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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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    Citations

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    Cited by:

    1. Claudio Fontana & Bernt Øksendal & Agnès Sulem, 2015. "Market Viability and Martingale Measures under Partial Information," Methodology and Computing in Applied Probability, Springer, vol. 17(1), pages 15-39, March.
    2. Bernt Øksendal & Agnès Sulem, 2014. "Forward–Backward Stochastic Differential Games and Stochastic Control under Model Uncertainty," Journal of Optimization Theory and Applications, Springer, vol. 161(1), pages 22-55, April.
    3. Takafumi Amaba, 2014. "A Discrete-Time Clark-Ocone Formula for Poisson Functionals," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 21(2), pages 97-120, May.
    4. 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.
    5. Aase, Knut K., 2005. "The perpetual American put option for jump-diffusions with applications," Discussion Papers 2005/12, Norwegian School of Economics, Department of Business and Management Science.
    6. Aase, Knut K., 2004. "The perpetual American put option for jump-diffusions: Implications for equity premiums," Discussion Papers 2004/19, Norwegian School of Economics, Department of Business and Management Science.
    7. Hu, Yaozhong & Øksendal, Bernt, 2019. "Linear Volterra backward stochastic integral equations," Stochastic Processes and their Applications, Elsevier, vol. 129(2), pages 626-633.
    8. Nacira Agram & Bernt Øksendal, 2015. "Malliavin Calculus and Optimal Control of Stochastic Volterra Equations," Journal of Optimization Theory and Applications, Springer, vol. 167(3), pages 1070-1094, December.
    9. N. Naguez & J. L. Prigent, 2017. "Optimal portfolio positioning within generalized Johnson distributions," Quantitative Finance, Taylor & Francis Journals, vol. 17(7), pages 1037-1055, July.
    10. Olfa Draouil & Bernt {O}ksendal, 2018. "Viable Insider Markets," Papers 1801.03720, arXiv.org.
    11. Leão, Dorival & Ohashi, Alberto, 2010. "Weak Approximations for Wiener Functionals," Insper Working Papers wpe_215, Insper Working Paper, Insper Instituto de Ensino e Pesquisa.
    12. 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.
    13. Tebaldi, Claudio, 2005. "Hedging using simulation: a least squares approach," Journal of Economic Dynamics and Control, Elsevier, vol. 29(8), pages 1287-1312, August.
    14. 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.
    15. 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.
    16. Bernt Oksendal & Agnès Sulem, 2011. "Portfolio optimization under model uncertainty and BSDE games," Working Papers inria-00570532, HAL.
    17. Aase, Knut K., 2005. "Using Option Pricing Theory to Infer About Equity Premiums," Discussion Papers 2005/11, Norwegian School of Economics, Department of Business and Management Science.
    18. Suzuki, Ryoichi, 2018. "Malliavin differentiability of indicator functions on canonical Lévy spaces," Statistics & Probability Letters, Elsevier, vol. 137(C), pages 183-190.
    19. Peng, Xingchun & Chen, Fenge & Hu, Yijun, 2014. "Optimal investment, consumption and proportional reinsurance under model uncertainty," Insurance: Mathematics and Economics, Elsevier, vol. 59(C), pages 222-234.
    20. Bernt {O}ksendal & Elin R{o}se, 2015. "A white noise approach to insider trading," Papers 1508.06376, arXiv.org.
    21. Haug, Jorgen, 2001. "Explicit characterizations of financial prices with history-dependent utility," Journal of Mathematical Economics, Elsevier, vol. 36(4), pages 337-356, December.
    22. Hans‐Peter Bermin, 2003. "Hedging Options: The Malliavin Calculus Approach versus the Δ‐Hedging Approach," Mathematical Finance, Wiley Blackwell, vol. 13(1), pages 73-84, January.

    More about this item

    JEL classification:

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

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