  We study the existence of solutions to backward stochastic differential
equations with drivers f(t,W,y,z) that are convex in z. We assume f to be
Lipschitz in y and W but do not make growth assumptions with respect to z. We
first show the existence of a unique solution (Y,Z) with bounded Z if the
terminal condition is Lipschitz in W and that it can be approximated by the
solutions to properly discretized equations. If the terminal condition is
bounded and uniformly continuous in W, we show the existence of a minimal
continuous supersolution by uniformly approximating the terminal condition with
Lipschitz terminal conditions. Finally, we prove existence of a minimal RCLL
supersolution for bounded lower semicontinuous terminal conditions by
approximating the terminal condition pointwise from below with Lipschitz
terminal conditions.
