Convergence, boundedness, and ergodicity of regime-switching diusion processes with infinite memory

doi:10.1007/s11464-020-0863-8

Front. Math. China

2021, Vol. 16

Issue (2) : 499-523 https://doi.org/10.1007/s11464-020-0863-8

RESEARCH ARTICLE

Convergence, boundedness, and ergodicity of regime-switching diusion processes with infinite memory

Jun LI^1,², Fubao XI^1,³(

)

¹. School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China
². Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou 730050, China
³. Beijing Key Laboratory on MCAACI, Beijing Institute of Technology, Beijing 100081, China

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Abstract

We study a class of diffusion processes, which are determined by solutions X(t) to stochastic functional differential equation with infinite memory and random switching represented by Markov chain Λ(t): Under suitable conditions, we investigate convergence and boundedness of both the solutions X(t) and the functional solutions X_t: We show that two solutions (resp., functional solutions) from different initial data living in the same initial switching regime will be close with high probability as time variable tends to infinity, and that the solutions (resp., functional solutions) are uniformly bounded in the mean square sense. Moreover, we prove existence and uniqueness of the invariant probability measure of two-component Markov-Feller process (X_t,Λ(t)); and establish exponential bounds on the rate of convergence to the invariant probability measure under Wasserstein distance. Finally, we provide a concrete example to illustrate our main results.

Keywords Regime-switching diffusion process infinite memory convergence boundedness Feller property invariant measure Wasserstein distance

Corresponding Author(s): Fubao XI

Issue Date: 01 June 2021

Cite this article:

Jun LI,Fubao XI. Convergence, boundedness, and ergodicity of regime-switching diusion processes with infinite memory[J]. Front. Math. China, 2021, 16(2): 499-523.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-020-0863-8
https://academic.hep.com.cn/fmc/EN/Y2021/V16/I2/499

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