A Clark-Ocone formula in UMD Banach spaces

Jan Maas,Jan Neerven

doi:10.1214/ecp.v13-1361

Abstract

Let $H$ be a separable real Hilbert space and let $\mathbb{F}=(\mathscr{F}_t)_{t\in [0,T]}$ be the augmented filtration generated by an $H$-cylindrical Brownian motion $(W_H(t))_{t\in [0,T]}$ on a probability space $(\Omega,\mathscr{F},\mathbb{P})$. We prove that if $E$ is a UMD Banach space, $1\le p<\infty$, and $F\in \mathbb{D}^{1,p}(\Omega;E)$ is $\mathscr{F}_T$-measurable, then $$ F = \mathbb{E} (F) + \int_0^T P_{\mathbb{F}} (DF)\,dW_H,$$ where $D$ is the Malliavin derivative of $F$ and $P_{\mathbb{F}}$ is the projection onto the ${\mathbb{F}}$-adapted elements in a suitable Banach space of $L^p$-stochastically integrable $\mathscr{L}(H,E)$-valued processes.

Highlights

A classical result of Clark [5] and Ocone [17] asserts that if F = (Ft)t∈[0,T ] is the augmented filtration generated by a Brownian motion (W (t))t∈[0,T ] on a probability space (Ω, F, P), every FT -measurable random variable F ∈ D1,p(Ω), 1 < p < ∞, admits a representation TF = E(F ) + E(DtF |Ft) dWt, where Dt is the Malliavin derivative of F at time t
The Clark-Ocone formula is used in mathematical finance to obtain explicit expressions for hedging strategies
H is a separable Hilbert space and E is a UMD Banach space. For this purpose we study the properties of the Malliavin derivative D of smooth E-valued random variables with respect to an isonormal process W on a separable Hilbert space H

Summary

Introduction

H is a separable Hilbert space and E is a UMD Banach space (the definition is recalled below) For this purpose we study the properties of the Malliavin derivative D of smooth E-valued random variables with respect to an isonormal process W on a separable Hilbert space H. As it turns out, D can be naturally defined as a closed operator acting from Lp(Ω; E) to Lp(Ω; γ(H, E)), where γ(H, E) is the operator ideal of γ-radonifying operators from a Hilbert space H to E. They thank Ben Goldys at UNSW and Alan McIntosh at ANU for their kind hospitality

Preliminaries

The Malliavin derivative

The divergence operator

The Skorokhod integral

A Clark-Ocone formula

Extension to L1