Singular stochastic control and optimal stopping

Abstract

The problem of controlling aBrownian motion by a process of bounded variation on a finite time-horizon, with convex, symmetric running and terminal state costs and linear cost of control, is related to a problem of optimal stopping for Brownian motion, with absorption at the origin. The connection between the two problems is established by probabilistic methods. First we establish an analogous relationship between a modified control problem, where we impose a reflecting barrier at the origin, and the aforementioned stopping problem. The remaining problem is then to show that the reflected and non-reflected control problems have the same value function. This is done by an application of the Tanaka-Meyer formula for semimartingales, which enables us to represent the absolute value of a state process in the non-reflected problem as a state process in the weak sense, we show that the converse is also true, in a certain sense, when the control law is given by a bounded Markovian drift. Fundamental to our approach is the approximation of control processes in a certain complete class for the reflected problem, by Lipschitz continuous control processes with a uniformly bounded time-derivative; as well as a new technique of constructing the solution to the discontinuous reflection problem