BS$Δ$Es and BSDEs with non-Lipschitz drivers: comparison, convergence and robustness

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 terminal condition $\xi$, there exist terminal conditions $\xi^N$ for the sequence of BS$\Delta$Es, converging to $\xi$ in $L^2$, such that for the solutions $Y^N$ and $Y$ of the corresponding BS$\Delta$Es and the limiting BSDE one has $\sup_{0\le t\le T} |Y^N_t - Y_t| \to 0$ in $L^2$. An important special case is when $f^N(t,\omega,y,z)$ and $f(t,\omega,y,z)$ are convex in $z.$ We show that in this situation, $\sup_{0\le t\le T} |Y^N_t - Y_t| \to 0$ in $L^2$ for every sequence of discrete terminal conditions $\xi^N$ converging to $\xi$ in $L^2$. 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 $Y^N$ is close to $Y$ in distribution too.