Exact solutions to two-dimensional homing problems

Abstract

Two-dimensional controlled Wiener and Bessel di usion processes are considered in circles. Both the in®nitesimal means and variances of the controlled processes depend on the control variables. The processes are controlled until they hit the circles for the ®rst time. The objective is to minimize the expected value of the time spent by the controlled processes inside the circles, while taking the control costs into account. Explicit and exact expressions are obtained for the optimal values of the control variables as well as for the value functions. The method of similarity solutions is used to solve the appropriate dynamic programming equation.