A decomposition approach for the discrete-time approximation of FBSDEs with a jump

We are concerned with the discretization of a solution of a Forward-Backward stochastic differential equation (FBSDE) with a jump process depending on the Brownian motion. In this paper, we study the cases of Lipschitz generators and the generators with a quadratic growth w.r.t. the variable z. We propose a recursive scheme based on a general existence result given in a companion paper and we study the error induced by the time discretization. We prove the convergence of the scheme when the number of time steps n goes to infinity. Our approach allows to get a convergence rate similar to that of schemes of Brownian FBSDEs.