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

Author

Listed:
  • Idris Kharroubi

    (CREST, CEREMADE)

  • Nicolas Langren'e

    (LPMA)

  • Huy^en Pham

    (CREST, LPMA)

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].

Suggested Citation

  • Idris Kharroubi & Nicolas Langren'e & Huy^en Pham, 2013. "A numerical algorithm for fully nonlinear HJB equations: an approach by control randomization," Papers 1311.4503, arXiv.org.
    • Handle: RePEc:arx:papers:1311.4503
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    File URL: http://arxiv.org/pdf/1311.4503
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    References listed on IDEAS

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    1. 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.
    2. repec:dau:papers:123456789/4273 is not listed on IDEAS
    3. repec:dau:papers:123456789/5524 is not listed on IDEAS
    4. 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.
    5. 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 113-147.
    6. Adrien Nguyen Huu & Nadia Oudjane, 2014. "Hedging Expected Losses on Derivatives in Electricity Futures Markets," Papers 1401.8271, arXiv.org.
    7. 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.
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    Cited by:

    1. Frank Bosserhoff & An Chen & Nils Sorensen & Mitja Stadje, 2021. "On the Investment Strategies in Occupational Pension Plans," Papers 2104.08956, arXiv.org.
    2. Steven Kou & Xianhua Peng & Xingbo Xu, 2016. "EM Algorithm and Stochastic Control in Economics," Papers 1611.01767, arXiv.org.
    3. 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.
    4. 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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