Ergodicity and First Passage Probability of Regime-Switching Geometric Brownian Motions

Ergodicity and First Passage Probability of Regime-Switching Geometric Brownian Motions??

Jinghai SHAO1

【摘 要】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,the author gives 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.【期刊名称】《数学年刊B辑（英文版）》【年(卷),期】2018(039)004【总页数】16

【关键词】Keywords Ergodicity,Regime-switching diffusions,Lyapunov functions,First passage probability

2010 MR Subject Classification 60A10,60J60,60J10Manuscript received September 17,2014.RevisedMay 6,2016.

1Center for Applied Mathematics,Tianjin University,Tianjin 300072,China.E-mail:shaojh@bnu.edu.cn

??This work was supported by the National Natural Science Foundation of China(Nos.11301030,11431014).

1 Introduction

In the study of mathematical finance,geometric Brownian motion(GBM for short)is used to model stock prices in the Black-Scholes model and is the most widely used model of stock price behavior.Let(Zt)be the solution of following stochastic differential equation(SDE for short):

with Z0=a>0,whereμ,σ are constants,and(Bt)is a one-dimensional Brownian motion.But