Some explicit expressions for GBM with Markovian switching and parameter estimations

Abstract

In this paper, we focus on the mean exit time and the scale function for the geometric Brownian motion with Markovian switching, in which the drift coefficients and the diffusion coefficients are associated with regime changes. The explicit expressions of mean exit time and scale function are obtained by solving the corresponding Poisson problem. Furthermore, we estimate parameters for the geometric Brownian motion with Markovian switching by composite likelihood and explore some properties of the estimates. Computer simulations are performed to illustrate our proposed algorithm, showing high accuracy of the estimators.

Keywords:

• Geometric Brownian motion
• Markovian switching
• mean exit time
• parameter estimations

Acknowledgments

The authors are grateful to the anonymous reviewers, Dr. Yaqing Sun and Zinan Zhang for their valuable comments and suggestions which led to improvements in this manuscript. The research of Z. Zhang was partially supported by the Natural Science Foundation of China [No. 12171081] and by the Fundamental Research Funds for the Central Universities with No.2232021G-13. The research of J. Tong was partially supported by the Natural Science Foundation of Shanghai [No. 19ZR1400600] and the Fundamental Research Funds for the Central Universities. The research of Z. Qin was partially supported by the Multi-Year Research Grant by University of Macau [MYRG2018-00210-FBA].