A Degenerate Variance Control Problem with Discretionary Stopping

Abstract

We consider an infinite horizon stochastic control problem with discretionary stopping. The state process is given by a one dimensional stochastic differential equation. The diffusion coefficient is chosen by an adaptive choice of the controller and it is allowed to take the value zero. The controller also chooses the quitting time to stop the system. Here we develop a martingale characterization of the value function and use it and the principle of smooth fit to derive an explicit optimal strategy when the drift coefficient of the state process is of the form b(x)=−θx where θ>0 is a constant.