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

TL;DR

This paper presents a deep learning approach to solve high-dimensional stochastic optimal control problems using the stochastic maximum principle, reformulating the problem via an extended Hamiltonian system and proposing three algorithms with demonstrated effectiveness.

Contribution

The paper introduces a novel deep learning framework for high-dimensional stochastic control problems based on the stochastic maximum principle and extended Hamiltonian systems.

Findings

Algorithms effectively solve high-dimensional problems

Numerical results validate approach's accuracy

Applicable to nonlinear PDEs via sub-linear expectations

Abstract

In this paper, we aim to solve the high dimensional stochastic optimal control problem from the view of the stochastic maximum principle via deep learning. By introducing the extended Hamiltonian system which is essentially an FBSDE with a maximum condition, we reformulate the original control problem as a new one. Three algorithms are proposed to solve the new control problem. Numerical results for different examples demonstrate the effectiveness of our proposed algorithms, especially in high dimensional cases. And an important application of this method is to calculate the sub-linear expectations, which correspond to a kind of fully nonlinear PDEs.