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.
Cheridito, P., & Stadje, M. A. (2012). Existence, minimality and approximations of solutions of BSDEs with convex drivers. Stochastic Processes and their Applications, 122(4), 1540-1565. https://doi.org/10.1016/j.spa.2011.12.008