Two-sided reflected Markov-modulated Brownian motion with applications to fluid queues and dividend payouts

TL;DR

This paper analyzes a two-sided reflected Markov-modulated Brownian motion with stochastic boundaries and jumps, providing a framework to compute its stationary distribution, with applications to fluid queues and dividend payout strategies.

Contribution

It extends existing models by deriving the stationary distribution for a Markov-modulated Brownian motion with stochastic boundaries and control, including the case with two states.

Findings

Derived the stationary distribution for the process.

Extended results to the case with controlled upper boundary.

Applied the model to dividend payout strategies.

Abstract

In this paper we study a reflected Markov-modulated Brownian motion with a two sided reflection in which the drift, diffusion coefficient and the two boundaries are (jointly) modulated by a finite state space irreducible continuous time Markov chain. The goal is to compute the stationary distribution of this Markov process, which in addition to the complication of having a stochastic boundary can also include jumps at state change epochs of the underlying Markov chain because of the boundary changes. We give the general theory and then specialize to the case where the underlying Markov chain has two states. Moreover, motivated by an application of optimal dividend strategies, we consider the case where the lower barrier is zero and the upper barrier is subject to control. In this case we generalized earlier results from the case of a reflected Brownian motion to the Markov modulated case.