Abstract

In this talk, I will present our recent results on numerical solution of high-dimensional Forward backward Stochastic Differential Equations (FBSDEs) and stochastic control problems via deep learning.In the field of numerically solving BSD Es, a well-known challenging problem was “the curse of dimensionality”. A recent important breakthrough in this research direction was made by E et al by using deep neural network. In this talk, we present a type of fully coupled high dimensional FBSDE in which the drift and diffusion coefficient are all given (nonlinear) functions of the backward variables (Y,Z). In order to solve this type of FBSDE , we systematically explore the dependence of the term $Z$ on state precesses $X,Y$ and even $Z$ itself. Three algorithms corresponding to different kinds of state feedback are developed via deep neural network and the numerical results demonstrate a remarkable performance. It is worth to notice that how to provide an efficient algorithm for this type of fully coupled nonlinear FBSDE was a largely open problem.The well-known nonlinear stochastic Hamiltonian system is a typical example of FBSD Es through which our algorithms have been successfully applied. We have also developed a direct stochastic optimal control approach for solving numerically this high dimensional problem. Two different algorithms suitable for different cases of the control problem are proposed. The numerical results demonstrate more stable convergence comparing with the FBSDE method for different Hamiltonian systems.Inspired by the deep learning method for solving FBSD Es, we also propose a method to solve high dimensional stochastic optimal control problem from the view of the stochastic maximum principle.Joint work with Prof. Shaolin Ji, Shige Peng and Dr. Xichuan Zhang.