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 (BS$Δ$Es) 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 BS$Δ$Es and BSDEs are governed by drivers $f^N(t,ω,y,z)$ 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 BS$Δ$Es are based on d-dimensional random walks $W^N$ approximating the d-dimensional Brownian motion W underlying the BSDE and that $f^N$ converges to f. Conditions are given under which for any bounded terminal condition $ξ$ for the BSDE, there exist bounded terminal conditions $ξ^N$ for the sequence of BS$Δ$Es converging to $ξ$, such that the corresponding solutions converge to the solution of the limiting BSDE. An important special case is when $f^N$ and f are convex in z. We show that in this situation, the solutions of the BS$Δ$Es converge to the solution of the BSDE for every uniformly bounded sequence $ξ^N$ converging to $ξ$. As a consequence, one obtains that the BSDE is robust in the sense that if $(W^N,ξ^N)$ is close to $(W,ξ)$ in distribution, then the solution of the Nth BS$Δ$E is close to the solution of the BSDE in distribution too.