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A numerical algorithm for fully nonlinear HJB equations: An approach by control randomization

Author

Listed:
  • Kharroubi Idris

    (CEREMADE, CNRS UMR 7534, Université Paris Dauphine, and CREST, France)

  • Langrené Nicolas

    (Laboratoire de Probabilités et Modèles Aléatoires, Université Paris Diderot, and EDF R&D, France)

  • Pham Huyên

    (Laboratoire de Probabilités et Modèles Aléatoires, Université Paris Diderot, and CREST-ENSAE, France)

Abstract

We propose a probabilistic numerical algorithm to solve Backward Stochastic Differential Equations (BSDEs) with nonnegative jumps, a class of BSDEs introduced in [`Feynman–Kac representation for Hamilton–Jacobi–Bellman IPDE', Ann. Probab., to appear] for representing fully nonlinear HJB equations. This includes in particular numerical resolution for stochastic control problems with controlled volatility, possibly degenerate. Our backward scheme, based on least-squares regressions, takes advantage of high-dimensional properties of Monte Carlo methods, and also provides a parametric estimate in feedback form for the optimal control. A partial analysis of the algorithm error is presented, as well as numerical tests on the problem of option superreplication with uncertain volatilities and/or correlations, including a detailed comparison with the numerical results from the alternative scheme proposed in [J. Comput. Finance 14 (2011), 37–71].

Suggested Citation

  • Kharroubi Idris & Langrené Nicolas & Pham Huyên, 2014. "A numerical algorithm for fully nonlinear HJB equations: An approach by control randomization," Monte Carlo Methods and Applications, De Gruyter, vol. 20(2), pages 145-165, June.
    • Handle: RePEc:bpj:mcmeap:v:20:y:2014:i:2:p:145-165:n:5
    DOI: 10.1515/mcma-2013-0024
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    References listed on IDEAS

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    1. Longstaff, Francis A & Schwartz, Eduardo S, 2001. "Valuing American Options by Simulation: A Simple Least-Squares Approach," The Review of Financial Studies, Society for Financial Studies, vol. 14(1), pages 113-147.
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    3. Aïd, René & Campi, Luciano & Langrené, Nicolas & Pham, Huyên, 2014. "A probabilistic numerical method for optimal multiple switching problems in high dimension," LSE Research Online Documents on Economics 63011, London School of Economics and Political Science, LSE Library.
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    6. Daniel Zanger, 2013. "Quantitative error estimates for a least-squares Monte Carlo algorithm for American option pricing," Finance and Stochastics, Springer, vol. 17(3), pages 503-534, July.
    7. Adrien Nguyen Huu & Nadia Oudjane, 2014. "Hedging Expected Losses on Derivatives in Electricity Futures Markets," Papers 1401.8271, arXiv.org.
    8. Longstaff, Francis A & Schwartz, Eduardo S, 2001. "Valuing American Options by Simulation: A Simple Least-Squares Approach," University of California at Los Angeles, Anderson Graduate School of Management qt43n1k4jb, Anderson Graduate School of Management, UCLA.
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