BSΔEs and BSDEs with non-Lipschitz drivers: Comparison, convergence and robustness

Abstract

We provide existence results and comparison principles for solutions of backward stochastic difference equations (BSΔEs) and then prove convergence of these to solutions of backward stochastic differential equations (BSDEs) when the mesh size of the time-discretizaton goes to zero. The BSΔEs and BSDEs are governed by drivers fN(t,ω,y,z) and f(t,ω,y,z), respectively. The new feature of this paper is that they may be non-Lipschitz in z. For the convergence results it is assumed that the BSΔEs are based on d-dimensional random walks WN approximating the d-dimensional Brownian motion W underlying the BSDE and that fN converges to f. Conditions are given under which for any bounded terminal condition ξ for the BSDE, there exist bounded terminal conditions ξN for the sequence of BSΔEs converging to ξ, such that the corresponding solutions converge to the solution of the limiting BSDE. An important special case is when fN and f are convex in z. We show that in this situation, the solutions of the BSΔEs converge to the solution of the BSDE for every uniformly bounded sequence ξN converging to ξ. As a consequence, one obtains that the BSDE is robust in the sense that if (WN,ξN) is close to (W,ξ) in distribution, then the solution of the Nth BSΔE is close to the solution of the BSDE in distribution too.