A quadratically convergent algorithm for first passage time distributions in the Markov-modulated Brownian motion

Abstract

ABSTRACTA 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 with Brownian Motion, the time-dependent analysis of the MMBM becomes easy once the first passage times between levels are determined. However, in the MMBM those distributions cannot be obtained explicitly, and we need efficient algorithms to compute them. In this article, we provide a powerful approach based on approximating the MMBM with a sequence of scaled Markov-modulated fluid flows without Brownian components that weakly converge to the MMBM. Our main result is a Riccati equation for an associated matrix of transforms that satisfies conditions for the Newton scheme to have quadratic convergence and thus yields a very practical tool. The solution of that Riccati equation determines needed first passage times in the MMBM without much additional work. The success of our approach, which is based essentially on first-order fluid flows and a stochastic limit process, is argued to be due to the way we have isolated certain terms involving the quadratic variation effects of the Brownian. As an illustration of our algorithm, we present a numerical example of time-dependent results for a MMBM considered by Asmussen for which he determined (only) the eventual first return probabilities which we use here as an accuracy check.