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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 Runge–Kutta method (SDE) ： ウィキペディア英語版 Runge–Kutta method (SDE) In mathematics of stochastic systems, the Runge–Kutta method is a technique for the approximate numerical solution of a stochastic differential equation. It is a generalization of the Runge–Kutta method for ordinary differential equations to stochastic differential equations (SDEs). Importantly, the method does not involve knowing derivatives the coefficient functions in the SDEs. ==Most basic scheme== Consider the Itō diffusion $X$ satisfying the following Itō stochastic differential equation :$=\; a(X\_)\; \backslash ,\; t\; +\; b(X\_)\; \backslash ,\; W\_,$ with initial condition $X\_0=x\_0$, where $W\_t$ stands for the Wiener process, and suppose that we wish to solve this SDE on some interval of time $()$. Then the basic Runge–Kutta approximation to the true solution $X$ is the Markov chain $Y$ defined as follows:〔P. E. Kloeden and E. Platen. ''Numerical solution of stochastic differential equations'', volume 23 of Applications of Mathematics. Springer--Verlag, 1992.〕 * partition the interval $()$ into $N$ subintervals of width $\backslash delta=T/N>0$: : * set $Y\_0:=x\_0$; * recursively compute $Y\_n$ for $1\backslash leq\; n\backslash leq\; N$ by :$Y\_\; :=\; Y\_\; +\; a(Y\_)\; \backslash delta\; +\; b(Y\_)\; \backslash Delta\; W\_\; +\; \backslash frac\; \backslash left(\; b(\backslash hat\_)\; -\; b(Y\_)\; \backslash right)\; \backslash left(\; (\backslash Delta\; W\_)^\; -\; \backslash delta\; \backslash right)\; \backslash delta^,$ where $\backslash Delta\; W\_\; =\; W\_\}$ and $\backslash hat\_\; =\; Y\_\; +\; a(Y\_n)\; \backslash delta\; +\; b(Y\_)\; \backslash delta^.$ The random variables $\backslash Delta\; W\_$ are independent and identically distributed normal random variables with expected value zero and variance $\backslash delta$. This scheme has strong order 1, meaning that the approximation error of the actual solution at a fixed time scales with the time step $\backslash delta$. It has also weak order 1, meaning that the error on the statistics of the solution scales with the time step $\backslash delta$. See the references for complete and exact statements. The functions $a$ and $b$ can be time-varying without any complication. The method can be generalized to the case of several coupled equations; the principle is the same but the equations become longer. 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「Runge–Kutta method (SDE)」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.