In this paper, we consider some properties of switching Brownian motion. Combining the analytic method and probabilistic method, some explicit expressions of density functions, the mean exit time and Laplace transform of exit time are given. This paper reveals how drift coefficients impact the first passage probabilities, scale functions and the mean exit time for switching Brownian motion.

Keywords:

AMSC: 91B02, 60J05, 60H10

References

1. A. Eizenberg and M. Freidlin, On the Dirichlet problem for a class of second order PDE systems with small parameter, Stoch. Stoch. Rep. 33 (1990) 111–148. Crossref, Google Scholar
2. L. C. Evans, Partial Differential Equations (American Mathematical Society, 1997). Google Scholar
3. X. Guo, An explicit solution to an optimal stopping problem with regime switching, J. Appl. Probab. 38 (2001) 464–481. Crossref, Web of Science, Google Scholar
4. P. Hieber and M. Scherer, A note on first-passage times of continuously time-changed Brownian motion, Stat. Probab. Lett. 82 (2012) 165–172. Crossref, Web of Science, Google Scholar
5. P. Hieber, First-passage times of regime switching models, Stat. Probab. Lett. 92 (2014) 148–157. Crossref, Web of Science, Google Scholar
6. P. Hieber, A correction note on: When the “Bull” meets the “Bear”-A first passage time problem for a hidden Markov process, Methodol. Comput. Appl. 16 (2014) 771–776. Crossref, Web of Science, Google Scholar
7. P. Khasminskii, Stochastic Stability of Differential Equations (Springer, Berlin, 2011). Google Scholar
8. X. Mao and C. Yuan, Stochastic Differential Equations with Markov Switching (Imperial College Press, 2006). Link, Google Scholar
9. D. Revuz and M. Yor, Continuous Martingales and Brownian Motion (Springer, 1999). Crossref, Google Scholar
10. A. V. Skorohod, Asymptotic Methods in Theory of Stochastic Differential Equations (American Mathematical Society, 1989). Google Scholar
11. J. Tong, Q. Meng, Z. Zhang and Y. Lu, A note on ergodicity for CIR model with Markov switching, Commun. Stat.-Simul. C. 50 (2021) 1445–1458. Crossref, Web of Science, Google Scholar
12. G. Yin and C. Zhu, Hybrid switching diffusions: Properties and Applications (Springer, Berlin, 2010). Crossref, Google Scholar
13. Z. Zhang, J. Tong and L. Hu, Ultracontractivity for Brownian motion with Markov switching, Stoch. Anal. Appl. 37 (2019) 1–13. Crossref, Web of Science, Google Scholar
14. Z. Zhang, M. Zhai, J. Tong and Q. Zhang, Some characterizations for Brownian motion with Markov switching, Nonlinear Anal.-Hybri. 42 (2021) 101086. Crossref, Web of Science, Google Scholar
15. Z. Zhang, X. Wang, J. Tong, T. Zhou and Z. Qin, Some explicit expressions for GBM with Markovian switching and parameter estimations, Commun. Stat.-Theor. M., https://doi.org/10.1080/03610926.2022.2100908. Google Scholar