Clark–Ocone theorem

In mathematics, the Clark–Ocone theorem (also known as the Clark–Ocone–Haussmann theorem or formula) is a theorem of . 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).

Let 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

More generally, the conclusion holds for any F in L²(C₀; R) that is differentiable in the sense of Malliavin.

The 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 L₀^2,1 be the Cameron–Martin space for C₀ (see abstract Wiener space. Let V : C₀ → L₀^2,1 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

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