Discrete time approximation of decoupled Forward-Backward SDE driven by pure jump L\'evy-processes

We present a new algorithms to discretize a decoupled forward backward stochastic differential equations driven by pure jump L\'evy process (FBSDEL in short). The method is built in two steps. Firstly, we approximate the FBSDEL by a forward backward stochastic differential equations driven by a Brownian motion and Poisson process (FBSDEBP in short), in which we replace the small jumps by a Brownian motion. Then, we prove the convergence of the approximation when the size of small jumps $\eps$ goes to 0. In the second step, we obtain the $L^p$ H\"older continuity of the solution of FBSDEBP and we construct two numerical schemes for this FBSDEBP. Based on the $L^p$ H\"older estimate, we prove the convergence of the scheme when the number of time steps $n$ goes to infinity. Combining these two steps leads to prove the convergence of numerical schemes to the solution of FBSDEL.