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We study the approximation of backward stochastic differential equations(BSDEs for short) with a constraint on the gains process. We first discretizethe constraint by applying a so-called facelift operator at times of a grid. Weshow that this discretely constrained BSDE converges to the continuouslyconstrained one as the mesh grid converges to zero. We then focus on theapproximation of the discretely constrained BSDE. For that we adopt a machinelearning approach. We show that the facelift can be approximated by anoptimization problem over a class of neural networks under constraints on theneural network and its derivative. We then derive an algorithm converging tothe discretely constrained BSDE as the number of neurons goes to infinity. Weend by numerical experiments. Mathematics Subject Classification (2010): 65C30,65M75, 60H35, 93E20, 49L25.

Discretization and Machine Learning Approximation of BSDEs with a Constraint on the Gains-Process