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Finance and Stochastics

, Volume 4, Issue 4, pp 465–496 | Cite as

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

  • Knut Aase
  • Bernt Øksendal
  • Nicolas Privault
  • Jan Ubøe
Original Paper

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

\(\)

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 
Mathematics Subject Classification (1991): 60H40, 60G20 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Knut Aase
    • 1
  • Bernt Øksendal
    • 2
  • Nicolas Privault
    • 3
  • Jan Ubøe
    • 4
  1. 1.Norwegian School of Economics and Business Administration Helleveien 30, N-5035 Bergen-Sandviken, Norway (e-mail: knut.aase@nhh.no) NO
  2. 2.Department of Mathematics, University of Oslo, Box 1053 Blindern, N-0316 Oslo, Norway (e-mail: oksendal@math.uio.no) NO
  3. 3.Department of Mathematics, Université de la Rochelle, Avenue Marillac, F-17042 La Rochelle Cedex 1, France (e-mail: nicolas.privault@univ-lr.fr) FR
  4. 4.Stord/Haugesund College, Skåregaten 103, N-5500, Haugesund, Norway (e-mail: jan.uboe@hsh.no) NO

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