Convergence of BS\Delta Es driven by random walks to BSDEs: the case of (in)finite activity jumps with general driver

In this paper we present a weak approximation scheme for BSDEs driven by a Wiener process and an (in)finite activity Poisson random measure with drivers that are general Lipschitz functionals of the solution of the BSDE. The approximating backward stochastic difference equations (BS\Delta Es) are driven by random walks that weakly approximate the given Wiener process and Poisson random measure. We establish the weak convergence to the solution of the BSDE and the numerical stability of the sequence of solutions of the BS\Delta Es. By way of illustration we analyse explicitly a scheme with discrete step-size distributions.