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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 Tanaka equation ： ウィキペディア英語版 Tanaka equation In mathematics, Tanaka's equation is an example of a stochastic differential equation which admits a weak solution but has no strong solution. It is named after the Japanese mathematician Hiroshi Tanaka. Tanaka's equation is the one-dimensional stochastic differential equation :$\backslash mathrm\; X\_t\; =\; \backslash sgn\; (X\_t)\; \backslash ,\; \backslash mathrm\; B\_t,$ driven by canonical Brownian motion ''B'', with initial condition ''X''0 = 0, where sgn denotes the sign function : (Note the unconventional value for sgn(0).) The signum function does not satisfy the Lipschitz continuity condition required for the usual theorems guaranteeing existence and uniqueness of strong solutions. The Tanaka equation has no strong solution, i.e. one for which the version ''B'' of Brownian motion is given in advance and the solution ''X'' is adapted to the filtration generated by ''B'' and the initial conditions. However, the Tanaka equation does have a weak solution, one for which the process ''X'' and version of Brownian motion are both specified as part of the solution, rather than the Brownian motion being given ''a priori''. In this case, simply choose ''X'' to be any Brownian motion $\backslash hat$ and define $\backslash tilde$ by :$\backslash tilde\_t\; =\; \backslash int\_0^t\; \backslash sgn\; \backslash big(\; \backslash hat\_s\; \backslash big)\; \backslash ,\; \backslash mathrm\; \backslash hat\_s\; =\; \backslash int\_0^t\; \backslash sgn\; \backslash big(\; X\_s\; \backslash big)\; \backslash ,\; \backslash mathrm\; X\_s,$ i.e. :$\backslash mathrm\; \backslash tilde\_t\; =\; \backslash sgn\; (X\_t)\; \backslash ,\; \backslash mathrm\; X\_t.$ Hence, :$\backslash mathrm\; X\_t\; =\; \backslash sgn\; (X\_t)\; \backslash ,\; \backslash mathrm\; \backslash tilde\_,$ and so ''X'' is a weak solution of the Tanaka equation. Furthermore, this solution is weakly unique, i.e. any other weak solution must have the same law. ==References== * (Example 5.3.2) 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「Tanaka equation」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.