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

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