Rate of Strong Convergence to Markov-modulated Brownian motion

TL;DR

This paper introduces a new stochastic fluid process sequence that converges strongly to a Markov-modulated Brownian motion, providing the first known proof of such strong convergence and establishing its rate.

Contribution

It constructs a novel stochastic fluid process sequence that converges strongly to MMBM and determines its convergence rate, improving upon previous approximation methods.

Findings

First proof of strong convergence to MMBM

Established convergence rate of o(n^{-1/2} log n)

Improved convergence rate for Brownian motion case

Abstract

In Latouche and Nguyen (2015), the authors constructed a sequence of stochastic fluid processes and showed that it converges weakly to a Markov-modulated Brownian motion (MMBM). Here, we construct a different sequence of stochastic fluid processes and show that it converges strongly to an MMBM. To the best of our knowledge, this is the first result on strong convergence to a Markov-modulated Brownian motion. We also prove that the rate of this almost sure convergence is $o(n^{-1/2} \log n)$. When reduced to the special case of standard Brownian motion, our convergence rate is an improvement over that obtained by a different approximation in \cite{gorostiza1980rate}, which is $o(n^{-1/2}(\log n)^{5/2})$.