Solving stochastic optimal control problem via stochastic maximum principle with deep learning method

In this paper, we aim to solve the high dimensional stochastic optimal control problem via deep learning. Through the stochastic maximum principle and its corresponding Hamiltonian system, we establish a framework in which the original control problem is reformulated as a new one. But the cost is that we must deal with an additional maximum condition. Three algorithms are proposed to solve the new control problem by deep learning. An important application of our proposed methods is that they can be used to calculate the sub-linear expectations, which correspond to a kind of fully nonlinear PDEs. Several numerical examples have been studied, and the results demonstrate rather optimistic performance, especially for high dimensional cases.