Probabilistic Solutions to Optimal Control Problems

Abstract

Let (x 1 (t), x 2 (t)) be a controlled two-dimensional diusion process. The problem of minimizing, or maximizing, the time spent by (x 1 (t), x 2 (t)) in a given subset of R 2 is solved, in two particular instances, by transforming the optimal control problems into purely probabilistic problems. In Section 2, (x 1 (t), x 2 (t)) is a two-dimensional Wiener process and the optimal control is obtained by transforming a nonlinear dynamic programming equation into the Kolmogorov backward equation for a two-dimensional geometric Brownian motion. In Section 3, the converse problem is solved. The problem of ®nding the maximal instantaneous reward that we can give for survival in the continuation region is also treated.