Optimal stopping and free boundary characterizations for some Brownian control problems

TL;DR

This paper characterizes the optimal control and free boundary in a two-dimensional Brownian control problem with state constraints, proving regularity results and confirming a conjecture about the control structure.

Contribution

It establishes the regularity of the value function and free boundary, and confirms a conjecture on the form of optimal controls in a class of singular control problems.

Findings

Value function is $C^1$ and $C^2$ in the no-action region.

Free boundary is Lipschitz continuous.

Optimal control involves Brownian motion with reflection at the free boundary.

Abstract

A singular stochastic control problem with state constraints in two-dimensions is studied. We show that the value function is $C^1$ and its directional derivatives are the value functions of certain optimal stopping problems. Guided by the optimal stopping problem, we then introduce the associated no-action region and the free boundary and show that, under appropriate conditions, an optimally controlled process is a Brownian motion in the no-action region with reflection at the free boundary. This proves a conjecture of Martins, Shreve and Soner [SIAM J. Control Optim. 34 (1996) 2133--2171] on the form of an optimal control for this class of singular control problems. An important issue in our analysis is that the running cost is Lipschitz but not $C^1$. This lack of smoothness is one of the key obstacles in establishing regularity of the free boundary and of the value function. We show that the free boundary is Lipschitz and that the value function is $C^2$ in the interior of the no-action region. We then use a verification argument applied to a suitable $C^2$ approximation of the value function to establish optimality of the conjectured control.