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White noise generalizations of the Clark-Haussmann-Ocone theorem with application to mathematical finance

Author

Listed:
  • Jan Ubøe

    (Stord/Haugesund College, Skåregaten 103, N-5500, Haugesund, Norway Manuscript)

  • Bernt Øksendal

    (Department of Mathematics, University of Oslo, Box 1053 Blindern, N-0316 Oslo, Norway)

  • Knut Aase

    (Norwegian School of Economics and Business Administration Helleveien 30, N-5035 Bergen-Sandviken, Norway)

  • Nicolas Privault

    (Department of Mathematics, Université de la Rochelle, Avenue Marillac, F-17042 La Rochelle Cedex 1, France)

Abstract

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

Suggested Citation

  • Jan Ubøe & Bernt Øksendal & Knut Aase & Nicolas Privault, 2000. "White noise generalizations of the Clark-Haussmann-Ocone theorem with application to mathematical finance," Finance and Stochastics, Springer, vol. 4(4), pages 465-496.
    • Handle: RePEc:spr:finsto:v:4:y:2000:i:4:p:465-496
    Note: received: December 1999; final version received: January 2000
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    JEL classification:

    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates

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