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

Kharroubi Idris (), Langrené Nicolas () and Pham Huyên ()
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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

Monte Carlo Methods and Applications, 2014, vol. 20, issue 2, 145-165

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].

Keywords: Backward stochastic differential equations; control randomization; HJB equation; uncertain volatility; empirical regressions; Monte Carlo (search for similar items in EconPapers)
Date: 2014
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DOI: 10.1515/mcma-2013-0024

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