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

Posted: 17 Nov 2000

See all articles by Knut K. Aase

Knut K. Aase

Norwegian School of Economics (NHH) - Department of Business and Management Science

Bernt Oksendal

University of Oslo - Department of Mathematics; Norwegian School of Economics (NHH) - Department of Finance

Nicolas Privault

Université de la Rochelle

Jan Uboe

Norwegian School of Economics (NHH) - Department of Business and Management Science

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

Suggested Citation

Aase, Knut K. and Oksendal, Bernt and Privault, Nicolas and Uboe, Jan, White Noise Generalizations of the Clark-Haussmann-Ocone Theorem with Application to Mathematical Finance. Finance and Stochastics, Vol. 4, No. 4. Available at SSRN: https://ssrn.com/abstract=239361

Knut K. Aase

Norwegian School of Economics (NHH) - Department of Business and Management Science ( email )

Helleveien 30
Bergen, NO-5045
Norway

Bernt Oksendal (Contact Author)

University of Oslo - Department of Mathematics ( email )

P.O. Box 1053
Blindern, N-0162, Os
Norway
+47-2285 5913 (Phone)
+47-2285 4349 (Fax)

Norwegian School of Economics (NHH) - Department of Finance

Helleveien 30
N-5045 Bergen
Norway

Nicolas Privault

Université de la Rochelle ( email )

Avenue Marillac
Department of Mathematics
F-17042 La Rochelle Cedex 1
France

HOME PAGE: http://perso.univ-lr.fr/nprivaul/

Jan Uboe

Norwegian School of Economics (NHH) - Department of Business and Management Science ( email )

Helleveien 30
Bergen, NO-5045
Norway

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