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].