Markov-modulated Brownian motion with two reflecting barriers

TL;DR

This paper analyzes a Markov-modulated Brownian motion with two reflecting barriers, providing a probabilistic proof of its stationary distribution and exploring its transient behavior and distribution at exponential epochs.

Contribution

It offers a simple probabilistic argument for the stationary distribution and extends the analysis to the distribution at exponential epochs and transient behavior.

Findings

Stationary distribution has a simple form under certain assumptions.

Probabilistic proof elucidates the structure of the stationary distribution.

Distribution at exponential epochs and transient behavior are characterized.

Abstract

We consider a Markov-modulated Brownian motion reflected to stay in a strip [0,B]. The stationary distribution of this process is known to have a simple form under some assumptions. We provide a short probabilistic argument leading to this result and explaining its simplicity. Moreover, this argument allows for generalizations including the distribution of the reflected process at an independent exponentially distributed epoch. Our second contribution concerns transient behavior of the reflected system. We identify the joint law of the processes t,X(t),J(t) at inverse local times.