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

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Author Info

  • 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.

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Bibliographic Info

Article provided by Springer in its journal Finance and Stochastics.

Volume (Year): 4 (2000)
Issue (Month): 4 ()
Pages: 465-496

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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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Cited by:
  1. Aase, Knut K., 2005. "The perpetual American put option for jump-diffusions with applications," Discussion Papers 2005/12, Department of Business and Management Science, Norwegian School of Economics.
  2. Aase, Knut K, 2005. "Using Option Pricing Theory to Infer About Historical Equity Premiums," University of California at Los Angeles, Anderson Graduate School of Management qt3dd602j5, Anderson Graduate School of Management, UCLA.
  3. Leão, Dorival & Ohashi, Alberto, 2010. "Weak Approximations for Wiener Functionals," Insper Working Papers wpe_215, Insper Working Paper, Insper Instituto de Ensino e Pesquisa.
  4. Tebaldi, Claudio, 2005. "Hedging using simulation: a least squares approach," Journal of Economic Dynamics and Control, Elsevier, vol. 29(8), pages 1287-1312, August.
  5. Wei Chen, 2013. "Fractional G-White Noise Theory, Wavelet Decomposition for Fractional G-Brownian Motion, and Bid-Ask Pricing Application to Finance Under Uncertainty," Papers 1306.4070, arXiv.org.
  6. Aase, Knut K., 2005. "Using Option Pricing Theory to Infer About Equity Premiums," Discussion Papers 2005/11, Department of Business and Management Science, Norwegian School of Economics.
  7. El-Khatib, Youssef & Abdulnasser, Hatemi-J, 2011. "On the calculation of price sensitivities with jump-diffusion structure," MPRA Paper 30596, University Library of Munich, Germany.
  8. Bernt Oksendal & Agnès Sulem, 2011. "Portfolio optimization under model uncertainty and BSDE games," Working Papers inria-00570532, HAL.
  9. Claudio Fontana & Bernt {\O}ksendal & Agn\`es Sulem, 2013. "Market viability and martingale measures under partial information," Papers 1302.4254, arXiv.org, revised Oct 2013.
  10. Aase, Knut K., 2004. "The perpetual American put option for jump-diffusions: Implications for equity premiums," Discussion Papers 2004/19, Department of Business and Management Science, Norwegian School of Economics.

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