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On the Malliavin approach to Monte Carlo approximation of conditional expectations

Author

Listed:
  • Bruno Bouchard
  • Ivar Ekeland
  • Nizar Touzi

Abstract

Given a multi-dimensional Markov diffusion X, the Malliavin integration by parts formula provides a family of representations of the conditional expectation E[g(X 2 )|X 1 ]. The different representations are determined by some localizing functions. We discuss the problem of variance reduction within this family. We characterize an exponential function as the unique integrated mean-square-error minimizer among the class of separable localizing functions. For general localizing functions, we prove existence and uniqueness of the optimal localizing function in a suitable Sobolev space. We also provide a PDE characterization of the optimal solution which allows to draw the following observation : the separable exponential function does not minimize the integrated mean square error, except for the trivial one-dimensional case. We provide an application to a portfolio allocation problem, by use of the dynamic programming principle. Copyright Springer-Verlag Berlin/Heidelberg 2004

Suggested Citation

  • Bruno Bouchard & Ivar Ekeland & Nizar Touzi, 2004. "On the Malliavin approach to Monte Carlo approximation of conditional expectations," Finance and Stochastics, Springer, vol. 8(1), pages 45-71, January.
    • Handle: RePEc:spr:finsto:v:8:y:2004:i:1:p:45-71
    DOI: 10.1007/s00780-003-0109-0
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    Citations

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    Cited by:

    1. Abbas-Turki Lokman A. & Bouselmi Aych I. & Mikou Mohammed A., 2014. "Toward a coherent Monte Carlo simulation of CVA," Monte Carlo Methods and Applications, De Gruyter, vol. 20(3), pages 195-216, September.
    2. Simon Lysbjerg Hansen, 2005. "A Malliavin-based Monte-Carlo Approach for Numerical Solution of Stochastic Control Problems: Experiences from Merton's Problem," Computing in Economics and Finance 2005 391, Society for Computational Economics.
    3. repec:dau:papers:123456789/5524 is not listed on IDEAS
    4. Bender, Christian & Parczewski, Peter, 2018. "Discretizing Malliavin calculus," Stochastic Processes and their Applications, Elsevier, vol. 128(8), pages 2489-2537.
    5. Elisa Alòs & Christian-Olivier Ewald, 2005. "A note on the Malliavin differentiability of the Heston volatility," Economics Working Papers 880, Department of Economics and Business, Universitat Pompeu Fabra.
    6. Belomestny, Denis & Kolodko, Anastasia & Schoenmakers, John G. M., 2009. "Regression methods for stochastic control problems and their convergence analysis," SFB 649 Discussion Papers 2009-026, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.
    7. Leão, Dorival & Ohashi, Alberto, 2012. "Weak Approximations for Wiener Functionals," Insper Working Papers wpe_276, Insper Working Paper, Insper Instituto de Ensino e Pesquisa.
    8. Lokman A. Abbas-Turki & Ioannis Karatzas & Qinghua Li, 2014. "Impulse Control of a Diffusion with a Change Point," Papers 1404.1761, arXiv.org.
    9. Bally Vlad & Caramellino Lucia & Zanette Antonino, 2005. "Pricing and hedging American options by Monte Carlo methods using a Malliavin calculus approach," Monte Carlo Methods and Applications, De Gruyter, vol. 11(2), pages 97-133, June.
    10. Moez Mrad & Nizar Touzi & Amina Zeghal, 2006. "Monte Carlo Estimation of a Joint Density Using Malliavin Calculus, and Application to American Options," Computational Economics, Springer;Society for Computational Economics, vol. 27(4), pages 497-531, June.
    11. Fujiwara, Hajime & Kijima, Masaaki, 2007. "Pricing of path-dependent American options by Monte Carlo simulation," Journal of Economic Dynamics and Control, Elsevier, vol. 31(11), pages 3478-3502, November.
    12. repec:dau:papers:123456789/5522 is not listed on IDEAS
    13. Björn Bick & Holger Kraft & Claus Munk, 2013. "Solving Constrained Consumption-Investment Problems by Simulation of Artificial Market Strategies," Management Science, INFORMS, vol. 59(2), pages 485-503, June.
    14. Andr�s Garc�a Mirantes & Javier Población & Gregorio Serna, 2012. "Analyzing the dynamics of the refining margin: implications for valuation and hedging," Quantitative Finance, Taylor & Francis Journals, vol. 12(12), pages 1839-1855, December.
    15. Bouchard, Bruno & Touzi, Nizar, 2004. "Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 111(2), pages 175-206, June.
    16. Christian Bender & Anastasia Kolodko & John Schoenmakers, 2008. "Enhanced policy iteration for American options via scenario selection," Quantitative Finance, Taylor & Francis Journals, vol. 8(2), pages 135-146.
    17. Crisan, D. & Manolarakis, K. & Touzi, N., 2010. "On the Monte Carlo simulation of BSDEs: An improvement on the Malliavin weights," Stochastic Processes and their Applications, Elsevier, vol. 120(7), pages 1133-1158, July.

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