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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Entropic value at risk ： ウィキペディア英語版 Entropic value at risk In financial mathematics and stochastic optimization, the concept of risk measure is used to quantify the risk involved in a random outcome or risk position. Many risk measures have hitherto been proposed, each having certain characteristics. The entropic value-at-risk (EVaR) is a coherent risk measure introduced by Ahmadi-Javid, which is an upper bound for the value at risk (VaR) and the conditional value-at-risk (CVaR), obtained from the Chernoff inequality. The EVaR can also be represented by using the concept of relative entropy. Because of its connection with the VaR and the relative entropy, this risk measure is called "entropic value-at-risk". The EVaR was developed to tackle some computational inefficiencies of the CVaR. Getting inspiration from the dual representation of the EVaR, Ahmadi-Javid〔〔 developed a wide class of coherent risk measures, called g-entropic risk measures. Both the CVaR and the EVaR are members of this class. == Definition == Let $(\backslash Omega,\backslash mathcal,P)$ be a probability space with $\backslash Omega$ a set of all simple events, $\backslash mathcal$ a $\backslash sigma$-algebra of subsets of $\backslash Omega$ and $P$ a probability measure on $\backslash mathcal$. Let $X$ be a random variable and $\backslash mathbf\_$ be the set of all Borel measurable functions $X:\backslash Omega\backslash rightarrow\; \backslash R$ whose moment-generating function $M\_X(z)$ exists for all $z\backslash geq\; 0$. The entropic value-at-risk (EVaR) of $X\backslash in\; \backslash mathbf\_$ with confidence level $1-\backslash alpha$ is defined as follows: In finance, the random variable $X\; \backslash in\; \backslash mathbf\_$, in the above equation, is used to model the ''losses'' of a portfolio. Consider the Chernoff inequality Solving the equation $e^M\_X(z)=\backslash alpha$ for $a$, results in $a\_X(\backslash alpha,z):=z^\backslash ln(M\_X(z)/\backslash alpha)$. By considering the equation (), we see that $\backslash text\_(X):=\backslash inf\_\backslash$, which shows the relationship between the EVaR and the Chernoff inequality. It is worth noting that $a\_X(1,z)$ is the ''entropic risk measure'' or ''exponential premium'', which is a concept used in finance and insurance, respectively. Let $\backslash mathbf\_$ be the set of all Borel measurable functions $X:\backslash Omega\backslash rightarrow\; \backslash R$ whose moment-generating function $M\_X(z)$ exists for all $z$. The dual representation (or robust representation) of the EVaR is as follows: where $X\backslash in\; \backslash mathbf\_$, and $\backslash Im$ is a set of probability measures on $(\backslash Omega,\backslash mathcal)$ with $\backslash Im=\backslash$. Note that $D\_(Q||P):=\backslash int\backslash frac(\backslash ln\backslash frac)dP$ is the relative entropy of $Q$ with respect to $P$, also called the Kullback–Leibler divergence. The dual representation of the EVaR discloses the reason behind its naming. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Entropic value at risk」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.