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

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  • Idris Kharroubi

    (CREST, CEREMADE)

  • Nicolas Langren\'e

    (LPMA)

  • Huy\^en Pham

    (CREST, LPMA)

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    Abstract

    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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    Bibliographic Info

    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

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    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, Anderson Graduate School of Management, UCLA 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, Paris Dauphine University 123456789/5524, Paris Dauphine University.
    3. Idris Kharroubi & Nicolas Langrené & Huyên Pham, 2013. "A numerical algorithm for fully nonlinear HJB equations: an approach by control randomization," Working Papers, HAL hal-00905899, HAL.
    4. 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, Society for Financial Studies, vol. 14(1), pages 113-47.
    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., World Scientific Publishing Co. Pte. Ltd., vol. 14(05), pages 709-722.
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    Cited by:
    1. Idris Kharroubi & Nicolas Langrené & Huyên Pham, 2013. "A numerical algorithm for fully nonlinear HJB equations: an approach by control randomization," Working Papers, HAL hal-00905899, HAL.
    2. Sakda Chaiworawitkul & Patrick S. Hagan & Andrew Lesniewski, 2014. "Semiclassical approximation in stochastic optimal control I. Portfolio construction problem," Papers 1406.6090, arXiv.org.

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