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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 Variance gamma process ： ウィキペディア英語版 Variance gamma process In the theory of stochastic processes, a part of the mathematical theory of probability, the variance gamma process (VG), also known as Laplace motion, is a Lévy process determined by a random time change. The process has finite moments distinguishing it from many Lévy processes. There is no diffusion component in the VG process and it is thus a pure jump process. The increments are independent and follow a Variance-gamma distribution, which is a generalization of the Laplace distribution. There are several representations of the VG process that relate it to other processes. It can for example be written as a Brownian motion $W(t)$ with drift $\backslash theta\; t$ subjected to a random time change which follows a gamma process $\backslash Gamma(t;\; 1,\; \backslash nu)$ (equivalently one finds in literature the notation $\backslash Gamma(t;\backslash gamma=1/\backslash nu,\backslash lambda=1/\backslash nu)$): :$$ X^(t; \sigma, \nu, \theta) \;:=\; \theta \,\Gamma(t; 1, \nu) + \sigma\,W(\Gamma(t; 1, \nu)) \quad. An alternative way of stating this is that the variance gamma process is a Brownian motion subordinated to a Gamma subordinator. Since the VG process is of finite variation it can be written as the difference of two independent gamma processes: :$$ X^(t; \sigma, \nu, \theta) \;:=\; \Gamma(t; \mu_p, \mu_p^2\,\nu) - \Gamma(t; \mu_q, \mu_q^2\,\nu) where :$$ \mu_p := \frac\sqrt} + \frac \quad\quad\text\quad\quad \mu_q := \frac\sqrt} - \frac \quad. Alternatively it can be approximated by a compound Poisson process that leads to a representation with explicitly given (independent) jumps and their locations. This last characterization gives an understanding of the structure of the sample path with location and sizes of jumps. On the early history of the variance-gamma process see Seneta (2000).〔 〕 == Moments == The mean of a variance gamma process is independent of $\backslash sigma$ and $\backslash nu$ and is given by :$E()\; =\; \backslash theta\; t$ The variance is given as :$Var()\; =\; (\backslash theta^2\; \backslash nu\; +\; \backslash sigma^2)t$ The 3rd central moment is :$E\; =\; (2\; \backslash theta^3\; \backslash nu^2\; +\; 3\; \backslash sigma^2\; \backslash theta\; \backslash nu)t$ The 4th central moment is :$E\; =\; (3\; \backslash sigma^4\; \backslash nu\; +\; 12\; \backslash sigma^2\; \backslash theta^2\; \backslash nu^2\; +\; 6\; \backslash theta^4\; \backslash nu^3)t\; +\; (3\; \backslash sigma^4\; +\; 6\; \backslash sigma^2\; \backslash theta^2\; \backslash nu\; +\; 3\; \backslash theta^4\; \backslash nu^2)t^2$ 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「Variance gamma process」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.