Matrix equations in Markov modulated Brownian motion: theoretical properties and numerical solution

Abstract

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.

Keywords:

• Doubling algorithm
• first passage time distribution
• Markov modulated Brownian motion
• matrix polynomials
• nonsymmetric algebraic Riccati equation
• quadratic convergence
• quadratic matrix equation

Additional information

Funding

Soohan Ahn was supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (grant number NRF-2018R1D1A1A09082881). Beatrice Meini was supported by INdAM-GNCS