TY - JOUR

T1 - Total shift during the first passages of Markov-modulated Brownian motion with bilateral ph-type jumps

T2 - Formulas driven by the minimal solution matrix of a Riccati equation

AU - Ahn, Soohan

N1 - Publisher Copyright:
© 2016 Taylor & Francis.

PY - 2016/7/2

Y1 - 2016/7/2

N2 - ABSTRACT: This article describes our study of the total shift during the first passages (one-sided and two-sided exit times) of Markov-modulated Brownian motion with bilateral ph-type jumps, which is referred to as MMBM. The total shift is defined as the value of a so-called shift process at the first passage epochs of the MMBM. The shift process, introduced by Bean and O’Reilly, behaves like a continuous Markovian fluid process; that is, it increases or decreases linearly with slopes regulated by the underlying Markov process that determines the path of the MMBM. Hence, the notion of total shift, which includes the first passage times of the MMBM as special cases, is useful for describing various performance measures of systems modeled by the MMBM. In this article, we present formulas for the Laplace–Stieltjes transform matrices of the total shift during various first passages of the MMBM. In particular, a Riccati equation is derived so that a matrix associated with the Laplace–Stieltjes transform of the total shift during the first return time of the MMBM is its minimal non-negative solution matrix. With this solution matrix, the Laplace–Stieltjes transform matrices can be obtained without much additional work. Furthermore, it is shown that the Riccati equation satisfies the conditions for the Newton scheme to have quadratic convergence, which enables us to use algorithms with quadratic convergence, such as Newton’s method and the Stochastic Doubling Algorithm, to compute the presented matrix-driven formulas. For the analyses, we take an approach based on approximating the MMBM with a sequence of scaled Markov-modulated fluid flows with bilateral ph-type jumps, referred to as MMFF, that weakly converge to the MMBM. Another contribution of this article is that duality results are derived in relation to the MMBM, which is an extension of the duality theorems developed by Ahn and Ramaswami for an MMFF without a jump.

AB - ABSTRACT: This article describes our study of the total shift during the first passages (one-sided and two-sided exit times) of Markov-modulated Brownian motion with bilateral ph-type jumps, which is referred to as MMBM. The total shift is defined as the value of a so-called shift process at the first passage epochs of the MMBM. The shift process, introduced by Bean and O’Reilly, behaves like a continuous Markovian fluid process; that is, it increases or decreases linearly with slopes regulated by the underlying Markov process that determines the path of the MMBM. Hence, the notion of total shift, which includes the first passage times of the MMBM as special cases, is useful for describing various performance measures of systems modeled by the MMBM. In this article, we present formulas for the Laplace–Stieltjes transform matrices of the total shift during various first passages of the MMBM. In particular, a Riccati equation is derived so that a matrix associated with the Laplace–Stieltjes transform of the total shift during the first return time of the MMBM is its minimal non-negative solution matrix. With this solution matrix, the Laplace–Stieltjes transform matrices can be obtained without much additional work. Furthermore, it is shown that the Riccati equation satisfies the conditions for the Newton scheme to have quadratic convergence, which enables us to use algorithms with quadratic convergence, such as Newton’s method and the Stochastic Doubling Algorithm, to compute the presented matrix-driven formulas. For the analyses, we take an approach based on approximating the MMBM with a sequence of scaled Markov-modulated fluid flows with bilateral ph-type jumps, referred to as MMFF, that weakly converge to the MMBM. Another contribution of this article is that duality results are derived in relation to the MMBM, which is an extension of the duality theorems developed by Ahn and Ramaswami for an MMFF without a jump.

KW - Bilateral ph-type jump

KW - Markov-modulated Brownian motion

KW - Markov-modulated fluid flow

KW - Riccati equation

KW - first passage time

KW - total shift

UR - http://www.scopus.com/inward/record.url?scp=84963584395&partnerID=8YFLogxK

U2 - 10.1080/15326349.2016.1165621

DO - 10.1080/15326349.2016.1165621

M3 - Article

AN - SCOPUS:84963584395

SN - 1532-6349

VL - 32

SP - 433

EP - 459

JO - Stochastic Models

JF - Stochastic Models

IS - 3

ER -