White Noise Generalizations of the Clark-Haussmann-Ocone Theorem with Application to Mathematical Finance

Abstract

We use a white noise approach to Malliavin calculus to prove the following white noise generalization of the Clark-Haussmann-Ocone formula \[F(\omega)=E[F]+\int_0^TE[D_tF|{\cal F}_t]\diamond W(t)dt\]

Here $E[F]$ denotes the generalized expectation, $D_tF(\omega)={{dF}\over{d\omega}}$ is the (generalized) Malliavin derivative, $\diamond$ is the Wick product and $W(t)$ is 1-dimensional Gaussian white noise. The formula holds for all $f\in{\cal G}^*\supset L^2(\mu)$, where ${\cal G}^*$ is a space of stochastic distributions and $\mu$ is the white noise probability measure. We also establish similar results for multidimensional Gaussian white noise, for multidimensional Poissonian white noise and for combined Gaussian and Poissonian noise. Finally we give an application to mathematical finance: We compute the replicating portfolio for a European call option in a Poissonian Black & Scholes type market.

JEL Classification: G12