Numerical Solution of Stochastic Control Problems Using the Finite Element Method

Martin Vieten
University of Wisconsin-Milwaukee
PhD Graduate Student

“The linear programming approach to stochastic control has provided insight on a wide variety of control problems, including singular control and optimal stopping. The specification of the dynamics in the linear programming set up is given by an operator-integral equation for the occupation measures of the controlled process. This structure can be exploited to establish a numerical approximation scheme by borrowing ideas from the finite element method. Research by Kaczmarek, Rus et. al. has indicated strong numerical performance of such an approximation, but failed to provide a complete convergence argument.
This thesis presents an adjusted finite element approximation scheme for the linear programming approach in stochastic control, and shows its convergence for models featuring an infinite time horizon, with either a bounded or unbounded state space, and singular behavior. Both the optimization of controls as well as the evaluation of cost criteria for fixed  controls are analyzed. The performance of the method is illustrated with several examples, including the classic `Modified Bounded Follower’, which has an analytic solution, models from finance, and more”.

Committee Members:
Profs. Richard Stockbridge (Advisor); Gerhard Dikta, Bruce Wade, Lei Wang, & Chao Zhu