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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 Change of variables (PDE) ： ウィキペディア英語版 Change of variables (PDE) Often a partial differential equation can be reduced to a simpler form with a known solution by a suitable change of variables. The article discusses change of variable for PDEs below in two ways: #by example; #by giving the theory of the method. ==Explanation by example== For example the following simplified form of the Black–Scholes PDE :$\backslash frac\; +\; \backslash frac\; S^2\backslash frac\; +\; S\backslash frac\; -\; V\; =\; 0.$ is reducible to the heat equation :$\backslash frac\; =\; \backslash frac$ by the change of variables:〔Ömür Ugur, ''An Introduction to Computational Finance'', Series in Quantitative Finance, v. 1, Imperial College Press, 298 pp., 2009〕 :$V(S,t)\; =\; v(x(S),\backslash tau(t))$ :$x(S)\; =\; \backslash ln(S)$ :$\backslash tau(t)\; =\; \backslash frac\; (T\; -\; t)$ :$v(x,\backslash tau)=\backslash exp(-(1/2)x-(9/4)\backslash tau)\; u(x,\backslash tau)$ in these steps: * Replace $V(S,t)$ by $v(x(S),\backslash tau(t))$ and apply the chain rule to get ::$\backslash frac\backslash left(-2v(x(S),\backslash tau)+2\; \backslash frac\; \backslash frac\; +S\backslash left(\backslash left(2\; \backslash frac\; +\; S\backslash frac\backslash right)$ \frac + S \left(\frac\right)^2 \frac\right)\right)=0. * Replace $x(S)$ and $\backslash tau(t)$ by $\backslash ln(S)$ and $\backslash frac(T-t)$ to get ::$\backslash frac\backslash left($ -2v(\ln(S),\frac(T-t)) -\frac(T-t))} +\frac(T-t))} +\frac(T-t))}\right)=0. * Replace $\backslash ln(S)$ and $\backslash frac(T-t)$ by $x(S)$ and $\backslash tau(t)$ and divide both sides by $\backslash frac$ to get ::$-2\; v-\backslash frac+\backslash frac+\; \backslash frac=0.$ * Replace $v(x,\backslash tau)$ by $\backslash exp(-(1/2)x-(9/4)\backslash tau)\; u(x,\backslash tau)$ and divide through by $-\backslash exp(-(1/2)x-(9/4)\backslash tau)\; u(x,\backslash tau)$ to yield the heat equation. Advice on the application of change of variable to PDEs is given by mathematician J. Michael Steele:〔J. Michael Steele, ''Stochastic Calculus and Financial Applications'', Springer, New York, 2001〕 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「Change of variables (PDE)」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.