White noise generalizations of the Clark-Haussmann-Ocone theorem, with application to mathematical finance
 
Knut Aase1),2)   Bernt Øksendal2),1)   Nicolas Privault3)   Jan Ubøe4)
1) Norwegian School of Economics and Business Administration, Helleveien 30, N-5035 Bergen - Sandviken, Norway.
2) Departement of Mathematics, University of Oslo, Box 1053 Blindern, N-0316 Oslo, Norway.
3) Département de Mathématiques, Université de La Rochelle, Avenue Marillac, F-17042 La Rochelle Cedex 1, France.
4) Stord/Haugesund College, Skåregaten 103, N-5 500, Haugesund, Norway.

Abstract:

We use a white noise approach to Malliavin calculus to prove the following white noise generalization of the Clark-Haussmann-Ocone formula

\begin{displaymath}F(\omega)=E[F]+\int_0^TE[D_tF\vert{\cal F}_t]\diamond W(t)dt\end{displaymath}

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 $f\in{\mathord{{\sl {\sf G}}}}^*\supset L^2(\mu)$, where ${\mathord{{\sl {\sf G}}}}^*$ 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.

Key words: White noise, Clark-Haussmann-Ocone formula, Markets with jumps, Hedging strategies.
Mathematics Subject Classification (1991): 60H40, 60G20.

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