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

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].

[1]  Adrien Nguyen Huu,et al.  Hedging Expected Losses on Derivatives in Electricity Futures Markets , 2014, 1401.8271.

[2]  Daniel Z. Zanger.  Quantitative error estimates for a least-squares Monte Carlo algorithm for American option pricing , 2013, Finance Stochastics.

[3]  Marco Avellaneda,et al.  Reducing variance in the numerical solution of BSDEs , 2013 .

[4]  Huyen Pham,et al.  Feynman-Kac representation for Hamilton-Jacobi-Bellman IPDE , 2012, 1212.2000.

[5]  H. Pham,et al.  A probabilistic numerical method for optimal multiple switching problem and application to investments in electricity generation , 2012, 1210.8175.

[6]  B. Bouchard,et al.  Monte-Carlo valuation of American options: facts and new algorithms to improve existing methods , 2012 .

[7]  J. Romo.  Pricing Digital Outperformance Options With Uncertain Correlation , 2011 .

[8]  Emmanuel Gobet,et al.  Approximation of discrete BSDE using least-squares regression , 2011 .

[9]  Pierre Henry-Labordere,et al.  Uncertain Volatility Model: A Monte-Carlo Approach , 2010 .

[10]  Nizar Touzi,et al.  A Probabilistic Numerical Method for Fully Nonlinear Parabolic PDEs , 2009, 0905.1863.

[11]  E. Gobet,et al.  Rate of convergence of an empirical regression method for solving generalized backward stochastic differential equations , 2006 .

[12]  H. Soner,et al.  Second‐order backward stochastic differential equations and fully nonlinear parabolic PDEs , 2005, math/0509295.

[13]  John N. Tsitsiklis,et al.  Regression methods for pricing complex American-style options , 2001, IEEE Trans. Neural Networks.

[14]  Francis A. Longstaff,et al.  Valuing American Options by Simulation: A Simple Least-Squares Approach , 2001 .

[15]  X. Zhou,et al.  Stochastic Controls: Hamiltonian Systems and HJB Equations , 1999 .

[16]  D. Talay.  Model Risk in Finance: Some Modeling and Numerical Analysis Issues , 2009 .

[17]  M. Mrad.  Méthodes numériques d'évaluation et de couverture des options exotiques multi-sous-jacents : modèles de marché et modèles à volatilité incertaine , 2008 .