Clark-Ocone theorem

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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 Itō integral with respect to that path.

[edit] Statement of the theorem

Let $C_{0} ([0, T]; \mathbb{R})$ (or simply $C 0$ for short) be classical Wiener space with Wiener measure $γ$ . Let $F : C_{0} \to \R$ be a $B C 1$ function, i.e. $F$ is bounded and Fréchet differentiable with derivative $\mathrm{D} F : C_{0} \to \mathrm{Lin} (C_{0}; \mathbb{R})$ bounded. Then

$F(\sigma) = \int_{C_{0}} F(p) \, \mathrm{d} \gamma(p) + \int_{0}^{T} \mathbb{E} \left\{ \left. \frac{\partial}{\partial t} \nabla_{H} F_{t} (-) \right| \mathcal{F}_{t} \right\} (\sigma) \, \mathrm{d} \sigma_{t}$ .

In the above

$F (σ)$ is the value of the function $F$ on some specific path of interest, $σ$ ;
$\int_{C_{0}} F(p) \, \mathrm{d} \gamma(p) = \mathbb{E}[F]$ is the expected value of $F$ over the whole of Wiener space $C 0$ ;
the second integral $\int_{0}^{T} \dots \, \mathrm{d} \sigma (t)$ is an Itō integral;
$\mathcal{F}_{t}$ is the natural filtration of Brownian motion $B : [0, T] \times \Omega \to \mathbb{R}$ : $\mathcal{F}_{t}$ is the smallest sigma algebra containing all $B_{s}^{-1} (A)$ for times $0 \leq s \leq t$ and Borel sets $A \subseteq \mathbb{R}$ ;
$\mathbb{E} \left\{ - | \mathcal{F}_{t} \right\}$ denotes conditional expectation with respect to the sigma algebra $\mathcal{F}_{t}$ ;
$\frac{\partial}{\partial t}$ denotes differentiation with respect to time, as usual;
finally, the symbol $\nabla_{H}$ denotes the H-gradient.

[edit] 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 Itō integrals as "divergences":

Let $B$ be a standard Brownian motion, and let $L_{0}^{2, 1}$ be the Cameron-Martin space for $C 0$ (see abstract Wiener space. Let $V : C_{0} \to L_{0}^{2,1}$ be such that

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

is in $L 2 (B)$ (i.e. is Itō integrable, and hence is an adapted process). Let $F : C_{0} \to \mathbb{R}$ be $B C 1$ as above. Then

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

i.e.

$\int_{C_{0}} \left\langle \nabla_{H} F (\sigma), V (\sigma) \right\rangle_{L_{0}^{2, 1}} \, \mathrm{d} \gamma (\sigma) = - \int_{C_{0}} F (\sigma) \mathrm{div} V (\sigma) \, \mathrm{d} \gamma (\sigma),$