Discrete time approximation of fully nonlinear HJB equations via BSDEs with nonpositive jumps

Abstract

We propose a new probabilistic numerical scheme for fully nonlinear equation of Hamilton–Jacobi–Bellman (HJB) type associated to stochastic control problem, which is based on the Feynman–Kac representation in [Kharroubi and Pham (2014)] by means of control randomization and backward stochastic differential equation with nonpositive jumps. We study a discrete time approximation for the minimal solution to this class of BSDE when the time step goes to zero, which provides both an approximation for the value function and for an optimal control in feedback form. We obtained a convergence rate without any ellipticity condition on the controlled diffusion coefficient. An explicit implementable scheme based on Monte Carlo simulations and empirical regressions, associated error analysis and numerical experiments are performed in the companion paper [ Monte Carlo Methods Appl. 20 (2014) 145–165].