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We propose a new multistep deep learning-based algorithm for the resolutionof moderate to high dimensional nonlinear backward stochastic differentialequations (BSDEs) and their corresponding parabolic partial differentialequations (PDE). Our algorithm relies on the iterated time discretisation ofthe BSDE and approximates its solution and gradient using deep neural networksand automatic differentiation at each time step. The approximations areobtained by sequential minimisation of local quadratic loss functions at eachtime step through stochastic gradient descent. We provide an analysis ofapproximation error in the case of a network architecture with weightconstraints requiring only low regularity conditions on the generator of theBSDE. The algorithm increases accuracy from its single step parent model andhas reduced complexity when compared to similar models in the literature.

Deep multi-step mixed algorithm for high dimensional non-linear PDEs and associated BSDEs