A Clark-Ocone formula in UMD Banach spaces

TL;DR

This paper extends the Clark-Ocone formula to UMD Banach spaces, providing a stochastic integral representation for F_T-measurable functions using Malliavin calculus and projection operators.

Contribution

It introduces a Clark-Ocone type formula in UMD Banach spaces, generalizing classical results to infinite-dimensional Banach space settings.

Findings

Established a Clark-Ocone formula for UMD Banach spaces.

Connected Malliavin derivatives with stochastic integrals in Banach spaces.

Provided a representation for F_T-measurable functions in this setting.

Abstract

Let H be a separable real Hilbert space and let F = (F_t)_{t\in [0,T]} be the augmented filtration generated by an H-cylindrical Brownian motion W_H on [0,T]. We prove that if E is a UMD Banach space, 1\leq p<\infty, and f\in D^{1,p}(E) is F_T-measurable, then f = \E f + \int_0^T P_F(Df) dW_H where D is the Malliavin derivative and P_F is the projection onto the F-adapted elements in a suitable Banach space of L^p-stochastically integrable L(H,E)-valued processes.