An Analysis of Monotone Follower Problems for Diffusion Processes

Abstract

We consider a singular stochastic control problem, which is called the Monotone Follower Stochastic Control Problem and give sufficient conditions for the existence and uniqueness of a local-time type optimal control. To establish this result we use a methodology that has not been employed to solve singular control problems. We first confine ourselves to local time strategies. Then we apply a transformation to the total reward accrued by reflecting the diffusion at a given boundary and show that it is linear in its continuation region. Now, the problem of finding the optimal boundary becomes a non-linear optimization problem: The slope of the linear function and an obstacle function need to be simultaneously maximized. The necessary conditions of optimality come from first order derivative conditions. We show that under some weak assumptions these conditions become sufficient. We also show that the local time strategies are optimal in the class of all monotone increasing controls. As a byproduct of our analysis, we give sufficient conditions for the value function to be $\mathbf{C}^2$ on all its domain. We solve two dividend payment problems to show that our sufficient conditions are satisfied by the examples considered in the mainstream literature. We show that our assumptions are satisfied not only when capital of a company is modeled by a Brownian motion with drift but also when we change the modeling assumptions and use a square root process to model the capital.