From ec3ac6655222d3cbd98b4194e63fbd26852d5f60 Mon Sep 17 00:00:00 2001 From: Stephen Mildenhall Date: Fri, 13 Jun 2025 23:41:59 +0100 Subject: [PATCH] README tidying; testdf massive... --- README-remember.md | 108 - README.md | 51 +- README.qmd | 99 - README.rst | 46 - greater_tables/greater_tables.py | 43 +- greater_tables/gtcore2.py | 139 +- greater_tables/gtformats.py | 9 + greater_tables/testdf.py | 533 +- greater_tables/tex_list.csv | 5804 +++++++ greater_tables/words-12.md | 25912 +++++++++++++++++++++++++++++ prefobnicate.tex | 76 - tests/img/favicon.ico | Bin 2753 -> 0 bytes tests/pmir-header.html | 2 - tests/prefobnicate.tex | 31 - tests/single-table.html | 356 - tests/styles.css | 199 - tests/tables.html | 3194 ---- tests/tables.ipynb | 5244 ------ tests/tables.pdf | Bin 127229 -> 0 bytes tests/tables.qmd | 412 - tests/tables.tex | 1537 -- tests/test_tables.py | 173 - tests/utilities.py | 348 - 23 files changed, 32146 insertions(+), 12170 deletions(-) delete mode 100644 README-remember.md delete mode 100644 README.qmd delete mode 100644 README.rst create mode 100644 greater_tables/tex_list.csv create mode 100644 greater_tables/words-12.md delete mode 100644 prefobnicate.tex delete mode 100644 tests/img/favicon.ico delete mode 100644 tests/pmir-header.html delete mode 100644 tests/prefobnicate.tex delete mode 100644 tests/single-table.html delete mode 100644 tests/styles.css delete mode 100644 tests/tables.html delete mode 100644 tests/tables.ipynb delete mode 100644 tests/tables.pdf delete mode 100644 tests/tables.qmd delete mode 100644 tests/tables.tex delete mode 100644 tests/test_tables.py delete mode 100644 tests/utilities.py diff --git a/README-remember.md b/README-remember.md deleted file mode 100644 index 8ad5381..0000000 --- a/README-remember.md +++ /dev/null @@ -1,108 +0,0 @@ - - -Here's how you subclass. - - -```python -class sGT(GT): - """ - Example standard GT with Steve House-Style defaults. - - Each application can create its own defaults by subclassing GT - in this way. - """ - - def __init__(self, df, caption="", guess_years=True, ratio_regex='lr|roe|coc', **kwargs): - """Create Steve House-Style Formatter. Does not handle list of lists input.""" - if isinstance(df, str): - df, aligners_ = GT.md_to_df(df) - if 'aligners' not in kwargs: - kwargs['aligners'] = aligners_ - kwargs['show_index'] = False - - nindex = df.index.nlevels - ncolumns = df.columns.nlevels - if 'ratio_cols' in kwargs: - ratio_cols = kwargs['ratio_cols'] - else: - if ratio_regex != '' and ncolumns == 1: - ratio_cols = df.filter(regex=ratio_regex).columns.to_list() - else: - ratio_cols = None - - if guess_years: - year_cols = sGT.guess_years(df) - else: - year_cols = kwargs.get('year_cols', None) - - # rule sizes - hrule_widths = (1.5, 1, 0) if nindex > 1 else None - vrule_widths = (1.5, 1, 0.5) if ncolumns > 1 else None - - table_hrule_width = 1 if nindex == 1 else 2 - table_vrule_width = 1 if ncolumns == 1 else ( - 1.5 if ncolumns == 2 else 2) - - # padding - nr, nc = df.shape - if 'padding_trbl' in kwargs: - padding_trbl = kwargs['padding_trbl'] - else: - pad_tb = 4 if nr < 16 else (2 if nr < 25 else 1) - pad_lr = 10 if nc < 9 else (5 if nc < 13 else 2) - padding_trbl = (pad_tb, pad_lr, pad_tb, pad_lr) - - font_body = 0.9 if nr < 25 else (0.8 if nr < 41 else 0.7) - font_caption = np.round(1.1 * font_body, 2) - font_head = np.round(1.1 * font_body, 2) - - pef_lower = -3 - pef_upper = 6 - pef_precision = 3 - - defaults = { - 'ratio_cols': ratio_cols, - 'year_cols': year_cols, - 'default_integer_str': '{x:,.0f}', - 'default_float_str': '{x:,.3f}', - 'default_date_str': '%Y-%m-%d', - 'default_ratio_str': '{x:.1%}', - 'cast_to_floats': True, - 'table_hrule_width': table_hrule_width, - 'table_vrule_width': table_vrule_width, - 'hrule_widths': hrule_widths, - 'vrule_widths': vrule_widths, - 'sparsify': True, - 'sparsify_columns': True, - 'padding_trbl': padding_trbl, - 'font_body': font_body, - 'font_head': font_head, - 'font_caption': font_caption, - 'pef_precision': pef_precision, - 'pef_lower': pef_lower, - 'pef_upper': pef_upper, - 'debug': False - } - defaults.update(kwargs) - super().__init__(df, caption=caption, **defaults) - - @staticmethod - def guess_years(df): - """Try to guess which columns (body or index) are years. - - A column is considered a year if: - - It is numeric (integer or convertible to integer) - - All values are within a reasonable range (e.g., 1800–2100) - """ - year_columns = [] - df = df.reset_index(drop=False, col_level=df.columns.nlevels - 1) - for i, col in enumerate(df.columns): - try: - series = pd.to_numeric(df[col], errors='coerce').dropna() - if series.dtype.kind in 'iu' and series.between(1800, 2100).all(): - year_columns.append(col) - except Exception: - continue - return year_columns - -``` diff --git a/README.md b/README.md index 8839198..465bd80 100644 --- a/README.md +++ b/README.md @@ -1,5 +1,8 @@ sort out the variety of readmes... this is the main one +https://shields.io/badges/read-the-docs + + # v 3.0 update * config files @@ -195,12 +198,56 @@ More coming soon. ## Documentation +![](https://img.shields.io/readthedocs/greater_tables_project) + Available on [readthedocs](https://greater-tables-project.readthedocs.io/en/latest). ## Versions -### 1.1.1 +3.0.0 +------- + +2.0.0 +------ + +1.1.1 +------- * Added logo, updated docs. -### 1.1.0 +1.1.0 +------ + +* added ``formatters`` argument to pass in column specific formatters by name as a number (``n`` converts to ``{x:.nf}``, format string, or function +* Added ```tabs`` argument to provide column widths +* Added ``equal`` argument to provide hint that column widths should all be equal +* Added ``caption_align='center'`` argument to set the caption alignment +* Added ``large_ok=False`` argument, if ``False`` providing a dataframe with more than 100 rows throws an error. This function is expensive and is designed for small frames. + + +1.0.0 +------ + +* Allow input via list of lists, or markdown table +* Specify overall float format for whole table +* Specify column alingment with 'llrc' style string +* ``show_index`` option +* Added more tests +* Docs updated +* Set tabs for width; use of width in HTML format. + + +0.6.0 +------ + +* Initial release + +Early development +------------------- + +* 0.1.0 - 0.5.0: Early development +* tikz code from great.pres_manager + + + + diff --git a/README.qmd b/README.qmd deleted file mode 100644 index 284763c..0000000 --- a/README.qmd +++ /dev/null @@ -1,99 +0,0 @@ ---- -format: - pdf: - include-in-header: prefobnicate.tex - html: - table-processing: false -jupyter: - keep-ipynb: true - jupytext: - formats: ipynb,qmd - text_representation: - extension: .qmd - format_name: quarto - format_version: '1.0' - jupytext_version: 1.16.4 - kernelspec: - display_name: Python 3 (ipykernel) - language: python - name: python3 ---- - -# Greater Tables - -Creating presentation quality tables from pandas dataframes is frustrating. It is hard to left-align text and right-align numbers using pandas `display` or `df.to_html`. The `great_tables` package does a really nice job with pandas and polars dataframes but does not support indexes or TeX output. - -This package provides consistent HTML and TeX table output with flexible type-based formatting, and table rules. Neither output relies on the pandas `to_html` or `to_latex` functions. TeX output uses Tikz tables for very tight control over layout and grid lines. The package is designed for use in Jupyter Lab notebooks Quarto documents. - -Usage: the main class `GT` should be subclassed to set appropriate defaults for your project. `sGT` provides an example. - -The project is currently in **beta** status. HTML output is better developed than TeX. - -## The Name - -Obviously, the name is a play on the `great_tables` package. But, I have been maintaining a set of macros called [GREATools](https://www.mynl.com/old/GREAT/home.html) (generalized, reusable, extensible actuarial tools) in VBA and Python since the late 1990s, and call all my macro packages "GREAT". - -## Installation - -```python -pip install greater-tables -``` - -## Examples - -The following example shows quite a hard table. It is formatted using the `sGT` class, which is a subclass of `GT` with a few defaults set. - -```{python} -#| echo: true -#| output: asis -import pandas as pd -import numpy as np -from greater_tables import sGT -level_1 = ["Group A", "Group A", "Group B", "Group B", 'Group C'] -level_2 = ['Sub 1', 'Sub 2', 'Sub 2', 'Sub 3', 'Sub 3'] - -multi_index = pd.MultiIndex.from_arrays([level_1, level_2]) - -start = pd.Timestamp.today().normalize() # Today's date, normalized to midnight -end = pd.Timestamp(f"{start.year}-12-31") # End of the year - -hard = pd.DataFrame( -{'x': np.arange(2020, 2025, dtype=int), -'a': np.array((100, 105, 2000, 2025, 100000), dtype=int), -'b': 10. ** np.linspace(-9, 9, 5), -'c': np.linspace(601, 4000, 5), -'d': pd.date_range(start=start, end=end, periods=5), -'e': 'once upon a time, risk is hard to define, not in Kansas anymore, neutrinos are hard to detect, $\\int_\\infty^\\infty e^{-x^2/2}dx$ is a hard integral'.split(',') -}).set_index('x') -hard.columns = multi_index -sGT(hard, 'A hard table.') -``` - -![HTML output](hard_table_html.png) - -![TeX output](hard_table_tex.png) - -The output illustrates: - -* Quarto or Jupyter automatically the class's `_repr_html_` method (or `_repr_latex_` for pdf/TeX/Beamer output), providing seamless integration across different output formats. -* Text is left-aligned, numbers are right-aligned. -* The index is displayed, was detected as likely years, and formatted without a comma separator. -* The first column of integers does have a comma thousands separator. -* The second column of floats spans several orders of magnitude and is formatted using Engineering format, n for nano through G for giga. -* The third column of floats is formatted with a comma separator and two decimals, based on the average absolute value. -* The fourth column of date times is formatted as ISO standard dates (not date times). -* The vertical lines separate the levels of the column multiindex. The subgroups are a little tricky. - -More coming soon. - - -## Documentation - -Available on [readthedocs](https://greater-tables-project.readthedocs.io/en/latest). - -## Versions - -### 1.1.1 -* Added logo, updated docs. - -### 1.1.0 diff --git a/README.rst b/README.rst deleted file mode 100644 index bc75fe1..0000000 --- a/README.rst +++ /dev/null @@ -1,46 +0,0 @@ -.. image:: https://img.shields.io/readthedocs/greater_tables_project - :alt: Read the Docs - -Release Notes -=============== - -1.1.0 ------- - -* added ``formatters`` argument to pass in column specific formatters by name as a number (``n`` converts to ``{x:.nf}``, format string, or function -* Added ```tabs`` argument to provide column widths -* Added ``equal`` argument to provide hint that column widths should all be equal -* Added ``caption_align='center'`` argument to set the caption alignment -* Added ``large_ok=False`` argument, if ``False`` providing a dataframe with more than 100 rows throws an error. This function is expensive and is designed for small frames. - - -1.0.0 ------- - -* Allow input via list of lists, or markdown table -* Specify overall float format for whole table -* Specify column alingment with 'llrc' style string -* ``show_index`` option -* Added more tests -* Docs updated -* Set tabs for width; use of width in HTML format. - - -0.6.0 ------- - -* Initial release - -Early development -------------------- - -* 0.1.0 - 0.5.0: Early development -* tikz code from great.pres_manager - -TODO -===== - -* Index aligners - - -https://shields.io/badges/read-the-docs \ No newline at end of file diff --git a/greater_tables/greater_tables.py b/greater_tables/greater_tables.py index 67dad1a..a921937 100644 --- a/greater_tables/greater_tables.py +++ b/greater_tables/greater_tables.py @@ -23,6 +23,7 @@ from rich import box from rich.table import Table from . hasher import df_short_hash +from . gtformats import GT_Format, TableFormat # turn this fuck-fest off pd.set_option('future.no_silent_downcasting', True) @@ -61,43 +62,6 @@ class Breakability(IntEnum): ACCEPTABLE = 10 -# specify text mode -Line = namedtuple('Line', ['begin', 'hline', 'sep', 'end', 'index_sep']) -DataRow = namedtuple('DataRow', ['begin', 'sep', 'end', 'index_sep']) -TableFormat = namedtuple('TableFormat', [ - 'lineabove', - 'linebelowheader', - 'linebetweenrows', - 'linebelow', - 'headerrow', - 'datarow', - 'padding', - 'with_header_hide' -]) - -# generic text format -GT_Format = TableFormat( - lineabove=Line('┍', '━', '┯', '┑', '┳'), - linebelowheader=Line('┝', '━', '┿', '┥', '╋'), - linebetweenrows=Line('├', '─', '┼', '┤', '╂'), - linebelow=Line('┕', '━', '┷', '┙', '┻'), - headerrow=DataRow('│', '│', '│', '┃'), - datarow=DataRow('│', '│', '│', '┃'), - padding=1, - with_header_hide=None -) - -# GT_Format = TableFormat( -# lineabove=Line('\u250d', '\u2501', '\u252f', '\u2511', '\u2533'), -# linebelowheader=Line('\u251d', '\u2501', '\u253f', '\u2525', '\u254b'), -# linebetweenrows=Line('\u251c', '\u2500', '\u253c', '\u2524', '\u2502'), -# linebelow=Line('\u2515', '\u2501', '\u2537', '\u2519', '\u253b'), -# headerrow=DataRow('\u2502', '\u2502', '\u2502', '\u2503'), -# datarow=DataRow('\u2502', '\u2502', '\u2502', '\u2503'), -# padding=1, -# with_header_hide=None -# ) - class GT(object): """ @@ -1716,7 +1680,10 @@ class GT(object): # dx = data in index # if this is the level that changes for this row # will use a top rule hence omit i = 0 which already has an hrule - if i > 0 and hrule == '' and j == index_change_level[i]: + # here have to be careful - if the index is not ! then not every row + # appears in the index change level. But if it DOES NOT appear then + # it isn't a change level so no rule required + if i > 0 and hrule == '' and i in index_change_level and j == index_change_level[i]: hrule = f'grt-hrule-{j}' # html.append(f'{c}') col_id = f'grt-c-{j}' diff --git a/greater_tables/gtcore2.py b/greater_tables/gtcore2.py index d9c108b..5c51ff7 100644 --- a/greater_tables/gtcore2.py +++ b/greater_tables/gtcore2.py @@ -18,18 +18,21 @@ import sys from textwrap import wrap from typing import Optional, Union, Literal import warnings +import yaml from bs4 import BeautifulSoup from cachetools import LRUCache import numpy as np import pandas as pd +from pandas.errors import IntCastingNaNError from pandas.api.types import is_datetime64_any_dtype, is_integer_dtype, \ is_float_dtype # , is_numeric_dtype +from pydantic import ValidationError from rich import box from rich.table import Table from . gtenums import Breakability, Alignment -from . gtformats import GT_Format, TableFormat +from . gtformats import GT_Format, TableFormat, Line, DataRow from . gtconfig import GTConfigModel from . hasher import df_short_hash @@ -242,7 +245,8 @@ class GT(object): **overrides, ): if config and config_path: - raise ValueError("Pass either 'config' or 'config_path', not both.") + raise ValueError( + "Pass either 'config' or 'config_path', not both.") if config: base_config = config @@ -251,7 +255,8 @@ class GT(object): raw = yaml.safe_load(config_path.read_text(encoding="utf-8")) base_config = GTConfigModel.model_validate(raw) except (ValidationError, OSError) as e: - raise ValueError(f"Failed to load config from {config_path}") from e + raise ValueError(f"Failed to load config from { + config_path}") from e else: base_config = GTConfigModel() @@ -402,7 +407,7 @@ class GT(object): self.raw_cols = raw_cols # figure the default formatter (used in conjunction with raw columns) - if config.default_formatter is None: + if self.config.default_formatter is None: self.default_formatter = self.default_formatter else: assert callable( @@ -413,13 +418,13 @@ class GT(object): return config.default_formatter(x) except ValueError: return str(x) - self.default_formatter = wrapped_config.default_formatter + self.default_formatter = wrapped_default_formatter # cast as much as possible to floats with warnings.catch_warnings(): warnings.simplefilter( "ignore", category=pd.errors.PerformanceWarning) - if config.cast_to_floats: + if self.config.cast_to_floats: for i, c in enumerate(self.df.columns): if c in self.raw_cols or c in self.date_cols: continue @@ -552,10 +557,10 @@ class GT(object): # self.default_float_formatter = None # self.hrule_widths = hrule_widths or (0, 0, 0) # if not isinstance(self.config.hrule_widths, (list, tuple)): - # self.config.hrule_widths = (self.config.hrule_widths,) + # self.config.hrule_widths = (self.config.hrule_widths,) # self.vrule_widths = vrule_widths or (0, 0, 0) # if not isinstance(self.config.hrule_widths, (list, tuple)): - # self.config.hrule_widths = (self.config.hrule_widths, ) + # self.config.hrule_widths = (self.config.hrule_widths, ) # self.table_hrule_width = table_hrule_width # self.table_vrule_width = table_vrule_width # self.font_body = font_body @@ -570,28 +575,25 @@ class GT(object): elif isinstance(tabs, (int, float)): self.tabs = (tabs,) elif isinstance(tabs, (np.ndarray, list, tuple)): - self.tabs = tabs # Already iterable, self.config.tabs = as is + self.tabs = tabs # Already iterable, self.tabs = as is else: self.tabs = [tabs] # Fallback for anything else # self.equal = equal - if config.padding_trbl is None: - if config.spacing == 'tight': - config.padding_trbl = (0, 5, 0, 5) - elif config.spacing == 'medium': - config.padding_trbl = (2, 10, 2, 10) - elif config.spacing == 'wide': - config.padding_trbl = (4, 15, 4, 15) + if self.config.padding_trbl is not None: + padding_trbl = self.config_padding_trbl + elif self.config.padding_trbl is None: + if self.config.spacing == 'tight': + padding_trbl = (0, 5, 0, 5) + elif self.config.spacing == 'medium': + padding_trbl = (2, 10, 2, 10) + elif self.config.spacing == 'wide': + padding_trbl = (4, 15, 4, 15) else: raise ValueError( 'config.spacing must be tight, medium, or wide or tuple of four ints.') - try: - self.padt, self.padr, self.padb, self.padl = config.padding_trbl - except ValueError: - # pydantics will see to this... - logger.error( - f'config.padding_trbl {config.padding_trbl=}, must be four ints, defaulting to medium padding') - self.padt, self.padr, self.padb, self.padl = 2, 10, 2, 10 + # pydantic will see to it this is OK + self.padt, self.padr, self.padb, self.padl = padding_trbl # because of the problem of non-unique indexes use a list and # not a dict to pass the formatters to to_html @@ -608,7 +610,7 @@ class GT(object): # cache for various things... self._cache = LRUCache(20) # config.sparsify - if config.sparsify and self.nindex > 1: + if self.config.sparsify and self.nindex > 1: self.df = GT.sparsify(self.df, self.df.columns[:self.nindex]) # for c in self.df.columns[:self.nindex]: # # config.sparsify returns some other stuff... @@ -857,7 +859,7 @@ class GT(object): self.default_float_formatter or self.make_float_formatter(self.df.iloc[:, i])) else: # print(f'{i} default') - self._df_formatters.append(self.config.default_formatter) + self._df_formatters.append(self.default_formatter) # self._df_formatters is now a list of length config.equal to cols in df if len(self._df_formatters) != self.df.shape[1]: raise ValueError( @@ -925,8 +927,10 @@ class GT(object): PADDING = 2 # per column if self.config.table_width_mode == 'explicit': # target width INCLUDES padding and column marks | - target_width = self.config.max_table_width - (PADDING + 1) * n_col - 1 - logger.info(f'Col padding effect {self.config.max_table_width=} ==> {target_width=}') + target_width = self.config.max_table_width - \ + (PADDING + 1) * n_col - 1 + logger.info(f'Col padding effect { + self.config.max_table_width=} ==> {target_width=}') elif self.config.table_width_mode == 'natural': target_width = natural + (PADDING + 1) * n_col + 1 elif self.config.table_width_mode == 'breakable': @@ -936,7 +940,8 @@ class GT(object): # extra space for the headers to relax, if useful if self.config.table_width_header_adjust > 0: - max_extra = int(self.config.table_width_header_adjust * target_width) + max_extra = int( + self.config.table_width_header_adjust * target_width) else: max_extra = 0 @@ -1004,7 +1009,14 @@ class GT(object): ans['recommended'], ans['natural_w_header']) # Ensure final constraint - ans['recommended'] = ans['recommended'].astype(int) + try: + ans['recommended'] = ans['recommended'].astype(int) + except IntCastingNaNError: + print('getting error') + print(ans['recommended']) + ans['recommended'] = pd.to_numeric( + ans['recommended'], errors='coerce').fillna(0).astype(int) + logger.info("Raw rec: %s\tTweaks: %s\tActual: %s\tTarget: %s\tOver/(U): %s", ans['raw_rec'].sum(), ans['header_tweak'].sum(), @@ -1517,7 +1529,7 @@ class GT(object): font-weight: bold; }} '''] - for i, w in enumerate(config.tabs): + for i, w in enumerate(tabs): style.append(f' #{self.df_id} .grt-c-{i} {{ width: {w}em; }}') style.append('') logger.info('CREATED CSS') @@ -1547,15 +1559,15 @@ class GT(object): colw, tabs = GT.estimate_column_widths( self.df, self.config.max_table_width, nc_index=self.nindex, scale=1, equal=self.config.equal) if self.config.debug: - print(f'Make html Input {self.config.tabs=}\nComputed {tabs=}') - if self.config.tabs is not None: - if len(tabs) == len(self.config.tabs): - tabs = self.config.tabs - elif len(self.config.tabs) == 1: - tabs = self.config.tabs * len(tabs) + print(f'Make html Input {self.tabs=}\nComputed {tabs=}') + if self.tabs is not None: + if len(tabs) == len(self.tabs): + tabs = self.tabs + elif len(self.tabs) == 1: + tabs = self.tabs * len(tabs) else: logger.error( - f'{self.config.tabs=} must be None, a single number, or a list of numbers of the correct length. Ignoring.') + f'{self.tabs=} must be None, a single number, or a list of numbers of the correct length. Ignoring.') # print('HTML ' + ', '.join([f'{c:,.2f}' for c in tabs])) # set column widths; tabs returns lengths of strings in each column @@ -1665,7 +1677,9 @@ class GT(object): # dx = data in index # if this is the level that changes for this row # will use a top rule hence omit i = 0 which already has an hrule - if i > 0 and hrule == '' and j == index_change_level[i]: + # appears in the index change level. But if it DOES NOT appear then + # it isn't a change level so no rule required + if i > 0 and hrule == '' and i in index_change_level and j == index_change_level[i]: hrule = f'grt-hrule-{j}' # html.append(f'{c}') col_id = f'grt-c-{j}' @@ -1752,8 +1766,12 @@ class GT(object): @staticmethod def apply_formatters_work(df, formatters): """Apply formatters to a DataFrame.""" - new_df = pd.DataFrame({i: map(f, df.iloc[:, i]) - for i, f in enumerate(formatters)}) + try: + new_df = pd.DataFrame({i: map(f, df.iloc[:, i]) + for i, f in enumerate(formatters)}) + except TypeError: + print('NASTY TYPE ERROR') + raise new_df.columns = df.columns return new_df @@ -1899,7 +1917,7 @@ class GT(object): row sep={row_sep}em, column sep={column_sep}em, nodes in empty cells, - nodes={{rectangle, scale={scale}, text badly ragged {config.debug}}}, + nodes={{rectangle, scale={scale}, text badly ragged {debug}}}, """ # put draw=blue!10 or so in nodes to see the node @@ -1928,12 +1946,12 @@ class GT(object): "ignore", category=pd.errors.PerformanceWarning) df = df.reset_index( drop=False, col_level=df.columns.nlevels - 1) - if config.sparsify: + if sparsify: if hrule is None: hrule = set() - for i in range(config.sparsify): - # TODO update to new config.sparsify!! - df.iloc[:, i], rules = GT.config.sparsify_old(df.iloc[:, i]) + for i in range(sparsify): + # TODO update to new sparsify!! + df.iloc[:, i], rules = GT.sparsify_old(df.iloc[:, i]) # don't want lines everywhere if len(rules) < len(df) - 1: hrule = set(hrule).union(rules) @@ -1960,15 +1978,15 @@ class GT(object): # estimate... originally called guess_column_widths, with more parameters colw, tabs = GT.estimate_column_widths(df, self.config.max_table_width, nc_index=nc_index, scale=self.config.tikz_scale, equal=self.config.equal) # noqa if self.config.debug: - print(f'Make TikZ Input {self.config.tabs=}\nComputed {tabs=}') - if self.config.tabs is not None: - if len(tabs) == len(self.config.tabs): - tabs = self.config.tabs - elif len(self.config.tabs) == 1: - tabs = self.config.tabs * len(tabs) + print(f'Make TikZ Input {self.tabs=}\nComputed {tabs=}') + if self.tabs is not None: + if len(tabs) == len(self.tabs): + tabs = self.tabs + elif len(self.tabs) == 1: + tabs = self.tabs * len(tabs) else: logger.error( - f'{self.config.tabs=} must be None, a single number, or a list of numbers of the correct length. Ignoring.') + f'{self.tabs=} must be None, a single number, or a list of numbers of the correct length. Ignoring.') # print('TIKZ ' + ', '.join([f'{c:,.2f}' for c in tabs])) # print(f'TIKZ {colw=}, {tabs=}') logger.info(f'tabs: {tabs}') @@ -1995,10 +2013,12 @@ class GT(object): latex = '' else: latex = f'[{latex}]' - config.debug = '' + debug = '' if self.config.debug: # color all boxes - config.debug = ', draw=blue!10' + debug = ', draw=blue!10' + else: + debug = '' sio.write(header.format(container_env=container_env, caption=caption, extra_defs=extra_defs, @@ -2006,7 +2026,7 @@ class GT(object): column_sep=column_sep, row_sep=row_sep, latex=latex, - debug=self.config.debug)) + debug=debug)) # table header # title rows, start with the empty spacer row @@ -2024,7 +2044,8 @@ class GT(object): if i == 1: # first column sets row height for entire row sio.write(f'\tcolumn {i:>2d}/.style={{' - f'nodes={{align={ad[al]:<6s}}}, text height=0.9em, text depth=0.2em, ' + f'nodes={{align={ + ad[al]:<6s}}}, text height=0.9em, text depth=0.2em, ' f'inner xsep={column_sep}em, inner ysep=0, ' f'text width={max(2, 0.6 * w):.2f}em}},\n') else: @@ -2053,7 +2074,7 @@ class GT(object): if isinstance(df.columns, pd.MultiIndex): for lvl in range(len(df.columns.levels)): nl = '' - sparse_columns[lvl], mi_vrules[lvl] = GT.config.sparsify_mi(df.columns.get_level_values(lvl), + sparse_columns[lvl], mi_vrules[lvl] = GT.sparsify_mi(df.columns.get_level_values(lvl), lvl == len(df.columns.levels) - 1) for cn, c, al in zip(df.columns, sparse_columns[lvl], align): # c = wfloat_format(c) @@ -2268,7 +2289,7 @@ class GT(object): # data all seems about the same width tabs.append(common_size) logger.info(f'Determined tab config.spacing: {tabs}') - if config.equal: + if equal: # see if config.equal widths makes sense dt = tabs[nl:] if max(dt) / sum(dt) < 4 / 3: @@ -2302,7 +2323,7 @@ class GT(object): @staticmethod def sparsify_old(col): """ - config.sparsify col values, col a pd.Series or dict, with items and accessor + sparsify col values, col a pd.Series or dict, with items and accessor column results from a reset_index so has index 0,1,2... this is relied upon. TODO: this doesn't work if there is a change in a higher level but not this level """ diff --git a/greater_tables/gtformats.py b/greater_tables/gtformats.py index cca9448..532827b 100644 --- a/greater_tables/gtformats.py +++ b/greater_tables/gtformats.py @@ -1,3 +1,4 @@ +# coding: utf-8 """ Define text table formats. @@ -80,3 +81,11 @@ GT_Format = TableFormat( # padding=1, # with_header_hide=None # ) + + +def default_formatter(x): + """ + + + + """ diff --git a/greater_tables/testdf.py b/greater_tables/testdf.py index be8acdb..a41657f 100644 --- a/greater_tables/testdf.py +++ b/greater_tables/testdf.py @@ -4,262 +4,226 @@ Make fake dataframes for testing. GPT from SJMM design. """ -# from pathlib import Path -# from dataclasses import dataclass, field -# from typing import Optional, Union -# from datetime import datetime, timedelta -# import hashlib -# import re - - -# import numpy as np -# import pandas as pd -# from faker import Faker - - -# @dataclass -# class TestDataFrameFactory: -# """ -# Factory for generating small synthetic pandas DataFrames for testing. - -# Attributes: -# colname_words: Optional list of strings to use for column names. -# default_word_count: Max number of words for string columns (default 3). -# seed: Optional random seed. If None, one is generated. -# """ -# colname_words: Optional[list[str]] = None -# default_word_count: int = 3 -# seed: Optional[int] = None -# _last_args: dict = field(default_factory=dict, init=False) - -# def __post_init__(self): -# self.faker = Faker() -# self.seed = int(self.seed if self.seed is not None else np.random.SeedSequence().entropy) -# self.rng = np.random.default_rng(self.seed) - -# def make(self, rows: int, columns: Union[int, str], index: Union[int, str] = 0, -# col_index: Union[int, str] = 0, missing: float = 0.0) -> pd.DataFrame: -# """ -# Generate a test DataFrame with the given specification. - -# 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'). -# col_index: Column index levels (same format as `index`). -# missing: Proportion of missing data in each column. - -# Returns: -# DataFrame -# """ -# self._last_args = dict(rows=rows, columns=columns, index=index, col_index=col_index, missing=missing) -# return self._generate(**self._last_args) - -# def another(self, new_seed: bool = True) -> pd.DataFrame: -# """ -# Generate another DataFrame with the last parameters. - -# Args: -# new_seed: If True, re-randomize the generator seed. - -# Returns: -# DataFrame -# """ -# if new_seed: -# self.seed = int(np.random.SeedSequence().entropy) -# self.rng = np.random.default_rng(self.seed) -# return self._generate(**self._last_args) - -# def random(self, index_levels: int = 1, column_levels: int = 1) -> pd.DataFrame: -# """ -# Generate a DataFrame with randomly chosen settings. - -# Args: -# index_levels: Number of index levels to use. -# column_levels: Number of column MultiIndex levels. - -# Returns: -# DataFrame -# """ -# rows = self.rng.integers(10, 50) -# col_types = self.rng.choice(['d', 'f', 'i', 's1', 's3', 's7', '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)) -# col_index = ''.join(self.rng.choice(['s', 'i', 'd'], size=column_levels)) -# return self.make(rows=rows, columns=''.join(col_types), index=index, col_index=col_index, missing=missing) - -# def _parse_colspec(self, spec: str) -> list[str]: -# return re.findall(r's\d+|[a-z]', spec) - - -# def _generate(self, rows: int, columns: Union[int, str], index: Union[int, str], -# col_index: Union[int, str], missing: float) -> pd.DataFrame: -# if isinstance(columns, int): -# col_types = ['s3'] * columns -# else: -# col_types = self._parse_colspec(columns) - -# colnames = self._make_column_names(len(col_types)) -# data = { -# name: self._generate_column(dt, rows) for name, dt in zip(colnames, col_types) -# } -# df = pd.DataFrame(data) -# df.index = self._make_index(index, rows, "i") -# df.columns = self._make_index(col_index, len(df.columns), "c") if isinstance(col_index, str) else df.columns -# df = self._insert_missing(df, missing) -# return df - -# def _make_column_names(self, n: int) -> list[str]: -# if self.colname_words: -# pool = self.colname_words -# else: -# pool = [self.faker.word() for _ in range(n * 2)] -# names = [] -# used = set() -# for word in pool: -# if len(names) >= n: -# break -# if word not in used: -# names.append(word) -# used.add(word) -# while len(names) < n: -# names.append(f"col_{len(names)}") -# return names - -# def _generate_column(self, dtype: str, n: int) -> pd.Series: -# if dtype.startswith('s'): -# max_words = int(dtype[1:]) if len(dtype) > 1 else self.default_word_count -# return pd.Series([" ".join(self.faker.words(self.rng.integers(max_words // 2 + 1, max_words + 1))) for _ in range(n)]) -# if dtype == 'f': -# return pd.Series(self.rng.normal(loc=100, scale=25, size=n)) -# if dtype == 'i': -# return pd.Series(self.rng.integers(1e9, 1e12, size=n), dtype='int64') -# if dtype == 'd': -# start_date = self.faker.date_between(start_date='-10y', end_date='today') -# return pd.Series(pd.date_range(start=start_date, periods=n, freq='D')) -# if dtype == 't': -# start_dt = datetime.now() - timedelta(days=365 * 2) -# return pd.Series([start_dt + timedelta(minutes=int(self.rng.integers(0, 2 * 365 * 24 * 60))) for _ in range(n)]) -# if dtype == 'h': -# return pd.Series([ -# hashlib.blake2b(f"val{i}".encode(), digest_size=32).hexdigest() -# for i in range(n) -# ]) -# if dtype == 'p': -# return pd.Series([str(Path(f"/data/{self.faker.word()}/{i}.dat")) for i in range(n)]) -# raise ValueError(f"Unknown dtype: {dtype}") - -# def _make_index(self, desc: Union[int, str], n: int, label_prefix: str) -> pd.Index: -# if isinstance(desc, int): -# return pd.RangeIndex(n, name=f"{label_prefix}0") -# levels = [] -# names = [] -# for j, dt in enumerate(desc): -# s = self._generate_column(dt, n) -# levels.append(s) -# names.append(f"{label_prefix}{j}") -# return pd.MultiIndex.from_arrays(levels, names=names) - -# def _insert_missing(self, df: pd.DataFrame, prop: float) -> pd.DataFrame: -# if prop <= 0: -# return df -# n_rows = df.shape[0] -# for col in df.columns: -# n_missing = max(1, int(np.floor(prop * n_rows))) -# missing_indices = self.rng.choice(n_rows, size=n_missing, replace=False) -# df.iloc[missing_indices, df.columns.get_loc(col)] = np.nan -# return df - - -# Reimport necessary modules after kernel reset -from pathlib import Path -from dataclasses import dataclass, field -from typing import Optional, Union from datetime import datetime, timedelta +from itertools import cycle +from math import prod +from pathlib import Path +from typing import Optional, Union import hashlib -import numpy as np -import pandas as pd -from faker import Faker +import random import re -@dataclass -class TestDataFrameFactory: - colname_words: Optional[list[str]] = None - default_word_count: int = 3 - seed: Optional[int] = None - _last_args: dict = field(default_factory=dict, init=False) +import numpy as np +import pandas as pd - def __post_init__(self): - self.faker = Faker() - self.seed = int(self.seed if self.seed is not None else np.random.SeedSequence().entropy) + +name_word_list = [ + "account", + "address", + "amount", + "balance", + "category", + "client", + "combined ratio", + "comment", + "currency", + "description", + "duration", + "email", + "entry", + "estimate", + "extension", + "failure", + "filename", + "identifier", + "location", + "loss ratio", + "note", + "operation", + "premium", + "processing", + "project", + "reference", + "remark", + "status", + "supplier", + "timestamp", + "transaction", + "type", + "user", + 'expense ratio', + 'loss date' +] + + +class TestDataFrameFactory: + """ + Create super-dooper test dataframes. + """ + + def __init__(self, seed: Optional[int] = None): + """ + Factory for generating small synthetic pandas DataFrames for testing. + + Attributes: + seed: Optional random seed. If None, one is generated. + """ + self._last_args = {} + self.seed = int( + seed if seed is not None else np.random.SeedSequence().entropy) + + # rng self.rng = np.random.default_rng(self.seed) + # word list for names of index levels + nwl = name_word_list[:] + random.shuffle(nwl) + 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') + 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() + tex_list = [i for i in tex_list if len(i) < 50] + random.shuffle(tex_list) + self._tex_gen = cycle(tex_list) + + # lengths of index (word count) sampled from: + self.index_value_lengths = [1]*10 + [2] * 4 + [3] + def make(self, rows: int, columns: Union[int, str], index: Union[int, str] = 0, col_index: Union[int, str] = 0, missing: float = 0.0) -> pd.DataFrame: - self._last_args = dict(rows=rows, columns=columns, index=index, col_index=col_index, missing=missing) - return self._generate(**self._last_args).sort_index() + """ + Generate a test DataFrame with the given specification. + + Data types + + d date + f float + h hash + i integer + l log float (greater range than float) + m year - month + p path (filename) + sx string length x + t time + x tex text - an equation + y year + + + 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'). + col_index: Column index levels (same format as `index`). + missing: Proportion of missing data in each column. + + Returns: + DataFrame + """ + self._last_args = dict(rows=rows, columns=columns, + index=index, col_index=col_index, missing=missing) + return self._generate(**self._last_args) def another(self, new_seed: bool = True) -> pd.DataFrame: + """ + Generate another DataFrame with the last parameters. + + Args: + new_seed: If True, re-randomize the generator seed. + + Returns: + DataFrame + """ if new_seed: self.seed = int(np.random.SeedSequence().entropy) self.rng = np.random.default_rng(self.seed) - return self._generate(**self._last_args).sort_index() + return self._generate(**self._last_args) - def random(self, index_levels: int = 1, column_levels: int = 1) -> pd.DataFrame: - rows = self.rng.integers(10, 50) - col_types = self.rng.choice(['d', 'f', 'i', 's3', 'h', 't', 'p'], size=self.rng.integers(3, 7)) + def random(self, index_levels: int = 0, column_levels: int = 0) -> pd.DataFrame: + """ + Generate a DataFrame with randomly chosen settings. + + Args: + index_levels: Number of index levels to use. + column_levels: Number of column MultiIndex levels. + + Returns: + DataFrame + """ + if index_levels == 0: + index_levels = random.choice([1, 1, 1, 1, 1, 2, 2, 3]) + if column_levels == 0: + column_levels = random.choice([1, 1, 1, 1, 1, 2, 2, 3]) + rows = self.rng.integers(5 * index_levels, 10 * index_levels) + col_types = self.rng.choice( + ['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)) - col_index = ''.join(self.rng.choice(['s', 'i', 'd'], size=column_levels)) + index = ''.join(self.rng.choice( + ['t', 'd', '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) def _generate(self, rows: int, columns: Union[int, str], index: Union[int, str], col_index: Union[int, str], missing: float) -> pd.DataFrame: - col_types = ['f'] * columns if isinstance(columns, int) else self._parse_colspec(columns) - colnames = self._make_column_names(len(col_types)) - data = { - name: self._generate_column(dt, rows) for name, dt in zip(colnames, col_types) - } - df = pd.DataFrame(data) - df.index = self._make_index(index, rows, "i") - df.columns = self._make_index(col_index, len(df.columns), "c") if isinstance(col_index, str) else df.columns + # if columns is an int then make up types + if isinstance(columns, int): + col_types = self.rng.choice( + ['d', 't', 'f', 'l', 'i', 's1', 's3', 's9', 'h', 'p', 'x'], size=columns) + 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 + else: + col_index_types = self._parse_colspec(col_index) + if isinstance(index, int): + index = ['s'] * index + else: + index = self._parse_colspec(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 + df.index = self._make_index(index, rows) df = self._insert_missing(df, missing) return df def _parse_colspec(self, spec: str) -> list[str]: return re.findall(r's\d+|[a-z]', spec) - def _make_column_names(self, n: int) -> list[str]: - if self.colname_words: - pool = self.colname_words - else: - pool = [self.faker.word() for _ in range(n * 2)] - names, used = [], set() - for word in pool: - if len(names) >= n: - break - if word not in used: - names.append(word) - used.add(word) - while len(names) < n: - names.append(f"col_{len(names)}") - return names - def _generate_column(self, dtype: str, n: int) -> pd.Series: if dtype.startswith('s'): - max_words = int(dtype[1:]) if len(dtype) > 1 else self.default_word_count - return pd.Series([" ".join(self.faker.words(self.rng.integers(1, max_words + 1))) for _ in range(n)]) + max_words = int(dtype[1:]) if len(dtype) > 1 else 1 + 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=100, scale=25, size=n)) + return pd.Series(self.rng.normal(loc=100000, scale=250000, 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 == 'i': - return pd.Series(self.rng.integers(1e9, 1e12, size=n), dtype='int64') + return pd.Series(self.rng.integers(-1e4, 1e6, size=n), dtype='int64') if dtype == 'd': - start_date = self.faker.date_between(start_date='-10y', end_date='today') + 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 == 't': start_dt = datetime.now() - timedelta(days=365 * 2) return pd.Series([ - start_dt + timedelta(minutes=int(self.rng.integers(0, 2 * 365 * 24 * 60))) + start_dt + + timedelta(minutes=int(self.rng.integers(0, 2 * 365 * 24 * 60))) for _ in range(n) ]) if dtype == 'h': @@ -268,45 +232,122 @@ class TestDataFrameFactory: for i in range(n) ]) if dtype == 'p': - return pd.Series([str(Path(f"/data/{self.faker.word()}/{i}.dat")) for i in range(n)]) + return pd.Series([str(Path(f"/data/{self.word()}/{i}.dat")) for i in range(n)]) + if dtype == 'x': + # tex + return pd.Series([self.tex() for i in range(n)]) raise ValueError(f"Unknown dtype: {dtype}") - def _make_index(self, desc: Union[int, str], n: int, label_prefix: str) -> pd.Index: + def _make_index(self, desc: Union[int, str, list[str]], n: int) -> pd.Index: if isinstance(desc, int): - return pd.RangeIndex(n, name=f"{label_prefix}0") + return pd.RangeIndex(n, name=self.index_name()) + if isinstance(desc, str): + desc = self._parse_colspec(desc) if len(desc) == 1: - s = self._generate_column(desc[0], n) - return pd.Index(s, name=f"{label_prefix}0") - return self._make_hierarchical_index(desc, n, label_prefix) + if desc[0] == 'i': + return pd.RangeIndex(n, name=self.index_name()) + elif desc[0] in ('d', 't', 'x'): + vals = self._generate_column(desc[0], n) + return pd.Index(vals, name=self.index_name()) + elif not all(i[0] == 's' for i in desc): + raise ValueError( + f'Inadmissible index spec: only string, int, and date types allowed, not {desc}.') + level_value_lengths = [1 if len(i) == 1 else int(i[1:]) for i in desc] + return self.make_index(rows=n, levels=len(desc), level_value_lengths=level_value_lengths, + p0=1, padding=2) - def _make_hierarchical_index(self, desc: str, n: int, label_prefix: str) -> pd.MultiIndex: - """ - Generate a nested hierarchical index of length `n` with `len(desc)` levels. - Levels are naturally nested, i.e., upper levels have fewer unique values. - """ - levels = [] + def index_name(self): + """Return a one-word index name.""" + return next(self._index_namer) - # generate lower-level (more detailed) values with full cardinality - detailed = self._generate_column(desc[-1], n) - levels.insert(0, detailed) + def word(self): + """Return a random word (cycles eventually).""" + return next(self._word_gen) - # generate higher levels with fewer unique values - for i, dt in enumerate(desc[:-1]): - u = 2 if i == 0 else 3 - unique_vals = self._generate_column(dt, u).unique() - repeated = self.rng.choice(unique_vals, size=n, replace=True) - levels.insert(0, repeated) - - names = [f"{label_prefix}{j}" for j in range(len(desc))] - return pd.MultiIndex.from_arrays(levels, names=names) + def tex(self): + """Return a blob of TeX.""" + return next(self._tex_gen) + @staticmethod + def random_date_within_last_n_years(n: int) -> pd.Timestamp: + today = datetime.today() + days = random.randint(0, n * 365) + return pd.Timestamp(today - timedelta(days=days)) def _insert_missing(self, df: pd.DataFrame, prop: float) -> pd.DataFrame: + """Insert missing values into dataframe.""" if prop <= 0: return df n_rows = df.shape[0] for col in df.columns: n_missing = max(1, int(np.floor(prop * n_rows))) - missing_indices = self.rng.choice(n_rows, size=n_missing, replace=False) + missing_indices = self.rng.choice( + n_rows, size=n_missing, replace=False) df.iloc[missing_indices, df.columns.get_loc(col)] = np.nan return df + + @staticmethod + def _is_prime(p: int) -> bool: + if p < 2: + return False + if p == 2: + return True + if p % 2 == 0: + return False + for i in range(3, int(p**0.5) + 1, 2): + if p % i == 0: + return False + return True + + @staticmethod + def _next_prime(p: int) -> int: + if p < 2: + return 2 + p += 1 if p % 2 == 0 else 2 # ensure odd start > p + while True: + if TestDataFrameFactory._is_prime(p): + return p + p += 2 + + @staticmethod + def primes_for_product(n: int, v: int, p0: int) -> list[int]: + """Return a list of distinct primes all >= p0 whose product is >= n.""" + primes = [] + p = TestDataFrameFactory._next_prime(max(p0 - 1, 1)) + while len(primes) < v: + primes.append(p) + p = TestDataFrameFactory._next_prime(p) + + while prod(primes := sorted(primes)) < n: + # increase one level until product is high enough + p = TestDataFrameFactory._next_prime(primes[-1]) + primes[-1] = p + # shuffle order + random.shuffle(primes) + return primes + + def make_index(self, rows: int, levels: int, + level_value_lengths: Union[list[int], None] = None, + p0: int = 1, + padding: int = 2): + """ + Make an Index with unique values, rows x len(level_value_lengths) cols. + + level_velue_lengths shows how many words long each value should be. + padding = over-sample by padding and select sample. + """ + if level_value_lengths is None: + level_value_lengths = random.sample( + self.index_value_lengths, levels) + else: + assert levels == len( + level_value_lengths), 'levels must equal len(level_value_lengths)' + level_choices = self.primes_for_product(rows * padding, levels, p0=p0) + r = [cycle([' '.join([self.word() for _ in range(w)]) for _ in range(k)]) + 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() + assert idx.is_unique + return idx diff --git a/greater_tables/tex_list.csv b/greater_tables/tex_list.csv new file mode 100644 index 0000000..2c5365c --- /dev/null +++ b/greater_tables/tex_list.csv @@ -0,0 +1,5804 @@ +,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/words-12.md b/greater_tables/words-12.md new file mode 100644 index 0000000..46006ad --- /dev/null +++ b/greater_tables/words-12.md @@ -0,0 +1,25912 @@ +aaron +aback +abacus +abandon +abandoned +abandoning +abandonment +abandons +abated +abba +abbas +abbot +abbott +abbreviate +abbreviated 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