Solving Elliptic Hamilton-Jacobi-Bellman Equations in a Value Space

Abstract

In this letter, we study the numerical solutions of a class of elliptic type of Hamilton-Jacobi-Bellman (HJB) equations, which arise from stochastic control and multiplayer stochastic game problems with random terminal time. Two closely related algorithms are proposed. One of them is to use value iteration on approximating Markov decision processes, and the other is to use a deep-learning approach solving Bellman equations. The convergence is shown by using a viscosity solution approach.