On the Singular Control of a Diffusion and Its Running Infimum or Supremum

TL;DR

This paper develops new integral operators and a verification theorem to solve singular control problems involving a diffusion process and its running extremum, with applications to optimal dividend strategies.

Contribution

Introduction of two novel integral operators and a verification theorem for two-dimensional singular control problems involving the process and its extremum.

Findings

Derived explicit solutions for a dividend optimization problem.

Established a general framework for control problems with extremum-dependent costs.

Validated the approach through application to a real-world financial problem.

Abstract

We study a class of singular stochastic control problems for a one-dimensional diffusion $X$ in which the performance criterion to be optimised depends explicitly on the running infimum $I$ (or supremum $S$) of the controlled process. We introduce two novel integral operators that are consistent with the Hamilton-Jacobi-Bellman equation for the resulting two-dimensional singular control problems. The first operator involves integrals where the integrator is the control process of the two-dimensional process $(X,I)$ or $(X,S)$; the second operator concerns integrals where the integrator is the running infimum or supremum process itself. Using these definitions, we prove a general verification theorem for problems involving two-dimensional state-dependent running costs, costs of controlling the process, costs of increasing the running infimum (or supremum) and exit times. Finally, we apply our results to explicitly solve an optimal dividend problem in which the manager's time-preferences depend on the company's historical worst performance.