde{\mathbb{Q}}$ -3578,$\mathsf E[X_i(1) \mid X(\mathbf{v}) = q_{\mathbf{v}}(p)]$ -3579,$t\downarrow 0$ -3580,$\mathcal{G}=\sigma(X)$ -3581,$\pi$ -3582,$\mathbf{g_4(s)=s^{0.9}}$ -3583,$h(x)=-d/dx(\log(S(x)))$ -3584,$x=8$ -3585,$X\_{2}$ -3586,$dS$ -3587,$\sum \alpha_i S\Delta (X\wedge a)$ -3588,"$g'(s) = \frac{1-w}{1-p_0}1_{[0, 1-p_0)}(s) + \frac{w}{1-p_1}1_{[0, 1-p_1)}(s)$" -3589,$\mathscr{E}$ -3590,$\Pr(A\le t)= 1/2 + \Pr(U\le t) /2 = 1/2 + t/2$ -3591,$pX$ -3592,$g(S(a))/S(a)$ -3593,$\mathsf E[X\mid \mathcal F'](\omega)$ -3594,$\rho_c(Y)=\mathsf E[Y]$ -3595,$\sum X_i(a)\Delta g(S)$ -3596,"$(p,q(1-g^{-1}(1-p)))$" -3597,$0\le \pi\le 0.5$ -3598,$\bar\delta=\bar\iota/(1+\bar\iota)$ -3599,$q^-(F(x))=x$ -3600,$1-g(s)$ -3601,$P=L + d(a-L)$ -3602,$p\not=0.75$ -3603,"$a=0, b=\alpha$" -3604,$\mathbf{q}$ -3605,$\{ X=\mathsf E[X] \}$ -3606,"$\bar S(a)=\int_0^a S(x)\,dx$" -3607,$X=X\wedge a + (X-a)^+=\sum_i X_i(a) + (X-a)^+$ -3608,$r_f = 0.01$ -3609,$X_2=X-X_1$ -3610,$c_1$ -3611,"$\displaystyle\int_\Omega g(X(\omega), \omega)\Pr(d\omega)$" -3612,$u^{(n-1)}$ -3613,$(r-\sigma^2/2)t$ -3614,$\tau_i=\tau$ -3615,$\tau=\tau_i=0$ -3616,$a=a(s)$ -3617,$f(L)=L$ -3618,$f(L) \le L$ -3619,$p=0.283$ -3620,$g'(s)=\alpha s^\alpha/s$ -3621,$n-4$ -3622,$xdF(x)$ -3623,$\mathsf{TVaR}_{0.8}(X)=25$ -3624,$X_0=X_1=0$ -3625,$Q_X$ -3626,$\mathsf{TVaR}_{p^\ast}(X)=\bar P$ -3627,$P_X(A)=0$ -3628,$L > a$ -3629,$f=0$ -3630,$f(x)dx=dp$ -3631,$P_X(A)=\mathsf P(X\in A)= F(b)-F(a)$ -3632,$Z(a')=g(S_X(a))/S_X(a))$ -3633,"$X_i(\omega), i=1,...,N$" -3634,$\Pr(X\ge x_0)=p_-$ -3635,$v_f(\mathsf E_Q[X_i] - \dfrac{\mathsf E_Q[X_i]}{\mathsf E_Q[X]}\mathsf E_Q[(X-A)^+])$ -3636,$\alpha(\mathsf Q)$ -3637,$\mu(\{p_1\})=w$ -3638,$\phi(p)$ -3639,$\rho(X)=\mathsf E[Xg'(S(X))]=\mathsf E[\sum_i X_i g'(S(X)))]=\sum_i \mathsf E[X_ig'(S(X))]$ -3640,$G\mathsf X$ -3641,$\omega=0$ -3642,$P-D$ -3643,$X>a$ -3644,$\iota=$ -3645,$\lim_{t\to 0}a(X_1; X+tX_1)=a(X_1;X)$ -3646,$e = P/C$ -3647,$\Pr(|X_n(\omega)-X(\omega)|>\epsilon)\to 0$ -3648,$\mathsf E[(a-X)^+]=\int_0^a F(x)dx$ -3649,$t^\star=1/2$ -3650,$t+1$ -3651,$1-B_p=B_{1-p}$ -3652,$\bar M(x)$ -3653,$X\not\preceq_n Y$ -3654,$0\le x < a$ -3655,$\mathsf E[ X_i \mid X(x) = q_{x}(p)]$ -3656,"$Z_2:=\sum_{t+d=2} Y_{t,d}$" -3657,$ since the contact function $ -3658,"$c_1+c_2=(c(1) + c(1,2) - c(2) + c(2) + c(1,2) -c(1))/2=c(1,2)$" -3659,"$(-\infty, \infty)$" -3660,$\Pr(B)=\Pr(A)$ -3661,$\mathsf{Q}(A)=\mathsf E[1_AZ]=0$ -3662,$a(X)\equiv a$ -3663,$x^{**}$ -3664,"$D^f\rho_{X\wedge a,X}(X_i)$" -3665,$X(p)=q(T(p))$ -3666,"$(1-S(x), x)$" -3667,$\tilde X_1+\tilde X_2\succeq^2 \tilde X_1$ -3668,$S_{\mathbf{v}}$ -3669,$\mathsf{VaR}$ -3670,$\bar S_i$ -3671,$\alpha_iSdX$ -3672,$\{X(\mathbf{v}) = q_{\mathbf{v}}(p)\}$ -3673,$c=0.5$ -3674,$K$ -3675,$g(p)/p-1$ -3676,$a(X_i; X)$ -3677,$\log(1+\mu t + \sigma dW_t)=\mu t + \sigma dW_t +o(dt)$ -3678,$\max(X)$ -3679,$x>\sup(X)$ -3680,$M=\inf\{ x\mid S(x)=0\}$ -3681,$\mathsf{VaR}_\pi(X)$ -3682,$\mathbf{\kappa_2}$ -3683,$-k<0$ -3684,$X_n=Y_1+\cdots +Y_n$ -3685,$^{}$ -3686,$\mathsf{CTE}_p(X)=(8+12+25)/3=15$ -3687,$p \ge 0.9$ -3688,$S_0=1000$ -3689,$1_{U0}$" -3707,$\prod_{n\ge N}(1-\frac{1}{n})=0$ -3708,$X\le 0$ -3709,$\mathsf E[1_A]$ -3710,$\rho(W)=\mathsf E[W]+\lambda\sigma(W)$ -3711,$g(s)=s^{0.8}$ -3712,$q \cdot X$ -3713,$p=0.1$ -3714,"$(p, q(1-g^{-1}(1-p)))=(p, q(\hat p))=(p, \hat q(p))$" -3715,$\mathsf P(X\le q_X(p))=p$ -3716,$\mathsf E[e^{X_t}]=e^{\mu t + \sigma^2t /2}$ -3717,"$\rho_1,\rho_2$" -3718,$P/(A-P)=P/Q$ -3719,$-\rho$ -3720,"$\rho_2(X)=\mathsf E[X] + \mathsf{cov}(X,Z)$" -3721,$\alpha_1(98)=0.1$ -3722,$1+2c(1-\Pr(Z>\mathsf E Z))$ -3723,$\pi=1.2613$ -3724,$8+11.1667=19.167$ -3725,$\gamma=0.421$ -3726,$\beta_i(x)/\alpha_i(x) < g(S(x))/S(x)$ -3727,$h(p)=s^3$ -3728,$\psi$ -3729,$f(R) = \mathsf E[f(X)]$ -3730,$\mathsf{VaR}_p(X)=\mu + \sigma \Phi^{-1}(p)$ -3731,$\mathsf E[X_1\tilde Z]=\mathsf E[X_2\tilde Z]=500$ -3732,$B_k$ -3733,"$Binomial(s,N)$" -3734,$x=S^{-1}(g^{-1}(s))$ -3735,$e^{-rt}$ -3736,$\mathsf{VaR}_{p^*}$ -3737,$=\mathrm{MV}(y-T(X))^+$ -3738,$p^{* }$ -3739,$\beta_Q=(a/Q)\beta_A + (P/Q)\beta_L$ -3740,$r\times n$ -3741,$F(2)=0.75$ -3742,$(80-11)\times 0.25$ -3743,"$S, S^{-1}$" -3744,$\mathsf{Q}'$ -3745,$q(0.1)=1$ -3746,"$k\mathsf E[(X_i-\mathsf E X_i)(X-\mathsf E X)]=k\mathsf{cov}(X_i,X)$" -3747,"$k=0,1,\dots,n-1$" -3748,$q(1-g^{-1}(1-p))$ -3749,$\tilde X_j$ -3750,$\bar F$ -3751,$\pm\infty$ -3752,"$c\in[0,1]$" -3753,$dg$ -3754,$p_Y<0.5$ -3755,$\mathsf E|X|<\infty$ -3756,"$(\mu,\sigma)$" -3757,$\mathsf E[Y\mid \mathcal F']$ -3758,"$(brR15 |- lee.south)+(-0.25,-0.25)$" -3759,$\pi=1.2497$ -3760,$\bar\iota$ -3761,$g(s) \ge 1$ -3762,$v(A\cup B) + v(A\cap B)\ge v(A) + v(B)$ -3763,$\bar P_\tau(a)=\bar P(a) + \tau(a-\bar P_\tau(a))$ -3764,$\nu>0$ -3765,$P_g\{X=M\}=g(0+)>0$ -3766,$\Delta=a'-a$ -3767,$\alpha_i(x)S(x)$ -3768,$\mathbf{\vert S\vert}$ -3769,$\mathsf E[X\mid\mathcal F_0]=\mathsf E[X]$ -3770,$r_h-\mu_L=r-r_L$ -3771,$-0.0012$ -3772,$\rho(X)$ -3773,$\mathsf Q$ -3774,$+1$ -3775,$\mathsf E[X](1+\pi)$ -3776,$\implies\mathsf{FATOU}$ -3777,$\mathsf E[X_0] + \mathsf{VaR}_p(X_1)$ -3778,$1-w$ -3779,$=1/(1-p)$ -3780,$Q_j = 1 - g(S_j)$ -3781,$A(X)$ -3782,$X\ge \mathsf{VaR}_p(X)$ -3783,$\mu(\{0\})=\phi(0)=g'(1)$ -3784,$p_+-p_-$ -3785,"$s\in (0,1]$" -3786,"$p\in (0,1)$" -3787,"$\lambda, \iota, \psi$" -3788,$\Pr(X_{-1}0.95}$ -3927,$g=u^2=0.01$ -3928,$100$ -3929,$X\wedge a \le X$ -3930,"$Y_{t,0}$" -3931,$s>s^\ast$ -3932,$g(s)-\hat g(s)$ -3933,$R:=\bar P_{act}-\bar S$ -3934,$Var[T]=s(1-s)/N$ -3935,$\sum w_i=1$ -3936,$\mathsf E[X] + d(\max(X)-\mathsf E[X])$ -3937,"$\{x_1,...,x_n\mid X < \max(X)-\epsilon\}$" -3938,$z=x$ -3939,$F_n(x)\to F(x)$ -3940,"$\rho(1000, 3000, 3500)$" -3941,$c=2.5$ -3942,$w(x)=e^{kx}$ -3943,$1_\omega(\omega')=1$ -3944,$g(s)=cs$ -3945,$f(t|s)$ -3946,$\displaystyle\int_0^\infty u(x)dF_X(x)$ -3947,$\Lambda\dfrac{\mu_{U}}{\sigma_U} = \dfrac{E( r_{U} ) - r_{f}}{\sigma_{r_{U}}} \left(\dfrac{\mu_{U}}{\sigma_{U}}\right)$ -3948,$f'(x_0)$ -3949,$y^{\ast}-x^{\ast} \ge \epsilon$ -3950,$\mathsf E[X_1]=\mathsf E[Y_{0}]$ -3951,$\mathbf{x_0}$ -3952,$\kappa_i(X)=\mathsf E[X_i\mid X]$ -3953,$g(S(x_{i+1}-))-g(S(x_{i}))$ -3954,$1-g$ -3955,"$d\,F(X)$" -3956,$Q(a)$ -3957,$(1+c)\mu$ -3958,$\mathsf{VaR}_{0.99}$ -3959,$dG(x)=g'(S(x))dF(x)$ -3960,$\rho(c) = c$ -3961,$n=100$ -3962,$\mathsf E[X] + \pi\mathsf{Var}^+(X)$ -3963,$\mathsf E_\mathsf{Q_2}[X_j]$ -3964,$\delta_p$ -3965,$\sigma$ -3966,$\mathsf E[X]=27.5$ -3967,$\mathsf E$ -3968,$\rho(X_0+\epsilon Y)=\mathsf E[(X_0+\epsilon Y)Z_\epsilon ]$ -3969,"$(s(t),m(t))$" -3970,$X>x$ -3971,$\sigma_U = 1$ -3972,"$(p,q(p))=(1-S(x),x)$" -3973,$f(P)=\mathsf E[f(X)]$ -3974,$w_i\ge 0$ -3975,$\int X_n\to 0$ -3976,"$r_f\ge 0, r>0$" -3977,$Z$ -3978,"$X_1,X_2$" -3979,$r_L$ -3980,"$\Omega=[0,1]\times [0,1]$" -3981,$x_i$ -3982,$P_i \ge \mathsf E[X_i]$ -3983,$a(X)$ -3984,$g(s)+g'(s)(1-s)\ge 1$ -3985,$\mathbf{\kappa_1}$ -3986,"$\mathsf{cov}(X_i,\sum_j X_j)=\mathsf{cov}(X_i,X_i)=\mathsf{Var}(X_i)>0$" -3987,$\mathsf E[X1_A] / \mathsf E[1_A]$ -3988,$\mathsf Q(A)=0$ -3989,$0-\rho(-H)$ -4010,"$Y_{0,1}$" -4011,$a(X;X)=\rho(X)=\sum_i a_i$ -4012,$\displaystyle\int_0^\infty xg'(S(x))f(x)dx$ -4013,$A(X)\not= B(X)$ -4014,$\lim_{y\uparrow x} f(y)$ -4015,$\displaystyle\int_\Omega X(\omega)\Pr^*(d\omega)$ -4016,$\psi^{-1}(p)$ -4017,$\mathcal Q\subset\mathcal M(\mathsf P)$ -4018,"$a(x_1,\dots,x_n):=a(X(x_1,\dots,x_n))$" -4019,$1-q$ -4020,$ds(t)/dt$ -4021,$X_{-3}=C'_1 + \cdots + C'_n$ -4022,$g(S(M-))/S(M-)$ -4023,$\mathsf{VaR}_{p_0}(X)=\sup X$ -4024,$p=1/2$ -4025,$y\not\in C$ -4026,$S=\Pr\{X>x\}$ -4027,$X_0=C_1 + \cdots + C_N$ -4028,$\mathsf E[g'(S(X))]=1$ -4029,"$2^0, 2^2, 2^4, ...$" -4030,$F_I$ -4031,$gdX$ -4032,$b_l \le 1 \le b_h=2-b_l$ -4033,$30+10t$ -4034,$m_1$ -4035,"$Y_{t,d}$" -4036,$F(x)=\sup\{ p\mid q(p) < x \}$ -4037,$\mathsf E[X \mid X \ge x] = \mathsf E[X 1_{X \ge x}] / \Pr(X \ge x)$ -4038,"$X_{i,j} \leftarrow \kappa_{i}(X_j)$" -4039,$1/g'(0)$ -4040,$1-g(1-p)$ -4041,$d\Pi = (r_h-\mu_L)\Pi dt$ -4042,$q(U_X) = m$ -4043,$\alpha_i(t)$ -4044,$U=X+Y$ -4045,$p^\ast$ -4046,"$\mathbf{D^f\rho_{X\wedge 30,X}(X_2)}$" -4047,"$0,0,0,1,2,5,8,12,23,40$" -4048,$0\le k < 2^m$ -4049,"$c=1,2,3$" -4050,$E[s]=0.1160$ -4051,$\mathbf{X\wedge a}$ -4052,"$\lambda([a,b]) = b-a$" -4053,$p^+$ -4054,$\mathsf E X + c{X-\tau }_p$ -4055,$S_X(t)=S_{X\wedge a}(t)$ -4056,$\mathsf E[\log(X)]$ -4057,$h(X)=X$ -4058,$D_1\supset D_2\supset \cdots \supset D_\infty$ -4059,$g''(s)=-\phi'(1-s)\le 0$ -4060,$\prec_1^*$ -4061,$X=100$ -4062,$X\wedge a(X)$ -4063,$\mathbf{X}$ -4064,$\times$ -4065,$\bar M(a)$ -4066,$\mathsf{LI}$ -4067,$(p_0 < p^\ast < p_1)$ -4068,"$c = 0.5,1.0,\dots,2.5$" -4069,$\sup X_n=1\not=\sup X=0$ -4070,$IL$ -4071,$\mathsf E[WX] \le \rho(X)$ -4072,$S(x)\leftrightarrow g(S(x))$ -4073,$\Pr(X=\mathsf{VaR}_p(X))=0$ -4074,$\lambda=\sum_i \lambda_i$ -4075,$\mathsf{TVaR}_{0.8}$ -4076,$Q = M/\iota$ -4077,$(a_i)_i$ -4078,$g(s)=d+sv$ -4079,$p\nu_p$ -4080,$\mathsf{TVaR}_p(X)=\mathsf E[X\mid X >\mathsf{VaR}_p(X)]=\sum_i\mathsf E[X_i\mid X>\mathsf{VaR}_p(X)]$ -4081,$f_i$ -4082,"$X_{0,2}$" -4083,$aq_X(p) \}$ -4108,$S(x)>>0$ -4109,$q_B \le q_C$ -4110,$\mathsf{TVaR}_{0.75}$ -4111,$g'(s) < \infty$ -4112,$\hat p$ -4113,$\kappa_i(q(1-g^{-1}(1-\tilde p)))$ -4114,$q^-(p)$ -4115,$\rho(X-\rho(X))=\rho(X)-\rho(X)=0$ -4116,$g_0$ -4117,$dt\to 0$ -4118,$\{X\in L^\infty \mid \rho(X)\le c \}$ -4119,"$Y_{2,2}$" -4120,$\mathsf P(f^{-1}(A))=\Pr(A)$ -4121,"$c_i=\displaystyle\int_0^1\dfrac{\partial c}{\partial x_i}(tx)\,dt$" -4122,$\rho(X_{-1}\wedge a_{ro})={{mvp_ro}}$ -4123,$\bar \iota = \dfrac{\bar M(a)}{\bar Q(a)}$ -4124,$\mathcal{N}_{X\wedge a}(X_i(a))$ -4125,$f'>0$ -4126,"$\bar M_{t,0}$" -4127,$E$ -4128,$p^\ast = 0.48732$ -4129,$r_P$ -4130,$S=g(S)=1$ -4131,$\mu_d = (6-d)^2$ -4132,$g(s)=0.9s + 0.1$ -4133,$\left( g(S(x_{(j)}))-g(S(x_{(j-1)})) \right) / ( x_{(j)}-x_{(j-1)} )$ -4134,$t^\star$ -4135,$1_Z$ -4136,$\Pr(E')=1-\Pr(E)$ -4137,$\omega < p^-$ -4138,$\rho_g(X)=\mathsf E[X]$ -4139,$q = s$ -4140,$a_i=\mathsf E_\mathsf{Q}[X_i]$ -4141,"$s\in (0,1)$" -4142,"$\omega\in [0,0.1)\cup [0.25, 0.35) \cup [0.5, 0.6) \cup [0.75, 0.85)$" -4143,$80-11=69$ -4144,$g'$ -4145,$\rho(X)+c$ -4146,$S(x)=(1+x)^{-\alpha}$ -4147,$r_M$ -4148,$U(2)=0$ -4149,$\alpha_i(x)$ -4150,$\sup X\le \sup Y$ -4151,$\mathsf E[X_1\mid X < 2^{-m}]$ -4152,$S(x)=\Phi((-x+\mu)/\sigma)$ -4153,$\tilde X_1 + \tilde X_2 \succeq^2 \tilde X_1$ -4154,$p=F(a)=1-S(a)$ -4155,$v\mathrm{EL}+da\ge \mathrm{EL}$ -4156,$X=X_s + X_c$ -4157,"$\mathsf{VaR}_{0.995}=64,861$" -4158,$\mathbf{X_{2}}$ -4159,$P = 3.1035$ -4160,$x=q(1-g^{-1}(1-p)))$ -4161,"$d=1,2,\dots$" -4162,$h=1$ -4163,"$k_1, k_2$" -4164,$p=0.95$ -4165,"$s^{\ast}=1/2, \lambda^{\ast}=0$" -4166,$\esssup(X)=1$ -4167,$1-p \ge g^{-1}(1-p) \implies 1-g^{-1}(1-p) \ge p \implies q(1-g^{-1}(1-p))>q(p)$ -4168,$\Pr(X_n\in A)=1$ -4169,"$x+y\wedge aX =\min(x+y,aX)$" -4170,$\mathsf{Q}(A)=\mathsf E_\mathsf{Q}[1_A]$ -4171,$H(X)x]=x+\mathsf E[X]$ -4178,$\mathbf{n}$ -4179,$\mu = t \nu$ -4180,$(1)(0.25)+(90)(0.25)=22.75$ -4181,$W = 0$ -4182,$\rho(X)=51.3887$ -4183,"$\{1,2 \}$" -4184,$\mathsf E[X] +\lambda\mathsf E[(X-\mathsf E X)^+]$ -4185,$d=i/(1+i)$ -4186,$\beta_2$ -4187,$Q=\nu a'$ -4188,$\{ X >q(p) \}$ -4189,"$g'>0, g''<0$" -4190,$Y=c\in \mathbb R$ -4191,$h(u)=1$ -4192,$\bar P_d=\mathsf E[Y_{d}]+\lambda\sigma(Y_{d})$ -4193,$\lim_{\epsilon \downarrow 0} (f(x+\epsilon)-f(x))/\epsilon$ -4194,$\omega_1$ -4195,$r>0$ -4196,"$\alpha_i(\mathbf{v}, x)$" -4197,$\omega\ge 0.4$ -4198,$\mathsf E_{\mathsf Q}[X_i]=\mathsf E[X_ig'(S(X))]$ -4199,$L(X)=(X-\mathsf E X)/\mathsf{SD}(X)$ -4200,$\bar q_{X_1+X_2}(s) \le 2\bar q(s)$ -4201,$f(t)=\rho(tX)$ -4202,$X_n\uparrow 1$ -4203,$\int S(x)dx$ -4204,$A\subset \Omega$ -4205,$\mathsf E[(X-x_l)^+]$ -4206,$(A-L)^+$ -4207,$P(x)/Q(x)$ -4208,$1+Z-\mathsf E Z$ -4209,"$\bar Q_{0,0}$" -4210,$r_U \Delta A - \Delta P$ -4211,$s_0=0$ -4212,$\mathsf E[(X-\mathsf E X)^+]={(X-\mathsf E X)^+}_1$ -4213,$g(S_{\mathsf{j}(a)})=0.5$ -4214,$-g''(s)=\alpha(\alpha-1)s^{\alpha-2}$ -4215,$\mathsf E[g'(S(X))]=\int_0^\infty g'(S(x))dF(x)=\int_0^\infty -\frac{d}{dx}g(S(x))dx=g(S(0))-g(S(\infty))=g(1)-g(0)=1$ -4216,$\bar Q(a) =a-\bar P_g(a)$ -4217,$\exp(a)$ -4218,$s\mapsto g(s)$ -4219,$\alpha X$ -4220,$c(S)\le c(T)$ -4221,$(1-\lambda)(1+\gamma)$ -4222,$1-\beta_i(t)g(S(t))$ -4223,$\mathsf E[X_ih(X)]=\mathsf E[\kappa_i(X)h(X)]$ -4224,$L_a^{a+da}$ -4225,"$a_{0,t}' := a_{0,t-1}-X_{0,t}$" -4226,$-Y$ -4227,$W_{t}$ -4228,$2^n$ -4229,$(1 - \nu F(a))$ -4230,$<$ -4231,$x=\sum_i \mathsf E[X_i\mid X=x]$ -4232,$g'\left (S_X(X)\right )$ -4233,"$X_1=(0,0,0,0,0,0,2,4,8,0)$" -4234,$R_L=R_f + \beta_L(R_M-R_f)$ -4235,$cv=0.137$ -4236,"$(2,2)$" -4237,$1/x$ -4238,$A(1_{X_1>x_1}+1_{X_2>x_2}) \le A(1_{X_1>x_1}) + A(1_{X_2>x_2})$ -4239,$\Delta=\Phi(d^*)$ -4240,$\alpha(\mathbb{Q})$ -4241,$F_g(x) = 1- g(S_X(x))$ -4242,$Z(1000)=(1-0)/(0.1-0)=10$ -4243,$\bar S(a)=\mathsf E[X\wedge a]$ -4244,$\tau=1$ -4245,$\rho(X_1) \ge D\rho_X(X_1)$ -4246,"$\mathcal Q =\{ \mathsf Q \mid \mathsf Q\ll \mathsf P,\ \alpha(\mathsf Q)=0 \}$" -4247,$\Pr(X < x)\ge 1/6$ -4248,"$X_j=\sum_i X_{i,j}$" -4249,$\mathsf{SD}$ -4250,$n>1$ -4251,$-\phi(d^*)<0$ -4252,$\rho(\tilde X+X)=\rho(\tilde X)+\rho(X)$ -4253,$a(\mathbf{v})$ -4254,$P(dx)$ -4255,$\mathsf Q(X>a)/P_X(X>a)=g(S(a))/S(a)$ -4256,$\rho(X-P)=\rho(X)-P$ -4257,$a+da$ -4258,$r_pq$ -4259,"$m\ge 1, n\ge 0$" -4260,$(1-g)$ -4261,$x=2$ -4262,$T_k$ -4263,$X\le c$ -4264,$t$ -4265,$X \succeq Y$ -4266,$\mathsf E_{\mathsf Q}[Y]$ -4267,$x=a$ -4268,"$\mathsf{biTVaR}_{p_0,p_1}^w(X)=\bar P$" -4269,$\log(X)$ -4270,$\nu^{-1}\mathsf E[\nu(X)]$ -4271,"$[0, 1-p)$" -4272,$\mathsf{VaR}_{0.95}(X)$ -4273,$S_t=a_0 + (1+c)\mu t - X_t$ -4274,$1/(1+r) = 0.893$ -4275,$D^n\rho_X(X_i)$ -4276,$A\subset \mathbb{R}$ -4277,$\bar P_t = \rho(Y_{t})$ -4278,$W_1$ -4279,$s/(1-s)$ -4280,$E_1$ -4281,"$f:[0,1]\to[0,1]$" -4282,$A\cap B$ -4283,"$(p,q(1-g^{-1}(1-p)))=(1-g(S(x)),x)$" -4284,$X_1=0$ -4285,"$\beta = \mathsf{cov}[r,r_M]/ \sigma^2_{r_M}$" -4286,"$f(x, \cdot)\in L_p(\Omega, \mathcal{F}, \mathcal{P})$" -4287,$\bar P(a)=\rho_g(L_0^a(X))$ -4288,$S_t=\exp(\mu t + \sigma W_t)$ -4289,$P_i(x)=\beta_i(x)g(S(x))$ -4290,$1-L/P = (P-L)/P$ -4291,"$x_{1,2}$" -4292,$w/(1-w)$ -4293,$\sup_\Omega |X_n - X| \to 0$ -4294,$\mathsf E_{\mathsf{Q}}[X\wedge a] \le \rho(X\wedge a)$ -4295,$\beta_i(a) g(S(a))$ -4296,$\mathsf E[Y]$ -4297,$\prec_n^*$ -4298,$2^{-t}$ -4299,$X_2/X$ -4300,$\Delta X=80-11=69$ -4301,$g(S(x))=\exp(-\alpha H(x))$ -4302,$\mathsf E[X_i \mid X=x]$ -4303,$\rho(\lambda X + (1-\lambda)\rho(X))$ -4304,$X\le Y+\Vert X-Y\Vert$ -4305,$(1-p)/(p\nu(p)^2)$ -4306,$I(F(x) < p)=\begin{cases} 1 & F(x)< p \\ 0 & F(x)\ge p\end{cases}$ -4307,$\mathsf E[U]=\mathsf E[X]$ -4308,$g(s)=1$ -4309,$x < x^\ast$ -4310,$\mathsf E[X] + c\mathsf E[(X-\mathsf E X)_+^2]$ -4311,$X\wedge a(X)\le Y\wedge a(Y)$ -4312,$t\uparrow 0$ -4313,"$\eta_{p,\alpha_1}(X) < \eta_{p,\alpha_2}(X)$" -4314,$\phi(t)=\int_0^t (1-p)^{-1}\mu(dp)$ -4315,$a_{\min}$ -4316,$\mathbf{t+1}$ -4317,$x\ge a$ -4318,$N=r_a$ -4319,$\int S(x)dx = \int xdF(x)$ -4320,$\mathsf{VaR}_{0.7}(X)=$ -4321,$\mathsf P(X=\sup(X))>0$ -4322,$\bar M_i(a) = \bar P_i(a) - \mathsf E[X_i(a)]$ -4323,$L(X)=(1-p)^{-1}1_{X\ge x_p}(X)$ -4324,$\mathsf E[X_i]$ -4325,$\mu = w \delta_{\alpha_1} + (1-w) \delta_{\alpha_2}$ -4326,$\mathsf{TVaR}_{0.95}(X)=\mathsf E[XZ]$ -4327,$A(-X)$ -4328,$\mathsf{j}(90)=6$ -4329,$a - P$ -4330,$q^-_X(0.95)$ -4331,$1100 \le x \le 1250$ -4332,$\sigma\sqrt{t}$ -4333,$(L-A)^+$ -4334,"$\int Zd\mathsf P = \int d\mathsf Q/d\mathsf P\, d\mathsf P = \int d\mathsf Q =1$" -4335,$\mathcal A$ -4336,$d\mathsf Q/d\mathsf P$ -4337,$\mathbf{X_{n}}$ -4338,$t_f$ -4339,$\hat q(p)$ -4340,$p<1$ -4341,$r < n$ -4342,$\mathcal S(X)=\mathsf{VaR}_p(X)$ -4343,"$(\omega',\omega'')$" -4344,$\mathbf{X_{2}(a)}$ -4345,"$u\in[0, 1-p]$" -4346,$k_i=a_i/v_i$ -4347,"$(s_j, g_j)$" -4348,"$(3,3)$" -4349,$T(p)$ -4350,$D/C$ -4351,"$s\in[0, 1-p]$" -4352,$p=1$ -4353,$a_{d}' = a_{d-1}-X_{d}$ -4354,$g'\left (S(X)\right )$ -4355,$p=0.75$ -4356,$\{\omega\mid X(\omega) = x_1\}$ -4357,$\tau a_i$ -4358,$d\downarrow 0$ -4359,$p\ge 1$ -4360,"$g(s)=\min(s/(1-p),1)$" -4361,$\phi'(p)\ge 0$ -4362,$\mathit{ROE}(s) = r_f + Ck(s)$ -4363,$q(p)\phi(p)$ -4364,$\mu_0=\mu_1$ -4365,$w_l=1-c\gamma$ -4366,$0.7 \le p < 0.8$ -4367,"$j=1,\dots, n$" -4368,$x=q_X(1-s)=\mathsf{VaR}_{1-s}(X)$ -4369,$\{p \ge p_-\}$ -4370,$(1-p)^{-1}$ -4371,$\rho(X)=\rho(\mathsf E[X]+X-\mathsf E[X])=\mathsf E[X] + \rho(X-\mathsf E[X])$ -4372,$\omega<1/n$ -4373,$\mathsf E[X_i\wedge a_i]$ -4374,$\tilde X = (x_{ij})$ -4375,$q(p)=c$ -4376,$a(X_i;X)\le \rho(X_i)$ -4377,$\rho(0) \ge 0$ -4378,$g'(s-)\ge 0$ -4379,$\exists$ -4380,$^1$ -4381,$x^{-\alpha}$ -4382,$k>1$ -4383,$D\rho_X(X_2)=45.1801$ -4384,$\mathsf E[X^2]$ -4385,$\mathsf E[|X|]<\infty$ -4386,$V(2)$ -4387,"$\rho_g(X)=\int_0^\infty g(S(x))\,dx$" -4388,$c(1)$ -4389,$X=X_{-1}+X_{0}$ -4390,$xf(x)dx$ -4391,$t=0.06405$ -4392,$Y_{0}=\sum_{d>0} X_{d}$ -4393,$a=Q+P$ -4394,$Y\preceq Z$ -4395,"$a_{0,0}:=a(Y_{0,0})$" -4396,$X_1+X_2\sim 2X$ -4397,$l$ -4398,$\WCE_p(X) := \sup\ \{ \mathsf E[X \mid A] \mid \Pr(A) > 1-p \}$ -4399,$r_h=r+\pi$ -4400,$\Delta Q_{ro}(a)$ -4401,$\bar Q_{0}$ -4402,$=1-\nu F(a)$ -4403,$X \le 0$ -4404,$\mathsf E[X_2(a)\mid X_1(a)=x] \le a-x$ -4405,$X^{-1}(A)\in\mathcal F$ -4406,$\sup(X\wedge a)=a$ -4407,$\mathbf{P_i} \in \mathbb{R}^2$ -4408,$\bar\nu=1/(1+\bar\iota)$ -4409,$\rho(X/n)=\rho(n(X/n))/n=\rho(X)/n$ -4410,$\mathsf x\mathsf{TVaR}$ -4411,$1_{U_X\ge p}=0$ -4412,$F(X) - F_X(X-)=0$ -4413,$\tilde X$ -4414,$m'(0) = (m_1-m_0)/s_1$ -4415,$B \in\mathcal B_p$ -4416,$1/16$ -4417,$\bar\iota(a)=\bar\iota$ -4418,$\mathsf E X + \inf_x \{\alpha_1\mathsf E[(x-X)^+] + \alpha_2\mathsf E[(X-x)^+] \}$ -4419,$\mathsf P(X=X(\omega_0))>0$ -4420,$X=X_i + (X-X_i)$ -4421,$\rho(X_1+X_2) \le \rho(X_1)+\rho(X_2)$ -4422,$(0)$ -4423,$\mu_i$ -4424,$\mathsf{VaR}_1=\esssup$ -4425,$\mathsf E[X_i/X\mid X>x]$ -4426,$#4$ -4427,$v=x$ -4428,$\phi(p)=g'(1-p)\ge 0$ -4429,"$(0,0)$" -4430,$s_2$ -4431,$F(x) < p \iff q^-(p) > x$ -4432,"$g(S)\,\Delta X$" -4433,$\delta+\nu$ -4434,$E'$ -4435,$\mathsf P(X\ge x_p)=1-p$ -4436,$\mu=7.8044$ -4437,$\rho(X)=\mathsf E[X] + c\mathsf{Var}(X)$ -4438,$a_{2}'$ -4439,$\Pr(X<2)=1/6<\Pr(X\le 2)=1/3$ -4440,$p>0$ -4441,$z$ -4442,$\mathsf{j}(91)=7$ -4443,$\zeta_s = 8$ -4444,$ag(0+)$ -4445,$\rho-\iota g>0$ -4446,$R = P-L$ -4447,$\mathrm{sgn}(z)|z|^{1/(q-1)}/\|z\|_p^{q/p}$ -4448,$o(dt)$ -4449,$q^-$ -4450,$A_4 = [0; \epsilon_1 + \epsilon_2]$ -4451,$\mathsf{Var}(Y) \ge \mathsf{Var}(X)$ -4452,$\esssup(X)=\sup\{x\mid \Pr(X>x)>0 \}$ -4453,$0\le \omega\le 1$ -4454,$q(p)=e^{\mu+z_p\sigma}$ -4455,$\mathsf E_\mu[\phi(\mathsf E_\pi u\circ f)]$ -4456,"$[f'_-(x_0), f'_+(x_0)]$" -4457,$a(\cdot)$ -4458,$\mathsf E[(a-X)^+]$ -4459,$L_p$ -4460,"$X\ge 0,(\tilde X-X)\ge 0$" -4461,$\rho(\lambda X)$ -4462,$\mathsf E[Z]\le 1$ -4463,$Pr(X_{-1} > a)$ -4464,$X\wedge a / X$ -4465,$\pi^{-1}\log\mathsf E[e^{\pi x}]$ -4466,$\tau=-1$ -4467,$\mathsf{TVaR}_{p^*}$ -4468,$Z>\mathsf E Z$ -4469,$X\ge x_p$ -4470,$\mathsf E[Y\mid \mathcal F']=\mathsf E[Y\mid X]=\mathsf E[X+Z\mid X]=X +\mathsf E[Z\mid X]=X$ -4471,$\mathbf{X_{2}/X}$ -4472,$A(1_{U>0.95})=A(1_{U\le 0.05})=g(0.05)=0.3017$ -4473,$\mathsf{TVaR}_{0.9}$ -4474,$2.576\sigma_d$ -4475,$\mathsf{TVaR}_1$ -4476,$\Pr(X_n=0)=1-1/n$ -4477,$D/L=\mathsf E[A\wedge L]/\mathsf E[L]$ -4478,$\bar P_n$ -4479,$F_0 = P_{act}-\mathsf E_{rn}[U]$ -4480,$\mathit{MV}_{gc}(a_{gc})=a_{gc}-\rho(X\wedge a_{gc})={{mv_gc}}$ -4481,$1\wedge \cdot$ -4482,"$g(s)= \displaystyle\int_0^s \phi(1-p)dp = \min(s/(1-\alpha), 1)$" -4483,$\ll$ -4484,$0\le \alpha \le 1$ -4485,$Z\succeq_2 \mathsf E[Z\mid X]$ -4486,$j=8$ -4487,$S_0=1-p_0$ -4488,$g_{ROC}$ -4489,$\rho_1(X)$ -4490,$f(s) \ge s$ -4491,$Q(a)=h(F(a))$ -4492,$\mathsf E[X_i g'(S(X))]$ -4493,"$\bar P_i(v_1, v_2, a) / v_i$" -4494,$q_C\le q_A$ -4495,$. Thus $ -4496,$k\mapsto k\rho(-X)$ -4497,$\mathsf{TVaR}_1=\sup$ -4498,$\lambda = \dfrac{E( r_{M} ) - r_{f}}{\sigma_{rM}}$ -4499,$g'(1)<1$ -4500,$u'''' \le 0$ -4501,$-X_i$ -4502,$ROE=(g-s)/(1-g)=m/(1-s-m)$ -4503,$X > a$ -4504,"$f(0,0)=0$" -4505,$\mathsf{Var}$ -4506,$l(kX)\le\rho(kX)$ -4507,$\lambda \ge 0$ -4508,"$0, 1/p$" -4509,$X\ge m$ -4510,$E(X_{0}(a))$ -4511,"$(0,1)$" -4512,$i=1\dots N$ -4513,$-(1-s)g''(1-s) + g(0+)\delta_1 + \sum_s s\Delta_s \delta_{1-s} + g'(1)\delta_0$ -4514,$\phi(0)=\mu(\{0\})$ -4515,$X_1=X_2=10$ -4516,$\mathbf{Z_5}$ -4517,$80=9.56 + 70.44$ -4518,"$\kappa_{i}(x) = \dfrac{\sum_{j:X_{j} = x} X_{i,j} p_j}{\sum_{j:X_{j} = x}p_j}$" -4519,$S(p)=1-p$ -4520,$x=q(\hat p)$ -4521,$g(s)\le 1$ -4522,$N\times d$ -4523,$X=a$ -4524,$P_{g}$ -4525,$x=q_{\mathbf{v}}(s)$ -4526,$dW_t$ -4527,$a_x=2$ -4528,$f(x)=\exp(-x/\mu)/\mu$ -4529,$\bar M_i(a)$ -4530,$Z\in \mathcal Q$ -4531,$U=4$ -4532,$f(x)=e^x$ -4533,$X_{-1}=C_1 + \cdots + C_N$ -4534,$M_i(x)+Q_i(x)$ -4535,$\pi-\lambda\mathsf E[X]$ -4536,$\mathbf{\sigma}$ -4537,$V=1_{X\le x^\ast}$ -4538,$\mathsf E[X(1_{U_X\ge p}-B)]=\mathsf E[(X-m)(1_{U_X\ge p}-B)]\ge 0$ -4539,$\bar Q_{2}$ -4540,$\bar P_g(a)=\rho(X\wedge a)$ -4541,$g''(s)=0$ -4542,$K=3$ -4543,$g(s)=\sqrt s$ -4544,"$\bar P_{0,0}$" -4545,"$(x,-x)$" -4546,$n=9$ -4547,$\hat q(p)=q(1-g^{-1}(1-p))$ -4548,$A(0)=0$ -4549,$\mathsf E_\mathsf{Q}[X_i] = \mathsf E_\mathsf{Q}[\mathsf E_\mathsf{Q}[X_i \mid X]]$ -4550,$\rho(X)\le\liminf \rho(X_n)$ -4551,$w(Z)/\mathsf E[w(Z)]$ -4552,$\mathsf E[YZ]\le 0$ -4553,$c$ -4554,$p^*=48.25/71=0.6796$ -4555,$d\tilde p=g'(1-p)dp=\phi(p)dp$ -4556,$BC$ -4557,$1 in a layer with loss probability $ -4558,"$a_i=\mathsf E[X_i] + k\mathsf{cov}(X_i, X)$" -4559,$d=1$ -4560,$s/g(s)\le 1$ -4561,$\mathsf E[X_i\tilde Z]=\rho_g(X)/2$ -4562,$1/\lambda$ -4563,$1-\alpha_i(x)S(x)$ -4564,$Z=Z_X$ -4565,"$(-\mathsf x*0.75, -2)$" -4566,$E[Z]$ -4567,$\sum_i X_i(a)=X\wedge a$ -4568,$\mathsf{P}(\{X\in A\})$ -4569,$M(x)/Q(x)$ -4570,"$(\Omega, \mathcal F, \Pr)$" -4571,$d\omega$ -4572,$\mathsf E[(X_i/X)g'(S(x)) \mid X > x]$ -4573,$\mathcal{Q}$ -4574,"$(x_B, g(S(x_B-))$" -4575,$X=q(U)$ -4576,$q_A(p) = \sup A$ -4577,$\lambda > 1$ -4578,$a \in \mathbb{A}$ -4579,$y\le q_C(p)$ -4580,$\mathsf{TVaR}_{0.95}(Y)=0.8\mathsf E[X]=2000$ -4581,$\rho(kX)$ -4582,$u=ug(1)=ug(1)+(1-u)g(0) \le g(u)$ -4583,$\Delta Q_{ro}(a) = a-a_{ro}$ -4584,$x_A=\partial x/\partial A$ -4585,$\mathsf{TVaR}_0$ -4586,$\lambda=0.73$ -4587,$Q^* > S$ -4588,$c\le 1$ -4589,$\omega=1$ -4590,$p<0.9$ -4591,$\tau=0.03$ -4592,$\Pr(A)=1-p$ -4593,"$\beta, \kappa$" -4594,$a=a_0+(1+c)\mu$ -4595,$f_{\mathbf{v}}$ -4596,$\mathsf E[X]+kR(X)$ -4597,$\frac{1}{1-p}\int_{1-p}^q \mathsf{VaR}_s(X)ds$ -4598,"$\bar L, \bar P, \bar M$" -4599,"$(\mathsf x*.75, -2)$" -4600,$\mathsf E[W]$ -4601,$\nu$ -4602,$\tau$ -4603,$x_l < x=\mathsf{VaR}$ -4604,$0\le p < 1$ -4605,$Z\mid X$ -4606,"$X:\Omega\to[0,\infty)\subset \mathbb R$" -4607,$\Pr(X < x) \le 0.4 \le \Pr(X\le x)$ -4608,"$[a, a+da]$" -4609,$f>0$ -4610,$S(x-)=0.1$ -4611,$\rho(X)\le b$ -4612,$\mathsf E[X_i/X|X>a]$ -4613,$s=0.45$ -4614,$(1-p)$ -4615,$Z(X(\omega))$ -4616,$\mathit{MV}_{ro}(a) = a-P(X_{-1}\wedge a)$ -4617,$\mathsf E[X^k]\le \mathsf E[Y^k]$ -4618,$9+1=10+0=10$ -4619,$g \circ S$ -4620,$\Pr(X=x_i)=\Pr(X>x_{i-1})-\Pr(X>x_i)=S(x_{i-1})-S(x_i)$ -4621,$r_f>0$ -4622,$X\wedge a$ -4623,$\mathsf E[(X-a)^+]/\mathsf E X$ -4624,$NT$ -4625,$p/\mathsf E[p]=p(1+r_f)$ -4626,$p\ge p_0$ -4627,$-\rho(-H)=\rho(H)$ -4628,$\mathcal Q(X)=\{ \mathsf Q\in\mathcal Q\mid \rho(X)=\mathsf E_\mathsf{Q}[X] \}$ -4629,$L_X(X)=\rho(X)$ -4630,"$\{(s_j, g_j)\} \cup \{(0,0), (1,1)\}$" -4631,$\displaystyle\int_0^\infty xdF(x)$ -4632,$g = s^{0.4}$ -4633,$\mathsf E_{\mathsf Q}[Y] = \mathsf E[YZ]$ -4634,"$0.06 \times (64,861 - 7,500)=3,442$" -4635,$a_1'=a_0-X_{1}$ -4636,$N(t)$ -4637,$v=S$ -4638,$\mathsf{VaR}_p(X)$ -4639,$u^{iv}<0$ -4640,$\lambda_1$ -4641,$X_1=c_1-Y/2$ -4642,$\alpha < 1$ -4643,$Y+W$ -4644,$\mathsf E[Z_A\mid X]$ -4645,$\bar q(s)=q(1-s)$ -4646,$L-f(L)$ -4647,$X=MX_2$ -4648,$a=a(X)$ -4649,$\alpha_i(a)$ -4650,$\bar\iota : 1$ -4651,$a < kP$ -4652,$X_i/X$ -4653,$\partial a/\partial v_i$ -4654,$U(-X)\ge U(-Y)$ -4655,$\rho(X)\le \lim\rho(X_n)$ -4656,$wq_Y(p)+(1-w)q_Z(p)$ -4657,$\Pr(X>\mathsf{VaR}_p(X))>1-p$ -4658,$p^-$ -4659,$h(0)=0$ -4660,$0\le p^\ast\le 1$ -4661,$\alpha\ge A(n)=\sum_s n_s(1-g(s))$ -4662,$af=1$ -4663,$N=n$ -4664,$q=1-p$ -4665,"$\{x_1,\dots,x_N\}$" -4666,$\Pr(\{\omega_2\})=2/3$ -4667,"$(0,0,0,0,0,0,0,0,5,5)$" -4668,$X_n(\omega)=0$ -4669,$\kappa_1(10) = \mathsf E[X_1\mid X=10]$ -4670,$1-s_j$ -4671,$\mathsf{TVaR}_p(X)-\mathsf{VaR}_p(X)=\sigma(\phi(\Phi^{-1}(p))/(1-p) - \Phi^{-1}(p))\to 1$ -4672,$\alpha_j'(x)<0$ -4673,$P=D$ -4674,$f(w) = \exp(-w)$ -4675,$1+r^*=(1+r)(1+\tau)$ -4676,$a=P+Q$ -4677,$X\wedge 10$ -4678,"$u\in[0,1]$" -4679,$L_0^l(X)$ -4680,$j=1$ -4681,$\mathbf{X_{1}/X}$ -4682,$g(s)=\mathsf{TVaR}_{.99}$ -4683,$m+1$ -4684,$\mathsf E[X_1\mid X_1+X_2=x]=mx/(m+n)$ -4685,$\mathsf E[X\wedge a]$ -4686,$9.67$ -4687,$L^*$ -4688,$\|\cdot \|_\rho=\rho(|\cdot |)$ -4689,"$(x_{2,1}, x_{2,2})$" -4690,"$(x,y)$" -4691,$p>1$ -4692,$\mathcal S(X)=\mathsf E[X]$ -4693,$\mathbf{X_{1}(a)}$ -4694,$\rho(X)=\mathsf E_\mathsf{Q}[X]-\alpha(\mathsf Q)$ -4695,$1-2c\Pr(Z>\mathsf E Z)$ -4696,$\mathsf{VaR}_1$ -4697,$p=\Phi^{-1}(4)=3.17\times 10^{-5}$ -4698,$g(s)=s^a$ -4699,$X_i\Delta g(S)$ -4700,$x'$ -4701,$\rho_g(X)=51.156$ -4702,$\rho(X)= \mathsf E_{\mathsf{Q}_X}[X]$ -4703,"$(s,g(s))=(0.2, 0.36)$" -4704,$\delta^{\star}$ -4705,$\mathsf Q^t\cdot X$ -4706,$0=\Pr(X<1)<1/6=\Pr(X\le 1)$ -4707,$s(0)=s_0=0$ -4708,$dS=-f(x)dx$ -4709,$1_{\{X>x\}}$ -4710,$\ge x$ -4711,$g'(1-p)$ -4712,$\mathbf{s_3}$ -4713,$Z(x)$ -4714,$0.495(r-i)$ -4715,$\tau(a-\bar P_\tau(a))$ -4716,$\mathsf E[X_2\mid X=x]$ -4717,"$\bar Q_{0,2}$" -4718,"$u'>0, u''>0$" -4719,$Y(\omega)=0$ -4720,$g(S(x))=s$ -4721,$P/S$ -4722,"$p\in[0,1]$" -4723,$X=F^{-1}(U)$ -4724,$>1$ -4725,$r\times m$ -4726,$P = \mathsf E[X] + \pi\mathsf{Var}^+(X)$ -4727,$s=0$ -4728,$\hat q(p)=x$ -4729,$\mathscr{O}(f)$ -4730,"$1/2,1/4,1/4$" -4731,$n-5$ -4732,$q(1-g^{-1}(1-p))/q(p)$ -4733,$Z-X$ -4734,$\mathbb{Q}'$ -4735,$s>0$ -4736,"$\pmb{j, p, S, \kappa_1, \Delta X, \Delta(X\wedge a)}$" -4737,$S(x)$ -4738,$0\le x < 1/6$ -4739,"$\mathbf{g(S)\,\Delta X'}$" -4740,"$\omega=0,1,\dots, 99$" -4741,$\tilde Z_X:=\mathsf E[Z\mid X]$ -4742,"$0,0,1,2,3,6,10,18,36,52$" -4743,$t=0$ -4744,$p=0.791$ -4745,$f(x+)$ -4746,$X_{2c}$ -4747,$\mathcal S$ -4748,$\mathsf E[u(w-X)] = u(w-c)$ -4749,$0\le \Pr(E)\le 1$ -4750,$q_{\mathbf{v}}(p)=\mathsf{VaR}_p(X(\mathbf{v}))$ -4751,$p(\omega)$ -4752,$\rho(X)=\mathsf E[Xe^{kX}]/\mathsf E[e^{kX}]$ -4753,$0\le p^*\le 1$ -4754,$r_N$ -4755,$\rho(X_g)-\rho(X_n)=51.1560-49.8986=1.2574$ -4756,$S(x)=0$ -4757,$5/6$ -4758,$s^\ast=1/2$ -4759,"$a_{0,t}:=a(Y_{0,t})$" -4760,$0\le p_0 \le p_1\le 1$ -4761,$0.2$ -4762,"$X_1,X$" -4763,$(1-r_0)\delta_1$ -4764,$\mathsf E[X_i \mid X]$ -4765,"$[0,1-p)$" -4766,$\mathsf E[Z]=g(1)-g(0)=1$ -4767,$\mathcal Q_2$ -4768,$\lambda\rho(X)$ -4769,$\rho\mapsto a^\rho(\ \cdot\ ;\ \cdot\ )$ -4770,$\mathcal D(X)=c\mathsf{Var}(X)$ -4771,$\le$ -4772,$u''' \ge 0$ -4773,$\rho(X\wedge a)$ -4774,$Y_2$ -4775,$X=X_1+...+X_n$ -4776,$g_j$ -4777,$1 < \alpha < 2$ -4778,$\| Z \|^*= \sup\ \{ \mathsf E[YZ] \mid \| Y \| \le 1 \}$ -4779,$\alpha_i(x)=\mathsf E\left[\frac{X_i}{X}\mid X > x \right]$ -4780,$\Delta \mathit{MV}_{ro}(a)$ -4781,$\phi(s)$ -4782,$\mathsf E_{\mathbb{Q}}$ -4783,$p\cdot X$ -4784,$\mathsf E_\mathsf{Q}[X_i]$ -4785,$p_0 \le p^\ast \le p_1$ -4786,$D\rho(\cdot)$ -4787,$\lambda=0.25$ -4788,$u'''>0$ -4789,${}^nS^{-1}_X(q)\le {}^nS_Y(q)$ -4790,$S_X$ -4791,"$(\Omega, \mathcal{F})$" -4792,$S\subset\Omega$ -4793,$P_1=\mathsf E[X_1g'(S_X(X))]$ -4794,$\hat q(p) > q(p)$ -4795,$\mathsf j(a)$ -4796,$X+c$ -4797,$\Phi^{-1}(0.995)=2.576$ -4798,$S(y_j-)-S(y_j)$ -4799,"$\{\, (\mathsf E_\mathsf{Q}[X_i], \mathsf E_\mathsf{Q}[X]) \mid \mathsf Q\in\mathcal Q \, \}$" -4800,$g(p)/p$ -4801,$\mathsf E[X] \le \bar P \le \sup X$ -4802,$\alpha_2(99)=0.9$ -4803,$\alpha_iS\Delta X$ -4804,$L_0^a(X)=X\wedge a$ -4805,$\mathbf{a_{1}'}$ -4806,$g(s)=s^{0.7}$ -4807,${}^2S^{-1}(t)=q\mathsf{TVaR}_q(X)$ -4808,$\bar P_{0}=\rho(Y_{0})$ -4809,$g'(s)=\phi(1-s)\ge 0$ -4810,$\mathsf{MONO}$ -4811,$\bar M_i(a)>0$ -4812,$g(S(0-))=1$ -4813,$\rho(0)=\rho(0+0)=\rho(0)+\rho(0)$ -4814,$\rho(X_0)$ -4815,$P(\hat s)=\mathsf E[\hat s]=s$ -4816,$\delta_p/\nu_p = \iota_p$ -4817,$\mathcal{Q}=\mathcal{M}$ -4818,"$d=1,\dots,N$" -4819,$x=0$ -4820,$\mathsf{j}$ -4821,"$E_1,\dots,E_N$" -4822,$(1-p)x_0$ -4823,$U\le p$ -4824,$\mathsf E[X_i\mid X]$ -4825,"$(x_1-\epsilon,x_1]$" -4826,$\mathsf E[X] + c\mathsf E[(X-\mathsf E X)^21_{X>\mathsf E[X]}]$ -4827,$\sigma=0.15$ -4828,$pl(p)$ -4829,$g'(0)$ -4830,$P = \mathsf{VaR}_\pi(X)$ -4831,$C_i$ -4832,$x\mapsto (x-a)^+$ -4833,$h\left(\displaystyle\int_\Omega g(X(\omega))\Pr(d\omega)\right)$ -4834,$\beta_L$ -4835,$D\rho_X(X_i)$ -4836,"$\alpha_1,\alpha_2$" -4837,$\ge\mathsf E[X_i]$ -4838,$\{X>x\}$ -4839,"$x_{2,2}$" -4840,$w=1$ -4841,$F_2\prec_2 F_1$ -4842,$Z_8$ -4843,$T^{-1}(A)$ -4844,$\mathsf{TVaR}_{p_0}(X)=\mathsf E[X \mid A]$ -4845,$\sup_\mathsf{Q} (\mathsf E_\mathsf{Q}[X] - l(Q))$ -4846,$g'(s) = rs^{r-1}$ -4847,$\alpha(\mathbb{Q})=\infty$ -4848,$\mathsf{Q}\in\mathscr{M}$ -4849,$\rho(X)/2$ -4850,$ro$ -4851,$\alpha(\mathsf Q) < \infty$ -4852,$\mathsf E[(X-x)^+]$ -4853,$x_p$ -4854,$X\Delta S$ -4855,$S=e^{\mu t}$ -4856,$\Delta gS$ -4857,$s^{th}$ -4858,$\mathbf{\beta_{1}g(S)\Delta X}$ -4859,$X=X_0+Y$ -4860,$20$ -4861,$\mathsf{TI}$ -4862,$b-a$ -4863,"$\tau(a-\rho_{a,\tau}(X))$" -4864,$\rho(X) < \infty$ -4865,$Y=c$ -4866,$\rho(W_0\wedge a_0)$ -4867,$g(0.01)=0.1$ -4868,$f(x)=(x-d)^+1_{\{x \le m \}}$ -4869,$X\circ T$ -4870,"$\mu=8.7, \sigma=2.5$" -4871,$p=0.05$ -4872,$\mathit{RV}$ -4873,$\bar P(\infty)=\mathsf E[q(U)\phi(U)]$ -4874,$n=7$ -4875,$X_4=X_5=10$ -4876,$n\to \infty$ -4877,$i\not=j$ -4878,$H(x)$ -4879,$a\le \rho(X)\le b$ -4880,$EL$ -4881,"$\{\mathsf E[X_i\,Z] \mid \rho(X)=\mathsf E[XZ] \}$" -4882,$\alpha_i'(x)<0$ -4883,$q_Z$ -4884,$dp=f(x)dx$ -4885,$v_{res}\sqrt{(1+v^2)/n}\approx v_{res}v/\sqrt{n}$ -4886,$\rho(X_i)\le 0$ -4887,$s=s_1+s_2$ -4888,$\displaystyle\int_0^1 X(1-g^{-1}(1-\tilde p))d\tilde p$ -4889,$F(x_0)= p_+>p_0$ -4890,$g''(s)<0$ -4891,"$(s,m)$" -4892,$U = A = 8.149$ -4893,$P(a)da$ -4894,$B(p)$ -4895,$Q=a-P$ -4896,"$2^1, 2^3, ...$" -4897,$c(S)= \rho\left( \sum_{i\in S} X_i \right)$ -4898,$\partial f_{\bar x}/\partial x_i$ -4899,$\log(x)$ -4900,$L_d^{d+l}$ -4901,$\alpha(\mathsf Q)\not=0$ -4902,$X\le a$ -4903,$\bar P_{d}=\rho(Y_{d})$ -4904,$\kappa_i(x)/x$ -4905,$\mathsf{TVaR}_p(X)=1=\mathsf E_\mathsf Q[X]$ -4906,$D^n\rho_X(X_2)=45.1838$ -4907,$f(x)=1$ -4908,$\tilde X_1=X_1 + \mathsf E[X_2\mid X_1]$ -4909,$X_0+\epsilon Y$ -4910,$g_\tau$ -4911,$\phi'(s)ds$ -4912,$\mathsf E[Z]=1$ -4913,$g'(1)>0$ -4914,$A=8.13$ -4915,$X_n$ -4916,$\kappa_i(x) = \mathsf E[X_i \mid X=x]$ -4917,$\mathsf E[F_2]=\mathsf E[F_0]$ -4918,$a=P+Q=EL+M+Q$ -4919,$\mathit{MV}_{ro}(a_{ro})$ -4920,$g(0-)f(\esssup(X))$ -4921,$g_1$ -4922,$g'(1-p)=\nu$ -4923,$X_1+X_2=X$ -4924,$|t|$ -4925,$\rho_h(X):=\mathsf E[X_h]$ -4926,$\prec_n$ -4927,$P(X\wedge a)=\bar P(a)$ -4928,$\bar x$ -4929,$x_h>x=\mathsf{VaR}$ -4930,$1+\gamma$ -4931,$S/P$ -4932,$X_0$ -4933,$b_h$ -4934,$\mathsf P(X>a)>0$ -4935,$(1+\gamma)^{t-x}$ -4936,$n > 2$ -4937,$\sigma(X)=\mathsf E[(X-\mathsf E X)^2]^{1/2}$ -4938,$=\displaystyle\int_0^\infty x \P(\{X \in dx \})$ -4939,$\phi'(p)=f(p)/(1-p)\ge 0$ -4940,$P = \mathsf E[X] + \pi \mathsf E[(X-\mathsf E[X])^+]$ -4941,$\mathsf{VaR}_{0.98}$ -4942,$\rho(X\wedge a)=\mathsf E_\mathsf{Q}[X\wedge a]$ -4943,$\sup X$ -4944,$h_f$ -4945,$\lambda>0$ -4946,${10\choose 5} = 252$ -4947,$T$ -4948,"$i,v$" -4949,$a_i':=\sum \alpha_i(1-S)\Delta (X\wedge a)$ -4950,$\mathsf E[X]=0.6$ -4951,$u_i$ -4952,$N=40$ -4953,$\mathsf E[Z\mid X]$ -4954,$Pr(X > a)$ -4955,$X_i(a')$ -4956,$t\mapsto s(t)$ -4957,$a_{1}$ -4958,$\int_0^1 f(p)dp = 1 - \alpha < 1$ -4959,$X=q(U_X)$ -4960,$t=w$ -4961,$B=\Omega$ -4962,"$1 million auto accident, a $" -4963,$E[X_2 | X]$ -4964,$3^{20}$ -4965,$\bar S_i(x)$ -4966,$\sum_\omega Z(\omega)\mathsf{P}(\omega)=\mathsf E[Z]$ -4967,$dX$ -4968,$D\rho_X(X_1)$ -4969,"$\int_0^\infty z(x)\,dF(x)=1$" -4970,"$X_{t+1,1}$" -4971,$\log$ -4972,$(1-g(s))q$ -4973,"$(0,0,\dots,0,10)$" -4974,"$\iota, \iota(p)$" -4975,$\mathsf E_\mathsf{Q}[0]=0$ -4976,$\mathsf E X$ -4977,$\mathsf{TVaR}_{0.5}(X_2)=45.5$ -4978,$t-2$ -4979,$Z_2$ -4980,$\prec_2$ -4981,$0\le x < X_1$ -4982,$a=\sum_i a_i$ -4983,$s<0.1$ -4984,"$a(x_1,x_2)=\sqrt{3x_1^2 + 4x_2^2}$" -4985,$E[u_j(W_j - X_j + Y_j - H[Y_j])]$ -4986,$1-p=0.9$ -4987,$h(s)=1-g(1-s)$ -4988,$(P-L)/A$ -4989,$X_1(10)$ -4990,$w_0$ -4991,$AR\succ BY$ -4992,$q_X\le q_Y$ -4993,$0 < \alpha\le 1$ -4994,"$\mathsf{biTVaR}_{0,1}^w$" -4995,"$a_i=\rho(X_i, p^*)$" -4996,$\phi(p)=g'(1-p)$ -4997,$1/(1+r)$ -4998,$\dfrac{1}{1+\iota} p$ -4999,$p(1-p)$ -5000,$\rho(X) = \int_0^\infty g(S(x))dx$ -5001,$\sum S\Delta(X\wedge a)$ -5002,$V^*$ -5003,$\partial a/\partial v_1$ -5004,"$A_1=[-k,-k]$" -5005,$p=0.25$ -5006,$a^{\star}(X)$ -5007,$\mathsf E[X]+k\mathsf{Var}(X)$ -5008,$0.8 \ge p < 0.9$ -5009,$\mathcal{G}$ -5010,$g'(s-)$ -5011,$k$ -5012,$\rho(X_n) \downarrow \rho(X)$ -5013,$q_X(U)$ -5014,$wq_X(p)+(1-w)q_Z(p)$ -5015,$\mathbf{X_{2c}}$ -5016,"$p\in [0,1]$" -5017,$g(s) \approx m_0+(1+m'(0))s$ -5018,"$Y_{0,0}:=\sum_{d>0} X_{0,d}$" -5019,"$Y_m=\max(X_1,\dots,X_m)$" -5020,$\mathsf{VaR}_{0.99}(X_1)=150$ -5021,$0.01$ -5022,$\mathbf{X_2(a)}$ -5023,$\mathsf E[X_2\mid X_1]$ -5024,$\mathsf E_\mathsf{Q}[\mathsf E[X_i \mid X]]$ -5025,$\alpha_i(x) = \mathsf E[X_i /X \mid X> t]\not=\mathsf E[X_i\mid X> t]/\mathsf E[X\mid X>t]$ -5026,"$t^\star \in [0,1]$" -5027,"$\{1,2,\dots, n\}$" -5028,$a < \max(X)$ -5029,$\mathcal N_X(X_i)$ -5030,$x^{\ast}:=\min(x)$ -5031,$0.5L_{250}^{500}(x)+0.75L_{500}^{750}+L_{750}^{1000}$ -5032,"$x_0, x_1, x_2$" -5033,$\sum (1-S)\Delta (X\wedge a)$ -5034,"$[0,\infty)\subset\mathbb{R}$" -5035,$\bar Z = F(\bar x)$ -5036,$^2$ -5037,$\rho_g(X)=\mathsf E_\mathsf{Q}[X]$ -5038,$\Pr(B\le t) = 1/2 + 1_{t>1/2}(1/2)$ -5039,$q_{X}(p)=\sqrt{2}\Phi^{-1}(p)$ -5040,$a = a(\mathbf{v}) = a(X(\mathbf{v}))$ -5041,$\mathbf{\beta_{2}}$ -5042,$s=1$ -5043,$S\cdot dX$ -5044,$s$ -5045,$S(x)=u$ -5046,$\sup_{\omega\in\Omega} (f(\omega)+g(\omega)) \le \sup_{\omega\in\Omega} f(\omega) + \sup_{\omega\in\Omega} g(\omega)$ -5047,"$0,1,1,1,2,3, 4,8, 12, 25$" -5048,$\triangleright$ -5049,$\mathsf{TVaR}_p(X)=51.156$ -5050,"$A\subset [0, \infty)$" -5051,"$\Delta\,g(S)$" -5052,$\mathbf{\beta_{1}}$ -5053,$\Pr(\{\omega \mid X_n(\omega)\to X(\omega) \})=1$ -5054,$f(x)\le f(y)$ -5055,$da$ -5056,"$(\mathsf x*1.2, 2)$" -5057,$S=1$ -5058,$\rho(X)=\mathsf E_{\mathsf{Q}}[X]$ -5059,$L_{250}^{\infty}$ -5060,$\mathsf E_{\mathsf{Q}}[X_i\mid X\le a](1-g(S(a))) + a\mathsf E_{\mathsf{Q}}[X_i/X\mid X >a]g(S(a))$ -5061,$0 = x_0< x_1<\cdots < x_n < \cdots$ -5062,$\var(\sum C_i)=\sum (m_i v_i)^2 = n(mv)^2$ -5063,$\mathbf{X(a)}$ -5064,"$\nu p\,da=\nu F(a)\,da$" -5065,$\mathsf xtext$ -5066,$\hat p:=1-g^{-1}(1-p)$ -5067,"$X(x_1, x_2)=(x_1+x_2)Y$" -5068,$1-F(x)=1-p$ -5069,$\mathcal F_t$ -5070,$\rho(X)=\rho(X-Y+Y)\le \rho(X-Y) + \rho(Y)$ -5071,$c \le 0$ -5072,$S(x_{(j)})(x_{(j+1)}-x_{(j)})$ -5073,$p=0.9$ -5074,$\mathsf E[X_iZ]$ -5075,$\rho(X+Y) \le \rho(X) + \rho(Y)$ -5076,$e^{X_t}$ -5077,$n\times r$ -5078,"$f'_\omega (\bar x, h)$" -5079,"$Y_{t,d+1}$" -5080,$F(b)-F(a)$ -5081,$a_{ro}:=\mathit{VaR}_{p}(X_{-1})=10743.5$ -5082,$\rho(X) - (-\rho(-X))=\rho(X)+\rho(-X)$ -5083,$Z_\epsilon$ -5084,$\{3\}$ -5085,$L(X)=e^{kX}/\mathsf E[e^{kX}]$ -5086,$\lim_{\epsilon \downarrow 0} (f(x-\epsilon)-f(x))/\epsilon$ -5087,"$1,9,4,4,2,$" -5088,$\mathsf{TVaR}_p( X )$ -5089,$g(S)$ -5090,$\mathsf{MON}'$ -5091,$\mathsf{TVaR}_{p_1}(X)$ -5092,$1_{X < q(1-s)}-(1-g)$ -5093,$g(x)=e^{2\pi i x\theta}$ -5094,$f=f(s)$ -5095,$\mathsf E[X \mid \mathcal F_0]$ -5096,$l=a$ -5097,"$H(A, L, t)$" -5098,$\mathsf{TVaR}_{0.75}=4\left( \frac{90}{8}+\frac{98}{16}+\frac{100}{16}\right)=94.5$ -5099,$\mathit{NPV}$ -5100,$E_k$ -5101,$g(s)=s^\rho$ -5102,$X\ge 0$ -5103,$1.2\times 10^9$ -5104,$f'(a)$ -5105,$\mathsf E[f(X-\pi P)] = f((1-\pi)P)$ -5106,$y\in A$ -5107,$0 < \lambda \le 1$ -5108,"$\mathsf{cov}(X_i,X)/\sigma_X$" -5109,$t_1$ -5110,$F(x)=\Pr(X\le x)$ -5111,$\lambda>1$ -5112,$g(S(x))=g(0)=0$ -5113,$D^n\rho_{X\wedge a}(X_i)$ -5114,$\tau < t+d$ -5115,$s_2=1$ -5116,$\mathsf E_\mathsf{Q}[X_i \mid X]=\mathsf E[X_i \mid X]$ -5117,$\mathsf E[X\mid \mathcal F_t]$ -5118,$\mathsf j(a)=\max \{ j:X_j < a \}$ -5119,$g'(S(x))\ge 1$ -5120,$1-\tilde p=g(S(x))$ -5121,$F_m\succ_m F_0$ -5122,"$X_{t,d+1}$" -5123,$A(-X)=-B(X)\not=-A(X)$ -5124,$g=1$ -5125,$0.99$ -5126,$f_t$ -5127,$\mathsf{Var}^+(X)$ -5128,$\rho_X(X_i) \ge \mathsf E[X_i]$ -5129,$E[YZ]$ -5130,$1-r_0$ -5131,$\Pr(X\le x)=0$ -5132,$\lambda=0$ -5133,$\beta_2g-\alpha_2S$ -5134,$x^*$ -5135,$\lambda t$ -5136,$\{X > \mathsf{VaR}_p(X)\}$ -5137,$r_f = 0.02$ -5138,$x=1$ -5139,"$[s_0, s_1]$" -5140,$(\beta g(S))'(x)=-\kappa_i(x)g'(S(x))f(x)/x$ -5141,"$a_{0,1}$" -5142,$X_{d}$ -5143,$q(p)=\inf\{x \mid F(x)\ge p \}$ -5144,"$([0,1], \mathcal B, \mathsf P)$" -5145,$S\ge (1-\epsilon)\mathsf E[X]$ -5146,$c(1)-c(\varnothing)=c(1)$ -5147,$\rho_a(0) = \rho(0 \wedge a(0)) = \rho(0 \wedge 0) = \rho(0) = 0$ -5148,$X_1=\mathsf E[X\mid \mathcal F_1]$ -5149,$\rho(X)\le \rho(\lambda X)/\lambda$ -5150,$c(\sum_{i\in S} X_i)$ -5151,$g(0)=0$ -5152,$\alpha_{1}$ -5153,$\var(\sum C'_i)=v_{res}^2 \sum c_i^2$ -5154,$0 < b \le 1$ -5155,$pX + (1-p)Z$ -5156,$\pi(X)$ -5157,$\mathsf E[Y \mid U]$ -5158,$\Pr(X>a)$ -5159,${}^nS^{-1}(q)$ -5160,$\sup X=\inf$ -5161,$Q^*$ -5162,$v-\nu^{\star}=(\iota^{\star}-i)/v\nu^{\star}$ -5163,$_{ro}$ -5164,$\iota=\delta/\nu$ -5165,$m'(1) = -m_2/(1-s_2)$ -5166,$D^n\rho_{X\wedge a}(\cdot)$ -5167,$(M-N)\times d$ -5168,$S(x_0)=1$ -5169,$10/11$ -5170,$f(L)=(L-a)^+$ -5171,$\mathsf{j}(a) = \max\{ j:X_j < a \}$ -5172,"$3.807=\lambda \sigma(W_{0,0})$" -5173,$\bar P(a)>\mathsf E[X\wedge a]$ -5174,$\mathsf E[X]=k/(k+\beta)$ -5175,"$\mathsf E[X_{t,d}\mid \mathcal F_{\tau}]$" -5176,$h(x)=\sqrt x$ -5177,$ for $ -5178,$S(x-)=1$ -5179,$\{ Z\not=0 \}$ -5180,$\iota=(g(s)-s)/(1-g(s))$ -5181,$\tau=0$ -5182,$(r-i)Q_t$ -5183,$\delta p$ -5184,$\mathsf{TVaR}_p = q(p)$ -5185,$\sigma=0.1980$ -5186,$X_1> x_1$ -5187,$\mathsf E[X\mid t+d]$ -5188,$1/(1-p)>1$ -5189,$\mathsf E_{\mathsf{Q}}[(X-a)^+] \le \rho((X-a)^+)$ -5190,$q_V(p)=0$ -5191,$(1-s)^{-1/2}/4$ -5192,$\mathsf E[p]\not=1$ -5193,$g(0-)$ -5194,$(s+\iota) / (1+\iota)$ -5195,${}^2S(t)=\mathsf E[(X-t)_+]$ -5196,$k = 1.4 + 1.8s$ -5197,$\Psi(x)=1-\exp(-e^x)$ -5198,$=\displaystyle\int_0^\infty S(x)dx$ -5199,$dp$ -5200,$da\to 0$ -5201,"$(lee.east |- lee.north)+(0.25,0.25)$" -5202,$G$ -5203,$X'=0$ -5204,$\rho_g$ -5205,$s > 0.5$ -5206,$\Pr(M=m)=\frac{r}{1+r}\frac{1}{(1+r)^m}$ -5207,$\rho=0.12$ -5208,$\beta_1g(S)dx$ -5209,$X(x)=x$ -5210,$g(S(x)) = S(x) + \delta(F(x))F(x)$ -5211,$L_X \in \mathcal L_\rho$ -5212,$g-S$ -5213,$x_0$ -5214,$\mathbf{a}$ -5215,$0=\rho(0)$ -5216,$Xm1=X_{-1}$ -5217,$\mathsf E[X\mid \mathcal F_{\tau}]$ -5218,$1-g^{-1}(1-p')$ -5219,$\alpha_i(x)S(x)=\mathsf E[(X_i/X)1_{X>t}]$ -5220,$\mathsf E[kX]=k\mathsf E[X]$ -5221,$B(1_{U>0.95})=B(1_{U\le 0.05})=h(0.05)=1-g(1-0.95)=0.0203$ -5222,$\phi(p)\ge 0$ -5223,$E(X_{-1}\wedge a)$ -5224,$n=8$ -5225,$R/Q$ -5226,$q < p$ -5227,$x=wy + (1-w)z$ -5228,"$B_3=[-k, \epsilon]$" -5229,$Q = 5.0449$ -5230,$\rho(X)=\max_k \mathsf E_{\mathsf Q_k}[X]$ -5231,$n'=7$ -5232,$g'(t)>0$ -5233,"$j=0,\dots, N-1$" -5234,$0\ < p < 1$ -5235,$(S_t-a)^+$ -5236,$\alpha+\beta = \iota^\ast/(1+\iota^\ast)$ -5237,$\sin(x)$ -5238,$\mathbf{P_i}$ -5239,$a_{gc}:=\mathit{VaR}_{p}(X)={{a_x}}$ -5240,$\mathsf{VaR}_{0.995}$ -5241,$P(X_{-1}(a_{gc}))={{mvp_gc}}$ -5242,$\rho''(x)=-U''(x)>0$ -5243,$\{\omega\mid X(\omega)=x\}$ -5244,$\tilde M_i(a) = \bar P_i(a) - \mathsf E[X_i(a)]$ -5245,$\kappa$ -5246,$\mathsf E[X_i \mid X=q(1-g^{-1}(1-p))]$ -5247,$e$ -5248,$\omega'=\omega$ -5249,$0.3 < s <0.4$ -5250,$g(s)=s^\alpha$ -5251,$X_1-X_2$ -5252,$a = \sum_i a_i$ -5253,$\rho(X)=\mathsf{VaR}_{0.995}(X)-\mathsf E[X]$ -5254,$\rho(X)=1$ -5255,$H(X)\le H(Y)$ -5256,$Y=X$ -5257,$\{\omega\in \Omega \mid (X\wedge a)=a \}$ -5258,$X\ge x_0$ -5259,$r=1$ -5260,"$\bar Q_{0,1}$" -5261,$Y\preceq_2 X$ -5262,$\rho(X)=k\mathsf{Var}(X)$ -5263,$\delta = \iota\nu$ -5264,$g'(1-s)=\phi(s)$ -5265,$q(U_X) < m$ -5266,$\alpha_1$ -5267,$A(X+Y)\le A(X)+A(Y)$ -5268,"$a_{0,t}' = a_{0,t}$" -5269,"$j=5,6$" -5270,$\mathsf Q_k$ -5271,$\lambda < 1$ -5272,$\mathcal E:=\{Y \circ T \mid T \text{ PPT} \}$ -5273,$Xp$ -5274,"$(lee.east |- lee.south)+(0.375,-0.25)$" -5275,$dF=-dS=$ -5276,$m(s) := (1-s)\wedge m(s)$ -5277,$\mu_{rU} = M/K = 0.133$ -5278,$y \wedge (x-a)^+$ -5279,$\mathcal A=\{X\mid \rho(X)\le 0 \}$ -5280,"$Y_{0,0}$" -5281,$\bar P_{1}$ -5282,$\alpha_1+\alpha_2=\beta_1+\beta_2=1$ -5283,$\mathbb{Q}'(\Omega_a) =\mathbb{Q}(\Omega_a)$ -5284,$a_l>b_l$ -5285,$X_0=0$ -5286,$\mathsf E[X\mid \mathcal F_t](\omega)=\sum_{i \le t} \omega_i/2^i+2^{-(t+1)}$ -5287,$\Delta Q_{gc}(a)$ -5288,$P_j=\sum_{i=0}^j p_i$ -5289,$\{y_j\}$ -5290,$X=3$ -5291,$\rho(X)=\bar P$ -5292,$\alpha(\mathsf Q)\ge 0$ -5293,$\mathsf E_{\mathsf{Q}}[X_i \mid X]$ -5294,$a_l$ -5295,$A$ -5296,$v(AB) + v(ABCD) = 3/2 > v(ABC) + v(BCD) = 4/3$ -5297,$\sum p_jX_j$ -5298,$\Pr(\{\omega_1\})=1/3$ -5299,$0.5+U/4$ -5300,$\mathbf{\alpha_2S\Delta X}$ -5301,$n=3$ -5302,$\bar\nu$ -5303,$p^*=1$ -5304,$r_K = \exp (\lambda) - 1$ -5305,$x<1$ -5306,$a(X)=a(\sum_i X_i) = \sum_i a_i$ -5307,$P(X_{-1}(a))=\bar P^a_0$ -5308,$\kappa_{1}$ -5309,$\{\omega\in\Omega \mid X(\omega) \le x\}\in\mathcal F$ -5310,$\mathsf{TVaR}_{0.6975}$ -5311,$F(q^-(p))=p$ -5312,$\mathsf E[XZ_j] = (5)(1/10)(8)+(5)(1/10)(9)=8.5=\mathsf{TVaR}_{0.8}(X)$ -5313,$B_2 \succ A_2$ -5314,$\hat{s}$ -5315,$\rho(X+\rho(X))=\rho(X)-\rho(X)=0$ -5316,$\mathsf{NORM}$ -5317,$Y\succeq X$ -5318,$\lim_{x\to\infty} xg(S(x))=0$ -5319,$\int xdF$ -5320,$t > 2/3$ -5321,$p=1-s_j$ -5322,$P_2\ge (\rho(X_1)-P_1) + \rho(\mathsf E[X_2\mid X_1])\ge \rho(\mathsf E[X_2\mid X_1])$ -5323,"$d,v\ge 0$" -5324,$X_1\le X_2$ -5325,$r_D$ -5326,$x=\max(X)$ -5327,$\rho(\tilde X_1)=\rho(X_1)+\rho(\mathsf E[X_2\mid X_1])$ -5328,$c=0$ -5329,$1/\lambda = \sum_j 1/\lambda_j$ -5330,$>0$ -5331,$\rho_a(X)>2\rho_a(X_1)$ -5332,$Z(200)=0$ -5333,$A=\{X>x\}$ -5334,$\mathsf E[Y_{d}]=\sum_{s>d} \mu_s$ -5335,$n\ge 0$ -5336,$\mathsf E[X_i(x)]$ -5337,$\bar P(a)\le a$ -5338,$\displaystyle\int_0^\infty g(S(x))dx$ -5339,$M(x)$ -5340,$\int_0^1 F^{-1}(p)dp$ -5341,$e_x=\sum_t {}_tp_{x}$ -5342,$g'\left (S_{X\wedge a}(X\wedge a)\right )$ -5343,$0 < g' \le 1$ -5344,$\mathit{NPV}_1$ -5345,$\bar F(a)=\int_0^a F(x)dx = a-\bar S(a) = \bar Q(a) + \bar M(a) = \mathsf E[(a-X)^+]$ -5346,$0.75+U/4$ -5347,$g_2$ -5348,$r_D=0$ -5349,$\displaystyle\int_\Omega X(\omega)\P(\omega)$ -5350,$p:=1-s$ -5351,$\bar\delta=\bar\iota\bar\nu$ -5352,$\rho(X)=\sup_{\mathsf Q\in\mathcal Q} \mathsf E_\mathsf{Q}[X]$ -5353,$\rho(aX)=a\rho(X)$ -5354,$P=\mathsf E[X]$ -5355,$f(x-)$ -5356,$A_i\cup A_i^c$ -5357,"$(s_0,g(s_0))$" -5358,$Q_0=0.25$ -5359,$3$ -5360,$X=\sum_t B_t/2^i$ -5361,$\iota(s)=(1-s)/(1-1)=\infty$ -5362,$Z_A=(1-p)^{-1}1_A$ -5363,$Q\circ T\in\mathcal{Q}$ -5364,$\mathsf E_{\mathsf Q}[X_i \mid X=x] = \mathsf E[X_iZ \mid X=x]/\mathsf E[Z \mid X=x] = \mathsf E[X_i \mid X=x]$ -5365,$\mathcal Q$ -5366,$t>\tau$ -5367,$w$ -5368,$\Delta X_j=X_{j+1}-X_j$ -5369,$\mathsf E[X_i \mid X = x]$ -5370,$1-g(S(t))$ -5371,$\mathbf{a_1'}$ -5372,$ to be the set of all sample points where the insurance event $ -5373,$1-1_{X>a}=1_{X\le a}$ -5374,$s=1-p$ -5375,$f(x)=x$ -5376,$\rho(X)=\mathsf E_{\mathbb{Q}}[X]$ -5377,$s \approx 0$ -5378,$j=9$ -5379,$k\le m$ -5380,$\epsilon$ -5381,$\bar Q(a)=a-\bar P(a)$ -5382,$#2$ -5383,$\rho(X) = \mathcal{N}_{\tilde X}(X)$ -5384,$p$ -5385,$3/4 \pm 1/4$ -5386,$10^{-2}$ -5387,$\mathcal B$ -5388,$\epsilon>0$ -5389,"$g(s) = \nu s + \delta, s>0$" -5390,$\rho(X) = \max_{\mathsf Q\in \mathcal Q} \ \mathsf E_\mathsf{Q}[X]$ -5391,$X(\omega)\ge a'$ -5392,$r=0.025$ -5393,$\{X=q_X(p)\}$ -5394,$m$ -5395,$\mathcal F_0$ -5396,$\alpha_i(x) = \mathsf E[X_i /X \mid X> x]\not=\mathsf E[X_i\mid X> x]/\mathsf E[X\mid X>x]$ -5397,$L_0$ -5398,$m\le 4$ -5399,$\mathsf{TVaR}_1(X)=\sup(X)$ -5400,$q(p)=\mathsf{VaR}_{p}(X)$ -5401,$\rho(X-Y)\le 0$ -5402,$\mathbf{d=0}$ -5403,$P_{i}(a)$ -5404,$\rho(X)=\mathsf{TVaR}_p(X)$ -5405,$\mathsf E[X]=\mathsf E[Y]$ -5406,"$\mathbf{v}=(v_1,v_2)$" -5407,$\kappa_i(t)=E[X_i \mid X=t]$ -5408,"$(s, g(s))$" -5409,"$(-1,1)$" -5410,$X'=\mathsf E[X\mid A]$ -5411,$\mathsf E[X]+\mathsf{SD}(X) \le \mathsf E[Y]+\mathsf{SD}(Y)$ -5412,$n\times 1$ -5413,$g'(S(x))<1$ -5414,$X_{1}$ -5415,$\rho(X)\le\lim \rho(X_n)$ -5416,$\mathsf{TVaR}_0(\cdot)=\mathsf E[\cdot]$ -5417,$q^+(p) := \sup\ \{x \mid F(x) \le p \} = \inf\ \{ x \mid F(x) > p \}$ -5418,$M-N$ -5419,"$i=2,3,4,5$" -5420,$\mathsf E[Z_j\mid X]$ -5421,$X_i(v_i)=v_iX_i(1)$ -5422,$X\le Y$ -5423,$S\Delta X'$ -5424,$\rho(X\wedge a)=0.909$ -5425,$(1+\gamma)F_0$ -5426,$\sigma=\sqrt{s(1-s)/N}$ -5427,$\iota(s)$ -5428,$a-\bar P(a)$ -5429,$F^{-1}$ -5430,$\mathsf E[X] + \pi \mathsf E[(X-\mathsf E[X])^+]$ -5431,$\kappa_2(X)$ -5432,$U$ -5433,"$Y_{t,1}$" -5434,"$k=1,2,\dots,n-1$" -5435,$1/(1+r_f) = \mathsf E[p]$ -5436,$g(S(x-))=1$ -5437,$X_0 + \epsilon Y$ -5438,"$\displaystyle\int_0^a \kappa_i(x)g'(S(x))f(x)\,dx + a\beta_i(a)g(S(a))$" -5439,$\kappa_i(x) = \mathsf E[X_i \mid X=x]=\mathsf E_{\mathsf Q}[X_i \mid X=x]$ -5440,$m(s)$ -5441,$x_0 \ge q^-(p)$ -5442,$X(\mathbf{v}) = \sum_i X_i(v_i)$ -5443,$a=9532.0$ -5444,$L_{250}^{1000}(x)$ -5445,"$\sigma=13,108$" -5446,$T_2 := ((n+1)-pN)x_n$ -5447,$\{ X>x \}$ -5448,$\iota = \dfrac{g(s)-s}{1-g(s)}$ -5449,$\Pr$ -5450,$S_{\mathbf{v}}(t)=\text{Pr}(X({\mathbf{v}})>t)$ -5451,$g(s) = s^r$ -5452,$\Delta X$ -5453,$=$ -5454,$R^2$ -5455,$S(x_4)$ -5456,$S_X(x) \ge S_{X_1}(x)$ -5457,$X+100$ -5458,"$\Omega=\{0,1,2,\dots \}$" -5459,$\kappa\ge K(n)=\sum_s n_s(1-g(s))k(s)$ -5460,$R(X)$ -5461,$g(S_6)\Delta X'_6$ -5462,$\rho(X-\rho(X))=0$ -5463,$g(0+) > 0$ -5464,"$X_i,X$" -5465,$p=0$ -5466,$r_h=\mu_L=0$ -5467,$g(0^+)>0$ -5468,$\mathrm{Pr}_{rn}\{P_{act}>P\}$ -5469,$\{ Z\mid \rho(X)=\mathsf E[XZ] \}$ -5470,"$(I, \mathcal B, \mathsf P)$" -5471,$\mathbb{R}^3$ -5472,$ is not continuous and $ -5473,$E'=\Omega\setminus E\in\mathcal F$ -5474,$a_x$ -5475,"$\{1,2,\dots,10000\}$" -5476,$\Pi$ -5477,$\mathsf E X + c\mathsf E[\vert X-\mathsf E X \vert^p]^{1/p}$ -5478,$ipl(p)$ -5479,$a'(x)=a(1)$ -5480,$-g''(t) = w \delta_{\alpha_1}/\alpha_1 + (1-w) \delta_{\alpha_2}/\alpha_2$ -5481,$\mathsf E[X\mid X\ge \mathsf{VaR}_p(X)]$ -5482,$a=\infty$ -5483,$\bar P_g$ -5484,$\sum_i a_i=\sum_i a(X_i;X)=\rho(X)$ -5485,$a^\rho$ -5486,$p_{\mathit{cl}}$ -5487,$h(1)=1$ -5488,$\mathbf{\omega_i}$ -5489,$\Delta_{1}$ -5490,$p^+=\mathsf P(X\le q_X(p))$ -5491,$\rho=\dfrac{M}{l} = \dfrac{1-\lambda}{\lambda}$ -5492,$\rho_1$ -5493,$S_{\mathbf{v}}(a)$ -5494,$^\circledR$ -5495,$\Pr(X>x)$ -5496,"$g(s)=\min(g_1(s), g_2(s))$" -5497,$a(X)=\mu+4\sigma$ -5498,$S\approx \mathsf E[X]$ -5499,$\mathbf p$ -5500,$\mathsf E[Y\mid\mathcal F']=\mathsf E[Y]$ -5501,$pq$ -5502,$\rho(X+Y)=\rho(X) + \rho(Y)$ -5503,$\mathsf x\mathsf{VaR}_p(X):=\mathsf{VaR}_p(X)-\mathsf E[X]$ -5504,$\Pr(q^-(F(X))\not=X)=0$ -5505,$g'=2/3$ -5506,$X\wedge a\Delta g$ -5507,$P=80$ -5508,$\rho(-H)=\rho(C)-1=-0.05$ -5509,$i$ -5510,$S_i(x)=\alpha_i(x)S(x)$ -5511,$X'(\omega) \le Y'(\omega)$ -5512,$B_t(\omega)=\omega_t$ -5513,$S(x)/P(x)$ -5514,$\mathbf{j}$ -5515,"$\int_0^s g'(t)\,dt=\nu s$" -5516,$L_X(v)=l(v)$ -5517,$\mu=\log(\theta)$ -5518,$\Pr(X > q_{\mathbf{v}}(p))=1-p$ -5519,$T_{(1)}=W$ -5520,$t\in\mathbb{R}$ -5521,"$(x_{1,i}, x_{2,k(i)})$" -5522,$\rho_g(X\wedge a)=\bar P(a)$ -5523,$g'>0$ -5524,$X\wedge a = \sum_i X_i(a)$ -5525,$t=-\log(1-p)$ -5526,$S_X(y)$ -5527,$\mathsf E[X\mid X=x]\equiv x$ -5528,$\sum_i x_i\Pr(X=x_i)$ -5529,$n=2^m+k$ -5530,$\mu t$ -5531,"$1/2, 1/4$" -5532,$\mathsf{CX}$ -5533,$\sigma^2 = \sum \sigma_i^2$ -5534,$\iota=M/Q$ -5535,$AB$ -5536,"$\displaystyle\int_0^a \beta_i(x)g(S(x))\,dx$" -5537,$\bullet$ -5538,$366.4$ -5539,"$\tilde X:[0,\infty)\to[0,\infty)$" -5540,$1-\alpha_i(t)S(t)$ -5541,$F_1$ -5542,$a=\mathsf{VaR}_p$ -5543,$(a'-X)^+$ -5544,$(\alpha_i S)'(x)=-\mathsf E[X_i\mid X=x]f(x)/x=-\kappa_i(x)f(x) / x$ -5545,$\mathbf{X_1pK}$ -5546,$\mathsf{FSD}$ -5547,$a={{a_x}}$ -5548,"$(0.2, 0.304)$" -5549,$e^{\mu_A}-1$ -5550,$-\rho(X-Y)\le \rho(Y)-\rho(X)$ -5551,$B^c_k$ -5552,$-$ -5553,$d+l$ -5554,$0.1005$ -5555,$r_i$ -5556,$\bar\delta a$ -5557,$c > 1/2$ -5558,"$\mathsf{PML}_{n, \lambda}(X)=\mathsf{PML}_{n, \lambda}$" -5559,$f(x)$ -5560,$h(1-p)=1-g(p)=1-\sqrt{0.9}=0.051$ -5561,$\mathbf{x}$ -5562,$Gn$ -5563,$\mathcal F$ -5564,$g_2(s)=\sqrt{s}$ -5565,$\bar P_0>\mathsf E[Y_{0}]$ -5566,$v_f=1/(1+r_f)$ -5567,$B\subset \Omega$ -5568,$\bar S(x)$ -5569,"$s_j,g_j\in[0,1]$" -5570,$\mu=21.315$ -5571,$a_{gc}=P(X_{-1}(a_{gc}))+P(X_{0}(a_{gc}))+\mathit{MV}_{gc}(a_{gc})$ -5572,$X0=X_{0}$ -5573,$X=(X\wedge a) + (X-a)^+$ -5574,$\mathsf E[L\wedge A]$ -5575,$(\mathsf{TVaR}_p - q(p))/(1-p)$ -5576,$X \preceq_n Y$ -5577,$\lambda_i$ -5578,$\mathsf{VaR}_{0.95}(X)=3395$ -5579,"$W_2=\sum_{t+d=2} Y_{t,d}$" -5580,$a\ge \sup(X)$ -5581,$a=Q+R$ -5582,$p/q-1=(p-q)/q>0$ -5583,$\alpha_1\ge \beta_1$ -5584,"$c_1=(c(1) + c(1,2)-c(2))/2$" -5585,$\Pr(X > a) \le \epsilon$ -5586,$Z\in D\rho(X_0)$ -5587,$\cdots$ -5588,$d\bar S(a)/da$ -5589,$\omega'=0$ -5590,$\rho(Y)=g(pq)$ -5591,"$\phi(s) = (1-p)^{-1}1_{[p, 1]}(s)$" -5592,$dg/ds$ -5593,$T_1 := X_{n+1} + \cdots + X_{N-1}$ -5594,$\kappa_i(x)=\mathsf E[ X_i \mid X = x]$ -5595,$\displaystyle\int_0^\infty xd(g\circ F)(x)$ -5596,$\mathsf{POS\ LOAD}$ -5597,$R_x$ -5598,$t\mapsto W_t$ -5599,$\mu+\lambda\sigma$ -5600,$\rho(X)\le\rho(0)=0$ -5601,$\kappa_2$ -5602,$k(i)$ -5603,$\chi( s ) = p - \log(s)$ -5604,$C$ -5605,$0\le x\le 1000$ -5606,"$\Omega=(0,1)$" -5607,$\mathsf E[X_iZ]=500$ -5608,$\mathsf E[X_i (X\wedge a)/X \mid X=x] = \mathsf E[X_i\mid X=x] (x\wedge a)/x$ -5609,$D(t)$ -5610,$w(x)=x$ -5611,$Z(X)$ -5612,$1 < x < 2$ -5613,$P/A$ -5614,$\mathsf{TVaR}_{p^*}(X_1)+\mathsf{TVaR}_{p^*}(X_2)=80$ -5615,$g(S(x))$ -5616,$s<0.20$ -5617,$M_i = \beta_ig-\alpha_iS$ -5618,"$[0,1,\dots,n]$" -5619,$a(X_i;X)\ge \mathsf E[X_i]$ -5620,$X\Delta g(S)$ -5621,$\mathsf Q\not\ll \mathsf P$ -5622,$q(p')=q(p)$ -5623,$\mathsf E[XZ_\epsilon]\to \mathsf E[XZ]$ -5624,$100G$ -5625,$g(x)$ -5626,$c-1$ -5627,$\mathbf{\Delta(X\wedge a)}$ -5628,$\lambda$ -5629,$C^1$ -5630,$q^-(F(x))\le x$ -5631,$h(p)p$ -5634,$a=f=1$ -5635,$R_L=(L-P)/P$ -5636,$\omega\mapsto \psi=F(X(\omega))$ -5637,$r-i$ -5638,$\sigma=0.4$ -5639,$y$ -5640,$d>0$ -5641,$\mathsf{TVaR}_p(X)= \sum_i X_iZ_i / 10$ -5642,$F_0=2$ -5643,$\rho(X+c) = \rho(X)+c$ -5644,$X\ge X+Y$ -5645,$X > x$ -5646,$c(X(\mathbf v))$ -5647,$\mathsf E_\mathsf{Q}[X_i \mid X=x]=\mathsf E[X_i g'(S(X))1_{\{X=x\}}] / \mathsf E[g'(S(X))1_{\{X=x\}}] = \mathsf E[X_i1_{\{X=x\}}]/\mathsf E[1_{\{X=x\}}]=\mathsf E[X_i\mid X=x]$ -5648,$\beta-\alpha$ -5649,"$(1+t)(1), (1+t)(2),\dots,(1+t)(10)$" -5650,$q = 1-p$ -5651,$\rho_g(X)=g(s)$ -5652,$\Delta_d=a_{d}'-a_{d}$ -5653,$\kappa_1$ -5654,$\mathsf E_\mathsf{Q}[X+c]=\mathsf E_\mathsf{Q}[X]+c$ -5655,$_{gc}$ -5656,$q(p')$ -5657,$f_i(x+y)=f_i(x)+f_i(y)$ -5658,$=\mathrm{MV}(T(X))$ -5659,$F(a-)=\lim_{x\uparrow a} F(x)$ -5660,$\int_\Omega X(\omega)\mathsf \Pr(d\omega)$ -5661,$g(S(x))>S(x)$ -5662,$s_0/2^{n}$ -5663,$\alpha f/(1-g)$ -5664,"$a_i=a(X_i, p^*)$" -5665,$\Delta X=X_1$ -5666,$V(U)$ -5667,"$f(x)=\int_0^1 f'(tx)\,dt$" -5668,$9$ -5669,$\mathsf E_{\mathsf Q}[X_i\mid X\le a](1-g(S(a))) + a\mathsf E_{\mathsf Q}[X_i/X\mid X >a]g(S(a))$ -5670,$S_{X_{-1}}(a)$ -5671,$S(y_j-)-S(y_j) =\Pr(X=y_j)$ -5672,$g(S_4)=0.5$ -5673,$S(x)>0$ -5674,$q(1)$ -5675,$x_{max}$ -5676,$a \ge 0$ -5677,$E[s|t]=0.08353$ -5678,$ag(S_{\mathsf{j}(a)})=(80)(0.5)=40$ -5679,$\rho(\tilde X\wedge a)\le a$ -5680,$\preceq$ -5681,$X'$ -5682,"$\mathsf{NORM,TI}$" -5683,"$X^+=\max(X,0)$" -5684,$h(s) < s$ -5685,$\mathsf E[X] + \pi \mathsf{SD}(X)$ -5686,$g(s)>s$ -5687,"$(s,g)$" -5688,$1_{U V(2)$ -5707,$\mathbf{Q=1-g(S)}$ -5708,$\mathsf E[Z_i\mid X] \ne \mathsf E[Z_j \mid X]$ -5709,$v_f(\mathsf E_\mathsf{Q}[X_i] - \mathsf E_\mathsf{Q}[X_i/X(X-a)^+])$ -5710,$D = L^* - L$ -5711,$Z_\mathit{lift}$ -5712,$\pi_1$ -5713,$p<0.01$ -5714,$f(s)$ -5715,$\mathbf{\rho(X\wedge a)}$ -5716,$\mathsf E[Z \mid X]\preceq_2 Z$ -5717,$\lambda X_1$ -5718,$\mathsf E[X]$ -5719,$h(X)$ -5720,$\rho_2(X_i)=0.5$ -5721,$Wx)=1-F(x)$ -5725,$\rho(X_n(t))$ -5726,$\int xf(x)dx$ -5727,$\mathsf E_\mathsf{Q}[X+tY]$ -5728,$\tau=0.156$ -5729,$\mathsf{VaR}_p(X) = \mathsf E[X] + \pi(X)\mathsf{SD}(X)$ -5730,$\log(\mathsf E[e^{\pi X}])/\pi$ -5731,"$t=2,3,\dots$" -5732,"$f:[0,1]\to\Omega$" -5733,"$x=x(A,L)=A/L$" -5734,$F(x)=p$ -5735,$X_2=c$ -5736,"$\mathbf{g(S)\, \Delta X}$" -5737,$\sum_{i} X_i(a) = X\wedge a$ -5738,$M = 0.6054$ -5739,$s^\ast = 1/2$ -5740,$W_j$ -5741,$a=\mathsf{TVaR}_p(X)$ -5742,$g(s)=1\wedge(s/0.35)$ -5743,$g'(s)=\alpha s^{\alpha-1}$ -5744,"$\mathbf{\omega_1},\dots,\mathbf{\omega_n}$" -5745,$\mathsf{TVaR}_{p_0}(X)$ -5746,"$A,B\subset \Omega$" -5747,$1/p$ -5748,$F_0$ -5749,$n/(n-1)=1/p$ -5750,$\displaystyle\int_0^\infty S(x)dx$ -5751,$a=\mathsf{VaR}_{1-g^{-1}(\tau)}(X)$ -5752,$(X(\omega_1)-X(\omega_2))(Y(\omega_1)-Y(\omega_2))\ge 0$ -5753,$ROE=-m'(1)/(1-m'(1))$ -5754,$\mathbf{F(x)=\Pr(X\le x)}$ -5755,$\delta>0$ -5756,$\mu(\{\alpha \})=1$ -5757,$\mathsf{Var}(U)>\mathsf{Var}(X)$ -5758,"$Y_{t,d=0}$" -5759,$(l-X)^+$ -5760,"$\rho(X)=\max(\rho_1(X), \rho_2(X))$" -5761,$9/6$ -5762,$j=2$ -5763,$\rho_1(X_i)=1$ -5764,$D^n\rho(\cdot)$ -5765,$\mathsf{FATOU}$ -5766,$p_0$ -5767,$\bar P=\bar P_1+\bar P_2$ -5768,$\mathsf{CTE}_p(X)=(12+25)/2=18.5$ -5769,$\rho(\tilde X_1)=\rho(X_1) + \mathsf E[X_2]$ -5770,$f=1$ -5771,$U_X = F(X-) + V(F(X) - F(X-))$ -5772,$ROL = EL + \lambda (\mathit{EL} (1 - \mathit{EL})/w)^{1/2}$ -5773,$q$ -5774,$v_{res}$ -5775,"$\{1,\dots,n \}$" -5776,$\Pr(X < x)=1/6=\Pr(X\le x)$ -5777,$\mathsf E_{\mathsf Q}[.]$ -5778,$\mathit{MV}_{gc}(a_{gc})=a_{gc}-P(X\wedge a_{gc})=5583.9$ -5779,$q_Y(U)$ -5780,$x^{\ast}$ -5781,$g''(s)=-s^{-3/2}/4$ -5782,$d\tilde p=g'(S(x))f(x)dx$ -5783,$N\times 1$ -5784,$F_X$ -5785,$\mathsf E[X]+\lambda\sigma(X)$ -5786,$\preceq_n$ -5787,$s \to 0$ -5788,$A\subseteq \Omega$ -5789,$r =$ -5790,$t=1$ -5791,"$(s_i,m_i)$" -5792,$F_X(x)\ge F_Y(x)$ -5793,$g'''>0$ -5794,$T=1$ -5795,$\mathsf x\mathsf{VaR}$ -5796,$\mathsf E X + c{(X-\mathsf E X)^+}_p$ -5797,$\mathcal F_{\tau}$ -5798,"$\mathbf X = (X_1, \dots, X_n)$" -5799,$\bar P_{act} = \bar P + F_0 > \bar P$ -5800,$(f)$ -5801,$y^2 - 2\sigma y=(y -\sigma)^2 -\sigma^2$ -5802,"$[0,t]$" diff --git a/greater_tables/tex_svg.py b/greater_tables/tex_svg.py new file mode 100644 index 0000000..95cb55c --- /dev/null +++ b/greater_tables/tex_svg.py @@ -0,0 +1,188 @@ +""" +Create and display svg files from tikz tex tables. + +Good for testing. + +From great2.blog +""" + +from datetime import datetime +import pandas as pd +from pathlib import Path +import re +import yaml +from itertools import count +from subprocess import Popen, PIPE +from IPython.display import display, Markdown, SVG + +from . hasher import txt_short_hash + + +class TikzProcessor(): + _tex_template_full = """\\documentclass[10pt, border=5mm]{{standalone}} + +% needs lualatex - uncomment for Wiley fonts +%\\usepackage{{fontspec}} +%\\setmainfont{{Stix Two Text}} +%\\usepackage{{unicode-math}} +%\\setmathfont{{Stix Two Math}} + +\\usepackage{{amsfonts}} +\\usepackage{{url}} +\\usepackage{{tikz}} +\\usepackage{{color}} +\\usetikzlibrary{{arrows,calc,positioning,shadows.blur,decorations.pathreplacing}} +\\usetikzlibrary{{automata}} +\\usetikzlibrary{{fit}} +\\usetikzlibrary{{snakes}} +\\usetikzlibrary{{intersections}} +\\usetikzlibrary{{decorations.markings,decorations.text,decorations.pathmorphing,decorations.shapes}} +\\usetikzlibrary{{decorations.fractals,decorations.footprints}} +\\usetikzlibrary{{graphs}} +\\usetikzlibrary{{matrix}} +\\usetikzlibrary{{shapes.geometric}} +\\usetikzlibrary{{mindmap, shadows}} +\\usetikzlibrary{{backgrounds}} +\\usetikzlibrary{{cd}} + +% really common macros +\\newcommand{{\\grtspacer}}{{\\vphantom{{lp}}}} + +\\def\\dfrac{{\\displaystyle\\frac}} +\\def\\dint{{\\displaystyle\\int}} + +\\begin{{document}} + +{tikz_begin}{tikz_code}{tikz_end} + +\\end{{document}} +""" + # -------------------------------------------- + _tex_template = """ +% really common macros +\\newcommand{{\\grtspacer}}{{\\vphantom{{lp}}}} + +\\def\\dfrac{{\\displaystyle\\frac}} +\\def\\dint{{\\displaystyle\\int}} + +\\begin{{document}} + +{tikz_begin}{tikz_code}{tikz_end} + +\\end{{document}} +""" + + def split_tikz(self): + """ + Split text to get the tikzpicture. Format is + + initial text pip then groups of four: + + 1. begin tag ``(1::4)`` + 2. tikz code ``(2::4)`` + 3. end tag ``(3::4)`` + 4. non-related text ``(4::4)`` + + """ + return re.split(r'(\\begin{tikz(?:cd|picture)}|\\end{tikz(?:cd|picture)})', self.txt) + + def __init__(self, txt, base_path='.', tex_engine='pdflatex'): + """ + TikzProcessor (from TikzConvertyer): process a tex tikz text string into svg. + The program + + * creates a pdf and svg from the tikz blob + + lualatex is more robust, but slower... + pdflatex can't handle the fancy wiley fonts + + """ + self.txt = txt + self.tex_engine = tex_engine + # directory for TeX and images + self.base_path = Path(base_path).resolve() + self.out_path = self.base_path / 'tikz' + self.out_path.mkdir(exist_ok=True) + self.file_path = self.out_path / txt_short_hash(txt) + + def process_tikz(self, verbose=False): + """ + Process the tikz into pdf and svg + """ + # container contains a tikzpicture + svg_path = self.file_path.with_suffix('.svg') + tex_path = self.file_path.with_suffix('.tex') + + # make tex code for a stand-alone document + tikz_begin, tikz_code, tikz_end = self.split_tikz()[ + 1:4] + tex_code = self._tex_template.format( + tikz_begin=tikz_begin, tikz_code=tikz_code, tikz_end=tikz_end) + tex_path.write_text(tex_code, encoding='utf-8') + print( + f'TIKZ: created temp file = {tex_path.name}') + pdf_file = tex_path.with_suffix('.pdf') + print(f'TIKZ: Update pdf file') + if self.tex_engine == 'pdflatex': + # faster with template + # TODO EVID hard coded template + template_path = Path('tikz_format.fmt') + assert template_path.exists() + template = str(template_path) + command = ['pdflatex', f'--fmt={template}', + f'--output-directory={str(tex_path.parent.resolve())}', + str(tex_path.resolve())] + else: + # for STIX fonts, no template + command = ['lualatex', + f'--output-directory={str(tex_path.parent.resolve())}', + str(tex_path.resolve())] + if verbose: + print(f'TIKZ: TeX Command={" ".join(command)}') + TikzProcessor.run_command(command) + # to recreate + (tex_path.parent / + f'make_tikz.bat').write_text(" ".join(command)) + if verbose: + print( + f'TIKZ: Creating svg file for Tikz (using new pdf2svg util)') + # https://github.com/jalios/pdf2svg-windows + command = [ + 'C:\\temp\\pdf2svg-windows\\dist-64bits\\pdf2svg', + str(pdf_file.resolve()), str(svg_path.resolve())] + # seems to return info on stderr? + if verbose: + print(f'PDF->SVG: {" ".join(command)}') + TikzProcessor.run_command(command, flag=False) + if not verbose: + # tidy up + tex_path.unlink() + tex_path.with_suffix('.aux').unlink() + tex_path.with_suffix('.log').unlink() + pdf_file.unlink() + + @staticmethod + def run_command(command, flag=True): + """ + Run a command and show results. Allows for weird xx behavior + + :param command: + :param flag: + :return: + """ + with Popen(command, stdout=PIPE, stderr=PIPE, universal_newlines=True) as p: + line1 = p.stdout.read() + line2 = p.stderr.read() + exit_code = p.poll() + if line1: + print('\n' + line1[-250:]) + if line2: + if flag: + raise ValueError(line2) + else: + print(line2) + return exit_code + + def display(self): + """display in Jupyter Lab.""" + display(SVG(self.file_path.with_suffix('.svg'))) diff --git a/greater_tables/tex_svg2.py b/greater_tables/tex_svg2.py new file mode 100644 index 0000000..7771c10 --- /dev/null +++ b/greater_tables/tex_svg2.py @@ -0,0 +1,133 @@ +""" +Create and display SVG files from TikZ pictures embedded in LaTeX. + +Good for testing. Outputs are cached by hash. PDF→SVG uses pdf2svg. + +GPT re-write of my old great2.blog code. +""" + +import re +from pathlib import Path +from subprocess import run, Popen, PIPE +from IPython.display import SVG, display + +from .hasher import txt_short_hash + + +class TikzProcessor: + # Full TeX preamble to generate a .fmt if needed + _tex_template_full = r"""\documentclass[10pt, border=5mm]{standalone} +\usepackage{amsfonts} +\usepackage{url} +\usepackage{tikz} +\usepackage{color} +\usetikzlibrary{arrows,calc,positioning,shadows.blur,decorations.pathreplacing} +\usetikzlibrary{automata,fit,snakes,intersections} +\usetikzlibrary{decorations.markings,decorations.text,decorations.pathmorphing,decorations.shapes} +\usetikzlibrary{decorations.fractals,decorations.footprints} +\usetikzlibrary{graphs,matrix,shapes.geometric} +\usetikzlibrary{mindmap,shadows,backgrounds,cd} +\dump +""" + + # Minimal template to embed user tikz + _tex_template = r""" +\newcommand{{\grtspacer}}{{\vphantom{{lp}}}} +\def\dfrac{{\displaystyle\frac}} +\def\dint{{\displaystyle\int}} +\begin{{document}} +{tikz_begin}{tikz_code}{tikz_end} +\end{{document}} +""" + + + def __init__(self, txt, base_path='.', tex_engine='pdflatex'): + self.txt = txt + self.tex_engine = tex_engine + self.base_path = Path(base_path).resolve() + self.out_path = self.base_path / 'tikz' + self.out_path.mkdir(exist_ok=True) + self.file_path = self.out_path / txt_short_hash(txt) + self.format_file = self.out_path / 'tikz_format.fmt' + + def split_tikz(self): + """Split text to extract the TikZ picture.""" + return re.split(r'(\\begin{tikz(?:cd|picture)}|\\end{tikz(?:cd|picture)})', self.txt) + + def ensure_format_file(self): + """Create format file for faster compilation if missing.""" + if self.format_file.exists(): + return + print('building format file...') + tmp = self.out_path / 'tikz_format.tex' + tmp.write_text(self._tex_template_full, encoding='utf-8') + self.run_command([ + 'pdflatex', + f'-ini', + f'-jobname={self.format_file.stem}', + '&pdflatex', + tmp.name, + ], raise_on_error=True, cwd=self.out_path) + # tmp.unlink() + (self.out_path / f'{self.format_file.stem}.log').unlink() + print('building format file...success', self.format_file.resolve()) + + def process_tikz(self, verbose=False): + """Compile TikZ to PDF and convert to SVG.""" + tikz_begin, tikz_code, tikz_end = self.split_tikz()[1:4] + tex_code = self._tex_template.format( + tikz_begin=tikz_begin, + tikz_code=tikz_code, + tikz_end=tikz_end + ) + + tex_path = self.file_path.with_suffix('.tex') + tex_path.write_text(tex_code, encoding='utf-8') + pdf_path = tex_path.with_suffix('.pdf') + svg_path = tex_path.with_suffix('.svg') + + self.ensure_format_file() + + tex_cmd = [ + 'pdflatex', + f'--fmt={self.format_file.stem}', + f'--output-directory={str(tex_path.parent)}', + str(tex_path) + ] + if verbose: + print("Running:", " ".join(tex_cmd)) + self.run_command(tex_cmd) + + (tex_path.parent / 'make_tikz.bat').write_text(" ".join(tex_cmd), encoding='utf-8') + + svg_cmd = [ + 'C:\\temp\\pdf2svg-windows\\dist-64bits\\pdf2svg', + str(pdf_path), + str(svg_path) + ] + if verbose: + print("Running:", " ".join(svg_cmd)) + self.run_command(svg_cmd, raise_on_error=False) + + if not verbose: + for ext in ('.tex', '.aux', '.log', '.pdf'): + path = tex_path.with_suffix(ext) + if path.exists(): + path.unlink() + + def display(self): + """Display the SVG in Jupyter.""" + display(SVG(self.file_path.with_suffix('.svg'))) + + @staticmethod + def run_command(command, raise_on_error=True, cwd=None): + """Run command with subprocess and show output.""" + with Popen(command, cwd=cwd, stdout=PIPE, stderr=PIPE, universal_newlines=True) as p: + stdout, stderr = p.communicate() + if stdout: + print(stdout.strip()[-250:]) + if stderr: + if raise_on_error: + raise RuntimeError(stderr.strip()) + else: + print(stderr.strip()) diff --git a/pyproject.toml b/pyproject.toml index 590151a..1be6c70 100644 --- a/pyproject.toml +++ b/pyproject.toml @@ -30,6 +30,9 @@ classifiers = [ include = ["greater_tables"] exclude = ["img", "tests", "docs"] +[tool.setuptools.package-data] +"greater_tables" = ["data/*.csv", "data/*.md"] + [tool.setuptools.dynamic] version = { attr = "greater_tables.__version__" }