Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-21T13:14:33.759Z Has data issue: false hasContentIssue false
  • English
  • Français

Long-time behavior of Lévy-driven Ornstein–Uhlenbeck processes with regime switching

Part of: Special processes Markov processes

Published online by Cambridge University Press:  04 May 2020

Zhongwei Liao  and
Jinghai Shao Open the ORCID record for Jinghai Shao [Opens in a new window]
Zhongwei Liao*
Affiliation:
South China Normal University
Jinghai Shao*
Affiliation:
Tianjin University
*
*Postal address: South China Research Center for Applied Mathematics and Interdisciplinary Studies, South China Normal University, Guangzhou 510631, China. Email address: zhwliao@hotmail.com
**Postal address: Center for Applied Mathematics, Tianjin University, Tianjin 300072, China. Email address: shaojh@tju.edu.cn
Get access Rights & Permissions [Opens in a new window]

Abstract

We investigate the long-time behavior of the Ornstein–Uhlenbeck process driven by Lévy noise with regime switching. We provide explicit criteria on the transience and recurrence of this process. Contrasted with the Ornstein–Uhlenbeck process driven simply by Brownian motion, whose stationary distribution must be light-tailed, both the jumps caused by the Lévy noise and the regime switching described by a Markov chain can derive the heavy-tailed property of the stationary distribution. The different role played by the Lévy measure and the regime-switching process is clearly characterized.

Keywords

Ornstein–Uhlenbeck processLévy processregime switchingrecurrence

MSC classification

Primary: 60J75: Jump processes
Secondary: 60K37: Processes in random environments 60J60: Diffusion processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 57 , Issue 1 , March 2020 , pp. 266 - 279
Copyright
© Applied Probability Trust 2020

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Supported in part by NNSFS of China (nos. 11771327, 11701588, 11831014).

References

Bardet, J., Guérin, H. & Malrieu, F. (2010). Long time behavior of diffusions with Markov switching. ALEA Lat. Am. J. Probab. Math. Stat. 7, 151–170.Google Scholar
Barndorff-Nielsen, O. E., Mikosch, T. & Resnick, S. I. (2001). Lévy Processes: Theory and Applications. Birkhäuser, Boston.CrossRefGoogle Scholar
Barndorff-Nielsen, O. E. & Shephard, N. (2001). Non-Gaussian Ornstein–Uhlenbeck-based models and some of their uses in financial economics. J. R. Statist. Soc. B [Statist. Methodology] 63, 167–207.CrossRefGoogle Scholar
Brown, S. J. & Dybvig, P. H. (1986). The empirical implications of the Cox, Ingersoll, Ross theory of the term structure of interest rates. J. Finance 41, 617–630.CrossRefGoogle Scholar
Carmona, P., Petit, F. & Yor, M. (1994). On exponential functionals of certain Lévy processes. Stoch. Stoch. Reports 47, 71–101.CrossRefGoogle Scholar
de Saporta, B. & Yao, J. F. (2005). Tail of a linear diffusion with Markov switching. Ann. Appl. Prob. 15, 992–1018.CrossRefGoogle Scholar
Hou, T. & Shao, J. (2019). Heavy tail and light tail of Cox–Ingersoll–Ross processes with regime-switching. Sci. China Math. https://doi.org/10.1007/s11425-017-9392-5.Google Scholar
Lindner, L. & Maller, R. (2005). Lévy integrals and the stationarity of generalised Ornstein–Uhlenbeck processes. Stoch. Process. Appl. 115, 1701–1722.CrossRefGoogle Scholar
Novikov, A. (2003). Martingales and first-exit times for the Ornstein–Uhlenbeck process with jumps. Teor. Veroyat. Primen. 48, 340–358; translation in Theory Prob. Appl. 48, 288–303 (2004).Google Scholar
Sato, K. & Yamazato, M. (1984). Operator-self-decomposable distributions as limit distributions of processes of Ornstein–Uhlenbeck type. Stoch. Process. Appl. 17, 73–100.CrossRefGoogle Scholar
Sato, K., Watanabe, T. & Yamazato, M. (1994). Recurrence conditions for multidimensional processes of Ornstein–Uhlenbeck type. J. Math. Soc. Japan 46, 245–265.CrossRefGoogle Scholar
Sato, K., Watanabe, T., Yamamuro, K. & Yamazato, M. (1996). Multidimensional process of Ornstein–Uhlenbeck type with nondiagonalizable matrix in linear drift terms. Nagoya Math. J. 141, 45–78.CrossRefGoogle Scholar
Shao, J. (2015). Criteria for transience and recurrence of regime-switching diffusion processes. Electron. J. Prob. 20, 1–15.CrossRefGoogle Scholar
Shiga, T. (1990). A recurrence criterion for Markov processes of Ornstein–Uhlenbeck type. Prob. Theory Relat. Fields 85, 425–447.CrossRefGoogle Scholar
Watanabe, T. (1998). Sato’s conjecture on recurrence conditions for multidimensional processes of Ornstein–Uhlenbeck type. J. Math. Soc. Japan 50, 155–168.CrossRefGoogle Scholar