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In mathematics, the Clark–Ocone theorem (also known as the Clark–Ocone–Haussmann theorem or formula) is a theorem of stochastic analysis. It expresses the value of some function ''F'' defined on the classical Wiener space of continuous paths starting at the origin as the sum of its mean value and an Itō integral with respect to that path. It is named after the contributions of mathematicians J.M.C. Clark (1970), Daniel Ocone (1984) and U.G. Haussmann (1978). ==Statement of the theorem== Let ''C''0((); R) (or simply ''C''0 for short) be classical Wiener space with Wiener measure ''γ''. Let ''F'' : ''C''0 → R be a BC1 function, i.e. ''F'' is bounded and Fréchet differentiable with bounded derivative D''F'' : ''C''0 → Lin(''C''0; R). Then : In the above * ''F''(''σ'') is the value of the function ''F'' on some specific path of interest, ''σ''; * the first integral, :: :is the expected value of ''F'' over the whole of Wiener space ''C''0; * the second integral, :: :is an Itō integral; * Σ∗ is the natural filtration of Brownian motion ''B'' : () × Ω → R: Σ''t'' is the smallest ''σ''-algebra containing all ''B''''s''−1(''A'') for times 0 ≤ ''s'' ≤ ''t'' and Borel sets ''A'' ⊆ R; * E() denotes conditional expectation with respect to the sigma algebra Σ''t''; * ∂/∂''t'' denotes differentiation with respect to time ''t''; ∇''H'' denotes the ''H''-gradient; hence, ∂/∂''t''∇''H'' is the Malliavin derivative. More generally, the conclusion holds for any ''F'' in ''L''2(''C''0; R) that is differentiable in the sense of Malliavin. 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「Clark–Ocone theorem」の詳細全文を読む スポンサード リンク
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