Optimal switching between a pair of Brownian motions.

Consider a pair of Brownian motions B sub s sup (1) on the interval [0,1] with absorption at the endpoints. The joint state space is the square E = [0,1]x[0,1]. The time evolution of the two processes can be controlled separately: i.e., we can let the B sub s sup (1) process run and freeze the B sub t sup (2) process to obtain horizontal Brownian motion, or we can let the B sub t sup (2) process run and freeze B sub s sup (1) giving us vertical Brownian motion. We assume that there is a pay-off function f(x sub 1, x sub 2) defined on the boundary of E. The objective is to find the optimal strategy for controlling the time evolution so as to maximize the expected pay-off obtained at the hitting time of the boundary. The optimal strategy is determined by a partition of the interior of the state space into two sets: horizontal control and vertical control. We will give a rather explicit characterization of these sets.