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

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.

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Acknowledgement

The author is grateful to Professors Mu-Fa Chen and Yong-Hua Mao for their valuable discussion.

Author information

Correspondence to Jinghai Shao.

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Shao, J. Ergodicity and First Passage Probability of Regime-Switching Geometric Brownian Motions. Chin. Ann. Math. Ser. B 39, 739–754 (2018). https://doi.org/10.1007/s11401-018-0093-5

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Keywords

  • Ergodicity
  • Regime-switching diffusions
  • Lyapunov functions
  • First passage probability

2000 MR Subject Classification

  • 60A10
  • 60J60
  • 60J10