Ergodicity and first passage probability of regime-switching geometric Brownian motions

TL;DR

This paper analyzes a regime-switching geometric Brownian motion, characterizing its recurrence, long-term behavior, and first passage probabilities, highlighting differences from standard models without switching.

Contribution

It provides a complete characterization of recurrence and long-term properties of regime-switching geometric Brownian motions, including explicit estimates of first passage probabilities.

Findings

Characterization of recurrence properties

Analysis of long-term moments

Estimates of first passage probabilities

Abstract

A regime-switching geometric Brownian motion is used to model a geometric Brownian motion with its coefficients changing randomly according to a Markov chain. In this work, we give a complete characterization of the recurrent property of this process. The long time behavior of this process such as its $p$-th moment is also studied. Moreover, the quantitative properties of the regime-switching geometric Brownian motion with two-state switching are investigated to show the difference between geometric Brownian motion with switching and without switching. At last, some estimates of its first passage probability are established.