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 aboveF is the value of the function F on some specific path of interest, σ;

the first integral,
the second integral,
Σ_∗ is the natural filtration of Brownian motion B : × Ω → R: Σ_t is the smallest σ-algebra containing all B_s⁻¹ 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 L² 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 Itô integrals as divergences:
Let B be a standard Brownian motion, and let L₀^2,1 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.