  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 Nth BS$\Delta$E is close to
the solution of the BSDE in distribution too.
