Probabilistic solutions to optimal control problems

Let (x(1)(t),x(2)(t)) be a controlled two-dimensional diffusion 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 finding the maximal instantaneous reward that we can give for survival in the continuation region is also treated.