BS$\Delta$Es 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$\Delta$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$\Delta$Es and BSDEs are governed by drivers $f^{N}(t,\omega,y,z)$ and $f(t,\omega,y,z),$ 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$\Delta$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 $\xi$ for the BSDE, there exist bounded terminal conditions $\xi^{N}$ for the sequence of BS$\Delta$Es converging to $\xi$, 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$\Delta$Es converge to the solution of the BSDE for every uniformly bounded sequence $\xi^{N}$ converging to $\xi$. As a consequence, one obtains that the BSDE is robust in the sense that if $(W^{N},\xi^{N})$ is close to $(W,\xi)$ in distribution, then the solution of the $N$th BS$\Delta$E is close to the solution of the BSDE in distribution too.