Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 01, pp. 1-16.

Title: Stability and rate of decay for solutions to stochastic differential equations
with Markov switching
       
Authors: Shuaishuai Lu (Jilin Univ., Changchun, China)
         Xue Yang  (Jilin Univ., Changchun, China)

Abstract:
In this article, we present the almost sure asymptotic stability and a general
rate of decay for solutions to stochastic differential equations (SDEs) 
with  Markov switching. By establishing a suitable Lyapunov function and using
an exponential Martingale inequality and the Borel-Cantelli theorem, 
we give sufficient conditions for the asymptotic stability.
Also, we obtain sufficient conditions for the construction of two kinds
of Lyapunov functions. Finally give two examples to illustrate the validity
of our results.

Submitted July 19, 2023. Published January 03, 2024.
Math Subject Classifications: 60H10, 34D99, 34F05.
Key Words: Almost sure asymptotic stability; Markov switching;
 exponential Martingale inequality; Lyapunov function; generalized Ito formula.