A leavable bounded-velocity stochastic control problem

This paper studies bounded-velocity control of a Brownian motion when discretionary stopping, or 'leaving', is allowed. The goal is to choose a control law and a stopping time in order to minimize the expected sum of a running and a termination cost, when both costs increase as a function of distance from the origin. There are two versions of this problem: the fully observed case, in which the control multiplies a known gain, and the partially observed case, in which the gain is random and unknown. Without the extra feature of stopping, the fully observed problem originates with Beneš (Stochastic Process. Appl. 2 (1974) 127-140), who showed that the optimal control takes the 'bang-bang' form of pushing with maximum velocity toward the origin. We show here that this same control is optimal in the case of discretionary stopping; in the case of power-law costs, we solve the variational equation for the value function and explicitly determine the optimal stopping policy. We also discuss qualitative features of the solution for more general cost structures. When no discretionary stopping is allowed, the partially observed case has been solved by Beneš et al. (Stochastics Monographs, Vol. 5, Gordon & Breach, New York and London, pp. 121-156) and Karatzas and Ocone (Stochastic Anal. Appl. 11 (1993) 569-605). When stopping is allowed, we obtain lower bounds on the optimal stopping region using stopping regions of related, fully observed problems.