On non-trivial barrier solutions of the dividend problem for a diffusion under constant and proportional transaction costs

TL;DR

This paper investigates complex optimal dividend strategies for diffusion processes under constant and proportional transaction costs, extending previous work by identifying non-trivial barrier solutions where simple strategies are suboptimal.

Contribution

It provides solutions for cases where simple barrier policies are not optimal, revealing more complex optimal dividend strategies under transaction costs.

Findings

Simple barrier strategies are not always optimal.

Complex, non-trivial barrier solutions can be optimal.

The solutions are more intricate but mathematically interesting.

Abstract

In Bai and Paulsen (SIAM J. Control optim. 48, 2010) the optimal dividend problem under transaction costs was analyzed for a rather general class of diffusion processes. It was divided into several subclasses, and for the majority of subclasses the optimal policy is a simple barrier policy; whenever the process hits an upper barrier $\bar{u}^*$, reduce it to $\bar{u}^*-\xi$ through a dividend payment. After transaction costs, the shareholder receives $k\xi-K$. It was proved that a simple barrier strategy is not always optimal, and here these more difficult cases are solved. The optimal solutions are rather complicated, but interesting.