Singular Stochastic Control, Linear Diffusions, and Optimal Stopping: A Class of Solvable Problems

Abstract

We consider a class of singular stochastic control problems arising frequently in applications of stochastic control. We state a set of conditions under which the optimal policy and its value can be derived in terms of the minimal r-excessive functions of the controlled diffusion, and demonstrate that the optimal policy is of the standard local time type. We then state a set of weak smoothness conditions under which the value function is increasing and concave, and demonstrate that given these conditions increased stochastic fluctuations decrease the value and increase the optimal threshold, thus postponing the exercise of the irreversible policy. In line with previous studies of singular stochastic control, we also establish a connection between singular control and optimal stopping, and show that the marginal value of the singular control problem coincides with the value of the associated stopping problem whenever 0 is not a regular boundary for the controlled diffusion.