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Application of GTH-like algorithm to Markov modulated Brownian motion with jumps

Sung-Chul Honga, Soohan Ahn1,a

aDepartment of Statistics, University of Seoul, Korea
Correspondence to: 1 Department of Statistics, University of Seoul, 163 Seoulsiripdaero, Dongdaemun-gu, Seoul 02504, South Korea. E-mail: sahn@uos.ac.kr
This work was supported by the 2018 Research Fund of the University of Seoul.
Received February 15, 2021; Revised April 12, 2021; Accepted April 13, 2021.
Abstract
The Markov modulated Brownian motion is a substantial generalization of the classical Brownian Motion. On the other hand, the Markovian arrival process (MAP) is a point process whose family is dense for any stochastic point process and is used to approximate complex stochastic counting processes. In this paper, we consider a superposition of the Markov modulated Brownian motion (MMBM) and the Markovian arrival process of jumps which are distributed as the bilateral ph-type distribution, the class of which is also dense in the space of distribution functions defined on the whole real line. In the model, we assume that the inter-arrival times of the MAP depend on the underlying Markov process of the MMBM. One of the subjects of this paper is introducing how to obtain the first passage probabilities of the superposed process using a stochastic doubling algorithm designed for getting the minimal solution of a nonsymmetric algebraic Riccatti equation. The other is to provide eigenvalue and eigenvector results on the superposed process to make it possible to apply the GTH-like algorithm, which improves the accuracy of the doubling algorithm.
Keywords : Markov modulated Brownian motion, Markovian arrival process, first passage time, doubling algorithm, nonsymmetric algebraic Riccati equation, GTH-like algorithm