Finite-Fuel Singular Control With Discretionary Stopping

Abstract

We discuss the finite-fuel, singular stochastic control problem of optimally tracking the standard Brownian motion started at , by an adapted process of bounded total variation , so as to minimize the total expected discounted cost over such processes and stopping times τ. Here , and are given real numbers. In its form this problem goes back to the seminal paper of Bene[sbreve], Shepp and Witsenhausen (1980). For fixed α>0 and δ>0 we characterize explicitly the optimal policy in the case λ>αδ (of the “act-or-stop” type, since the continuation cost is relatively large), and in the case with (of the “act, stop, or wait” type, since the relative continuation cost is relatively small). In the latter case, an associated free-boundary problem is solved exactly. The case , of “moderate” relative continuation cost, is suggested as an open question