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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 Financial models with long-tailed distributions and volatility clustering ： ウィキペディア英語版 Financial models with long-tailed distributions and volatility clustering Financial models with long-tailed distributions and volatility clustering have been introduced to overcome problems with the realism of classical financial models. These classical models of financial time series typically assume homoskedasticity and normality cannot explain stylized phenomena such as skewness, heavy tails, and volatility clustering of the empirical asset returns in finance. In 1963, Benoit Mandelbrot first used the stable (or $\backslash alpha$-stable) distribution to model the empirical distributions which have the skewness and heavy-tail property. Since $\backslash alpha$-stable distributions have infinite $p$-th moments for all $p>\backslash alpha$, the tempered stable processes have been proposed for overcoming this limitation of the stable distribution. On the other hand, GARCH models have been developed to explain the volatility clustering. In the GARCH model, the innovation (or residual) distributions are assumed to be a standard normal distribution, despite the fact that this assumption is often rejected empirically. For this reason, GARCH models with non-normal innovation distribution have been developed. Many financial models with stable and tempered stable distributions together with volatility clustering have been developed and applied to risk management, option pricing, and portfolio selection. == Infinitely divisible distributions == A random variable $Y$ is called ''infinitely divisible'' if, for each $n=1,2,\backslash dots$, there are independent and identically-distributed random variables : $Y\_,\; Y\_,\; \backslash dots,\; Y\_\; \backslash ,$ such that : $Y\backslash stackrel\backslash sum\_^n\; Y\_,\; \backslash ,$ where $\backslash stackrel$ denotes equality in distribution. A Borel measure $\backslash nu$ on $\backslash mathbb$ is called a ''Lévy measure'' if $\backslash nu()=0$ and : If $Y$ is infinitely divisible, then the characteristic function $\backslash phi\_Y(u)=E()$ is given by : $$ \phi_Y(u) =\exp \left( i\gamma u- \frac \sigma^2 u^2 + \int_^\infty (e^-1-iux1_ ) \, \nu(dx) \right), \sigma\ge0,~~\gamma\in\mathbb where $\backslash sigma\backslash ge0$, $\backslash gamma\backslash in\backslash mathbb$ and $\backslash nu$ is a Lévy measure. Here the triple $(\backslash sigma^2,\; \backslash nu,\; \backslash gamma)$ is called a ''Lévy triplet of'' $Y$. This triplet is unique. Conversely, for any choice $(\backslash sigma^2,\; \backslash nu,\; \backslash gamma)$ satisfying the conditions above, there exists an infinitely divisible random variable $Y$ whose characteristic function is given as $\backslash phi\_Y$. 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「Financial models with long-tailed distributions and volatility clustering」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.