Numerical Solution of Stochastic Control Problems Using the Finite Element Method

Abstract

Based on linear programming formulations for infinite horizon stochastic control problems, a numerical technique in fashion of the finite element method is developed. The convergence of the approximate scheme is shown and its performance is illustrated on multiple examples. This thesis begins with an introduction of stochastic optimal control and a review of the theory of the linear programming approach. The analysis of existence and uniqueness of solutions to the linear programming formulation for fixed controls represents the first contribution of this work. Then, an approximate scheme for the linear programming formulations is established. To this end, a novel discretization of the involved measures and constraints using finite dimensional function subspaces is introduced. Its convergence is proven using weak convergence of measures, and a detailed analysis of the approximate relaxed controls. The applicability of the established method is shown through a collection of examples from stochastic control. The considered examples include models with bounded or unbounded state space, models featuring continuous and singular control as well as discounted or long-term average cost criteria. Analyses of various model parameters are given, and in selected examples, the approximate solutions are compared to available analytic solutions. A summary and an outlook on possible research directions is given.