The dividend problem with a finite horizon

TL;DR

This paper characterizes the optimal dividend strategy over a finite horizon using Hamilton-Jacobi-Bellman equations and introduces a novel connection between singular control and optimal stopping problems involving reflected diffusions.

Contribution

It provides a new characterization of the value function and optimal strategy for the finite-horizon dividend problem, linking singular control with reflected diffusion and stopping problems.

Findings

Unique classical solution to the HJB equation for the problem.

Optimal strategy involves Skorokhod reflection at a time-dependent boundary.

New connection established between control problems and reflected diffusions.

Abstract

We characterise the value function of the optimal dividend problem with a finite time horizon as the unique classical solution of a suitable Hamilton-Jacobi-Bellman equation. The optimal dividend strategy is realised by a Skorokhod reflection of the fund's value at a time-dependent optimal boundary. Our results are obtained by establishing for the first time a new connection between singular control problems with an absorbing boundary and optimal stopping problems on a diffusion reflected at $0$ and created at a rate proportional to its local time.