It is well-known that decision-making problems from stochastic control can be formulated by means of a forward–backward stochastic differential equation (FBSDE). Recently, the authors of Ji et al. (2022) proposed an efficient deep learning algorithm based on the stochastic maximum principle (SMP). In this paper, we provide a convergence result for this deep SMP-BSDE algorithm and compare its performance with other existing methods. In particular, by adopting a strategy as in Han and Long (2020), we derive a-posteriori estimate, and show that the total approximation error can be bounded by the value of the loss functional and the discretization error. We present numerical examples for high-dimensional stochastic control problems, both in the cases of drift- and diffusion control, which showcase superior performance compared to existing algorithms.

A novel discretization is presented for decoupled forward–backward stochastic differential equations (FBSDE) with differentiable coefficients, simultaneously solving the BSDE and its Malliavin sensitivity problem. The control process is estimated by the corresponding linear BSDE driving the trajectories of the Malliavin derivatives of the solution pair, which implies the need to provide accurate Γ estimates. The approximation is based on a merged formulation given by the Feynman–Kac formulae and the Malliavin chain rule. The continuous time dynamics is discretized with a theta-scheme. In order to allow for an efficient numerical solution of the arising semidiscrete conditional expectations in possibly high dimensions, it is fundamental that the chosen approach admits to differentiable estimates. Two fully-implementable schemes are considered: the BCOS method as a reference in the one-dimensional framework and neural network Monte Carlo regressions in case of high-dimensional problems, similarly to the recently emerging class of Deep BSDE methods (Han et al. (2018 Solving high-dimensional partial differential equations using deep learning. Proc. Natl. Acad. Sci., 115, 8505–8510); Huré et al. (2020 Deep backward schemes for high-dimensional nonlinear PDEs. Math. Comp., 89, 1547–1579)). An error analysis is carried out to show L ² convergence of order 1/2, under standard Lipschitz assumptions and additive noise in the forward diffusion. Numerical experiments are provided for a range of different semilinear equations up to 50 dimensions, demonstrating that the proposed scheme yields a significant improvement in the control estimations.