TY - JOUR

T1 - Matrix equations in Markov modulated Brownian motion

T2 - theoretical properties and numerical solution

AU - Ahn, Soohan

AU - Meini, Beatrice

N1 - Publisher Copyright:
© 2019, © 2019 Taylor & Francis Group, LLC.

PY - 2020/4/2

Y1 - 2020/4/2

N2 - A Markov modulated Brownian motion (MMBM) is a substantial generalization of the classical Brownian motion and is obtained by allowing the Brownian parameters to be modulated by an underlying Markov chain of environments. As in Brownian motion, the stationary analysis of the MMBM becomes easy once the distributions of the first passage time between levels are determined. Asmussen (Stochastic Models, 1995) proved that such distributions can be obtained by solving a suitable quadratic matrix equation (QME), while, more recently, Ahn and Ramaswami (Stochastic Models, 2017) derived the distributions from the solution of a suitable algebraic Riccati equation (NARE). In this paper we provide an explicit algebraic relation between the QME and the NARE, based on a linearization of a matrix polynomial. Moreover, we discuss the doubling algorithms such as the structure-preserving doubling algorithm (SDA) and alternating-directional doubling algorithm (ADDA), with shifting technique, which are used for finding the sought of the NARE.

AB - A Markov modulated Brownian motion (MMBM) is a substantial generalization of the classical Brownian motion and is obtained by allowing the Brownian parameters to be modulated by an underlying Markov chain of environments. As in Brownian motion, the stationary analysis of the MMBM becomes easy once the distributions of the first passage time between levels are determined. Asmussen (Stochastic Models, 1995) proved that such distributions can be obtained by solving a suitable quadratic matrix equation (QME), while, more recently, Ahn and Ramaswami (Stochastic Models, 2017) derived the distributions from the solution of a suitable algebraic Riccati equation (NARE). In this paper we provide an explicit algebraic relation between the QME and the NARE, based on a linearization of a matrix polynomial. Moreover, we discuss the doubling algorithms such as the structure-preserving doubling algorithm (SDA) and alternating-directional doubling algorithm (ADDA), with shifting technique, which are used for finding the sought of the NARE.

KW - Doubling algorithm

KW - Markov modulated Brownian motion

KW - first passage time distribution

KW - matrix polynomials

KW - nonsymmetric algebraic Riccati equation

KW - quadratic convergence

KW - quadratic matrix equation

UR - http://www.scopus.com/inward/record.url?scp=85077872787&partnerID=8YFLogxK

U2 - 10.1080/15326349.2019.1704785

DO - 10.1080/15326349.2019.1704785

M3 - Article

AN - SCOPUS:85077872787

SN - 1532-6349

VL - 36

SP - 251

EP - 284

JO - Stochastic Models

JF - Stochastic Models

IS - 2

ER -