Clark–Ocone theorem

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

${\displaystyle F(\sigma )=\int _{C_{0}}F(p)\,\mathrm {d} \gamma (p)+\int _{0}^{T}\mathbf {E} \left[\left.{\frac {\partial }{\partial t}}\nabla _{H}F(-)\right|\Sigma _{t}\right](\sigma )\,\mathrm {d} \sigma _{t}.}$

In the above

• F(σ) is the value of the function F on some specific path of interest, σ;
• the first integral,

${\displaystyle \int _{C_{0}}F(p)\,\mathrm {d} \gamma (p)=\mathbf {E} [F]}$

is the expected value of F over the whole of Wiener space C₀;

• the second integral,

${\displaystyle \int _{0}^{T}\cdots \,\mathrm {d} \sigma (t)}$

is an Itō integral;

• Σ_∗ is the natural filtration of Brownian motion B : [0, T] × 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[·|Σ_t] 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(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 Itô 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

${\displaystyle {\dot {V}}={\frac {\partial V}{\partial t}}:[0,T]\times C_{0}\to \mathbb {R} }$

is in L²(B) (i.e. is Itō integrable, and hence is an adapted process). Let F : C₀ → R be BC¹ as above. Then

${\displaystyle \int _{C_{0}}\mathrm {D} F(\sigma )(V(\sigma ))\,\mathrm {d} \gamma (\sigma )=\int _{C_{0}}F(\sigma )\left(\int _{0}^{T}{\dot {V}}_{t}(\sigma )\,\mathrm {d} \sigma _{t}\right)\,\mathrm {d} \gamma (\sigma ),}$

i.e.

${\displaystyle \int _{C_{0}}\left\langle \nabla _{H}F(\sigma ),V(\sigma )\right\rangle _{L_{0}^{2,1}}\,\mathrm {d} \gamma (\sigma )=-\int _{C_{0}}F(\sigma )\operatorname {div} (V)(\sigma )\,\mathrm {d} \gamma (\sigma )}$

or, writing the integrals over C₀ as expectations:

${\displaystyle \mathbb {E} {\big [}\langle \nabla _{H}F,V\rangle {\big ]}=-\mathbb {E} {\big [}F\operatorname {div} V{\big ]},}$

where the "divergence" div(V) : C₀ → R is defined by

${\displaystyle \operatorname {div} (V)(\sigma ):=-\int _{0}^{T}{\dot {V}}_{t}(\sigma )\,\mathrm {d} \sigma _{t}.}$

The interpretation of stochastic integrals as divergences leads to concepts such as the Skorokhod integral and the tools of the Malliavin calculus.

• Integral representation theorem for classical Wiener space, which uses the Clark–Ocone theorem in its proof
• Integration by parts operator
• Malliavin calculus

• Nualart, David (2006). The Malliavin calculus and related topics. Probability and its Applications (New York) (Second ed.). Berlin: Springer-Verlag. ISBN 978-3-540-28328-7.

• Friz, Peter K. (2005-04-10). "An Introduction to Malliavin Calculus" (PDF). Archived from the original (PDF) on 2007-04-17. Retrieved 2007-07-23.