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

  • Idris Kharroubi

    (CREST, CEREMADE)

  • Nicolas Langren\'e

    (LPMA)

  • Huy\^en Pham

    (CREST, LPMA)

Registered author(s):

    We propose a probabilistic numerical algorithm to solve Backward Stochastic Differential Equations (BSDEs) with nonnegative jumps, a class of BSDEs introduced in [9] for representing fully nonlinear HJB equations. In particular, this allows us to numerically solve 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 error of the scheme is provided, as well as numerical tests on the problem of superreplication of option with uncertain volatilities and/or correlations, including a detailed comparison with the numerical results from the alternative scheme proposed in [7].

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    File URL: http://arxiv.org/pdf/1311.4503
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    Paper provided by arXiv.org in its series Papers with number 1311.4503.

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    Date of creation: Nov 2013
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    Handle: RePEc:arx:papers:1311.4503
    Contact details of provider: Web page: http://arxiv.org/

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    1. 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.
    2. Fahim, Arash & Touzi, Nizar & Warin, Xavier, 2011. "A Probabilistic Numerical Method for Fully Nonlinear Parabolic PDEs," Economics Papers from University Paris Dauphine 123456789/5524, Paris Dauphine University.
    3. Longstaff, Francis A & Schwartz, Eduardo S, 2001. "Valuing American Options by Simulation: A Simple Least-Squares Approach," Review of Financial Studies, Society for Financial Studies, vol. 14(1), pages 113-47.
    4. Idris Kharroubi & Nicolas Langrené & Huyên Pham, 2013. "A numerical algorithm for fully nonlinear HJB equations: an approach by control randomization," Working Papers hal-00905899, HAL.
    5. Jacinto Marabel, 2011. "Pricing Digital Outperformance Options With Uncertain Correlation," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 14(05), pages 709-722.
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