A note on ergodicity for CIR model with Markov switching

Abstract

Recently, Zhang et al. show that if ${\sum }_{i=1}^N{\pi }_i\beta (i)\ne 0,$ then the Cox-Ingersoll-Ross (CIR) model with Markov switching (see below, the SDE (1.2)) is ergodic in the Wasserstein distance if and only if ${\sum }_{i=1}^N{\pi }_i\beta (i)>0.$ In this article, we will show that if ${\sum }_{i=1}^N{\pi }_i\beta (i)=0,$ the Cox-Ingersoll-Ross (CIR) model with Markov switching is non-ergodic. Explicit expressions for the mean and variance of the CIR model with Markov switching are obtained. As a byproduct, the explicit expressions for mean of stationary distribution and second-order moments for such model are presented. Besides, we find the necessary and sufficient conditions of weak stationarity for CIR model with Markov switching.

Keywords:

• Cox-Ingersoll-Ross (CIR) model
• Markov chain
• Stationary distribution

Acknowledgments

The authors are grateful to the anonymous reviewer and Dr. Xinghu Jin for their valuable comments and suggestions which led to improvements in this manuscript. The research of J. Tong was partially supported by the Fundamental Research Funds for the Central Universities and the National Natural Science Foundation of China (Nos. 11401093 and 11571071). The research of Z. Zhang was partially supported by the Humanities and Social Sciences Fund of Ministry of Education of China (No. 17YJA910004).