A decomposition approach for the discrete-time approximation of FBSDEs with a jump I : the Lipschitz case

We study the discrete-time approximation for solutions of forward-backward stochas- tic dierential equations (FBSDEs) with a jump. In this part, we study the case of Lipschitz generators, and we refer to the second part of this work [15] for the quadratic case. Our method is based on a result given in the companion paper [14] which allows to link a FBSDE with a jump with a recursive system of Brownian FBSDEs. Then we use the classical results on discretization of Brownian FBSDEs to approximate the recursive system of FBSDEs and we recombine these approximations to get a dis- cretization of the FBSDE with a jump. This approach allows to get a convergence rate similar to that of schemes for Brownian FBSDEs.