Existence, minimality and approximation of solutions to BSDEs with convex drivers

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 the existence of a minimal RCLL supersolution for bounded lower semicontinuous terminal conditions by approximating the terminal condition pointwise from below with Lipschitz terminal conditions.