词条 | Clark–Ocone theorem |
释义 |
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 theoremLet C0([0, T]; R) (or simply C0 for short) be classical Wiener space with Wiener measure γ. Let F : C0 → R be a BC1 function, i.e. F is bounded and Fréchet differentiable with bounded derivative DF : C0 → Lin(C0; R). Then In the above
is the expected value of F over the whole of Wiener space C0;
is an Itō integral;
More generally, the conclusion holds for any F in L2(C0; R) that is differentiable in the sense of Malliavin. Integration by parts on Wiener spaceThe Clark–Ocone theorem gives rise to an integration by parts formula on classical Wiener space, and to write Itō integrals as divergences: Let B be a standard Brownian motion, and let L02,1 be the Cameron–Martin space for C0 (see abstract Wiener space. Let V : C0 → L02,1 be a vector field such that is in L2(B) (i.e. is Itō integrable, and hence is an adapted process). Let F : C0 → R be BC1 as above. Then i.e. or, writing the integrals over C0 as expectations: where the "divergence" div(V) : C0 → R is defined by The interpretation of stochastic integrals as divergences leads to concepts such as the Skorokhod integral and the tools of the Malliavin calculus. See also
References
| last = Nualart | first = David | title = The Malliavin calculus and related topics | series = Probability and its Applications (New York) | edition = Second |publisher = Springer-Verlag | location = Berlin | year = 2006 | isbn = 978-3-540-28328-7 }} External links
|url=http://www.statslab.cam.ac.uk/~peter/malliavin/Malliavin2005/mall.pdf |title=An Introduction to Malliavin Calculus |accessdate=2007-07-23 |last=Friz |first=Peter K. |date=2005-04-10 |format=PDF |archiveurl=https://web.archive.org/web/20070417205303/http://www.statslab.cam.ac.uk/~peter/malliavin/Malliavin2005/mall.pdf |archivedate=2007-04-17 |deadurl=yes |df= }}{{DEFAULTSORT:Clark-Ocone theorem}} 2 : Theorems regarding stochastic processes|Theorems in measure theory |
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