TY - JOUR

T1 - Time-dependent and stationary analyses of two-sided reflected Markov-modulated Brownian motion with bilateral ph-type jumps

AU - Ahn, Soohan

N1 - Publisher Copyright:
© 2016 The Korean Statistical Society

PY - 2017/3/1

Y1 - 2017/3/1

N2 - 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.

AB - 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.

KW - Markov renewal

KW - Markov-modulated Brownian motion

KW - Markov-modulated fluid flow

KW - Skip-free level crossing

KW - Two-sided reflection

KW - Weak convergence

UR - http://www.scopus.com/inward/record.url?scp=84996835581&partnerID=8YFLogxK

U2 - 10.1016/j.jkss.2016.06.002

DO - 10.1016/j.jkss.2016.06.002

M3 - Article

AN - SCOPUS:84996835581

SN - 1226-3192

VL - 46

SP - 45

EP - 69

JO - Journal of the Korean Statistical Society

JF - Journal of the Korean Statistical Society

IS - 1

ER -