Ergodicity and transience of SDEs driven by -stable processes with Markovian switching

ABSTRACT

In this paper, we focus on ergodicity and transience of SDEs driven by symmetric $\mathit{\alpha }$-stable processes ($\mathit{\alpha }\in (1,2)$) with Markovian switching. Some sufficient conditions for ergodicity of several classes of stable processes with Markovian switching are given. Furthermore, an almost necessary and sufficient condition for ergodicity of Ornstein–Uhlenbeck type driven by symmetric $\mathit{\alpha }$-stable processes with Markovian switching is presented. As applications, some examples are provided to illustrate our results.

KEYWORDS:

• Ergodicity
• Foster-Lyapunov criteria
• symmetric $\mathit{\alpha }$-stable process
• Markovian switching
• M-matrix

AMS SUBJECT CLASSIFICATIONS:

• 60J60
• 60J27
• 93E15

Acknowledgements

The authors are grateful to the anonymous reviewers for their valuable comments and suggestions which led to improvements in this manuscript.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

The research of Z. Zhang was partially supported by the National Natural Science Foundation of China [grant number 11471071], Natural Science Foundation of Shanghai [grant number 14ZR1401200] and Institute of Nonlinear Science of Donghua University. The research of J. Tong was partially supported by the National Natural Science Foundation of China [grant number 11401093], [grant number 11571071].