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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 Russo–Vallois integral ： ウィキペディア英語版 Russo–Vallois integral In mathematical analysis, the Russo–Vallois integral is an extension to stochastic processes of the classical Riemann–Stieltjes integral :$\backslash int\; f\; \backslash ,\; dg=\backslash int\; fg\text{'}\; \backslash ,\; ds$ for suitable functions $f$ and $g$. The idea is to replace the derivative $g\text{'}$ by the difference quotient :$g(s+\backslash varepsilon)-g(s)\backslash over\backslash varepsilon$ and to pull the limit out of the integral. In addition one changes the type of convergence. ==Definitions== Definition: A sequence $H\_n$ of stochastic processes converges uniformly on compact sets in probability to a process $H,$ :$H=\backslash text\backslash lim\_H\_n,$ if, for every $\backslash varepsilon>0$ and $T>0,$ :$\backslash lim\_\backslash mathbb(\backslash sup\_|H\_n(t)-H(t)|>\backslash varepsilon)=0.$ One sets: :$I^-(\backslash varepsilon,t,f,dg)=\backslash int\_0^tf(s)(g(s+\backslash varepsilon)-g(s))\backslash ,ds$ :$I^+(\backslash varepsilon,t,f,dg)=\backslash int\_0^t\; f(s)(g(s)-g(s-\backslash varepsilon))\; \backslash ,\; ds$ and :$()\_\backslash varepsilon\; (t)=\backslash int\_0^t(f(s+\backslash varepsilon)-f(s))(g(s+\backslash varepsilon)-g(s))\backslash ,ds.$ Definition: The forward integral is defined as the ucp-limit of :$I^-$: $\backslash int\_0^t\; fd^-g=\backslash text\backslash lim\_I^-(\backslash varepsilon,t,f,dg).$ Definition: The backward integral is defined as the ucp-limit of :$I^+$: $\backslash int\_0^t\; f\; \backslash ,\; d^+g\; =\; \backslash text\backslash lim\_I^+(\backslash varepsilon,t,f,dg).$ Definition: The generalized bracket is defined as the ucp-limit of :$()\_\backslash varepsilon$: $()\_\backslash varepsilon=\backslash text\backslash lim\_()\_\backslash varepsilon\; (t).$ For continuous semimartingales $X,Y$ and a cadlag function H, the Russo–Vallois integral coincidences with the usual Ito integral: :$\backslash int\_0^t\; H\_s\; \backslash ,\; dX\_s=\backslash int\_0^t\; H\; \backslash ,\; d^-X.$ In this case the generalised bracket is equal to the classical covariation. In the special case, this means that the process :$():=()\; \backslash ,$ is equal to the quadratic variation process. Also for the Russo-Vallois Integral an Ito formula holds: If $X$ is a continuous semimartingale and :$f\backslash in\; C\_2(\backslash mathbb),$ then :$f(X\_t)=f(X\_0)+\backslash int\_0^t\; f\text{'}(X\_s)\; \backslash ,\; dX\_s\; +\; \backslash int\_0^t\; f\text{'}\text{'}(X\_s)\; \backslash ,\; d()\_s.$ By a duality result of Triebel one can provide optimal classes of Besov spaces, where the Russo–Vallois integral can be defined. The norm in the Besov space :$B\_^\backslash lambda(\backslash mathbb^N)$ is given by :$||f||\_^\backslash lambda=||f||\_\; +\; \backslash left(\backslash int\_0^\backslash infty\; )^q\; \backslash ,\; dh\backslash right)^$ with the well known modification for $q=\backslash infty$. Then the following theorem holds: Theorem: Suppose :$f\backslash in\; B\_^\backslash lambda,$ :$g\backslash in\; B\_^,$ :$1/p+1/p\text{'}=1\backslash text1/q+1/q\text{'}=1.$ Then the Russo–Vallois integral :$\backslash int\; f\; \backslash ,\; dg$ exists and for some constant $c$ one has :$\backslash left|\; \backslash int\; f\; \backslash ,\; dg\; \backslash right|\; \backslash leq\; c\; ||f||\_^\backslash alpha\; ||g||\_^.$ Notice that in this case the Russo–Vallois integral coincides with the Riemann–Stieltjes integral and with the Young integral for functions with finite p-variation. 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「Russo–Vallois integral」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.