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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 Doob's martingale inequality ： ウィキペディア英語版 Doob's martingale inequality In mathematics, Doob's martingale inequality is a result in the study of stochastic processes. It gives a bound on the probability that a stochastic process exceeds any given value over a given interval of time. As the name suggests, the result is usually given in the case that the process is a non-negative martingale, but the result is also valid for non-negative submartingales. The inequality is due to the American mathematician Joseph L. Doob. ==Statement of the inequality== Let ''X'' be a submartingale taking non-negative real values, either in discrete or continuous time. That is, for all times ''s'' and ''t'' with ''s'' < ''t'', :$X\_\backslash leq\backslash mathbf\; \backslash left(X\_\; \backslash big|\; \backslash mathcal\_\; \backslash right).$ (For a continuous-time submartingale, assume further that the process is càdlàg.) Then, for any constant ''C'' > 0 and ''p'' ≥ 1, :$\backslash mathbf\; \backslash left(\backslash sup\_\; X\_\; \backslash geq\; C\; \backslash right)\; \backslash leq\; \backslash frac^\; \backslash right\; )\}[X\_T]\; =\; \backslash int\_\; X\_\; (\backslash omega)\; \backslash ,\; \backslash mathrm\; \backslash mathbf\; (\backslash omega)$ in the sense of Lebesgue integration. $\backslash mathcal\_$ denotes the σ-algebra generated by all the random variables ''Xi'' with ''i'' ≤ ''s''; the collection of such σ-algebras forms a filtration of the probability space. 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「Doob's martingale inequality」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.