Time-dependent and stationary analyses of two-sided reflected Markov-modulated Brownian motion with bilateral ph-type jumps

Abstract

A Markov-modulated Brownian motion with bilateral ph-type jumps, referred to as MMBM, is a generalization of the Lévy process. In this paper, we study the time-dependent behavior of the two-sided reflected MMBM (TR-MMBM) with boundaries 0 and β>0. In contrast to previous research on the subject, we propose a different approach based on the observation that the TR-MMBM can be realized as the limit of a sequence of two-sided reflected Markov-modulated fluid flows with bilateral ph-type jumps (TR-MMFF), which are MMBMs without a Brownian component. Therefore, the TR-MMBM can be analyzed via methods for the TR-MMFFs, through limiting arguments based on the weak convergence and continuous mapping theorems. Along these lines, we first analyze time-dependent behaviors of the sequence of TR-MMFFs using a new methodology that adopts the so-called completed graph and also using Markov renewal and skip-free level crossing arguments. Then, relying on the appropriate stochastic limit arguments, we finally present the Laplace transform of the time-dependent distribution of the TR-MMBM with respect to time. In addition, we show that the stationary distribution of the TR-MMBM can be obtained directly from the Laplace transform.