Clark–Ocone theorem


In mathematics, the Clark–Ocone theorem 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 Itô integral with respect to that path. It is named after the contributions of mathematicians J.M.C. Clark, Daniel Ocone and U.G. Haussmann.

Statement of the theorem

Let C0 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. Then
In the above
More generally, the conclusion holds for any F in L2 that is differentiable in the sense of Malliavin.

Integration by parts on Wiener space

The Clark–Ocone theorem gives rise to an integration by parts formula on classical Wiener space, and to write Itô integrals as divergences:
Let B be a standard Brownian motion, and let L02,1 be the Cameron–Martin space for C0 . Let F : C0 → R be BC1 as above. Then
i.e.
or, writing the integrals over C0 as expectations:
where the "divergence" div : 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.