Discrete time approximation of HJB equations via BSDEs with nonpositive jumps

We propose a new probabilistic numerical scheme for fully nonlinear equations of Hamilton-Jacobi-Bellman (HJB) type associated to stochastic control problems, which is based on the a recent Feynman-Kac representation 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.