Stability of solutions of Caputo fractional stochastic differential equations

Abstract

In this paper, we study the stability of Caputo-type fractional stochastic differential equations. Stochastic stability and stochastic asymptotical stability are shown by stopping time technique. Almost surly exponential stability and pth moment exponentially stability are derived by a new established Itô’s formula of Caputo version. Numerical examples are given to illustrate the main results.

Averaging principle for non‐Lipschitz fractional stochastic evolution equations with random delays modulated by two‐time‐scale Markov switching processes

“…Therefore, the study of fractional calculus and fractional differential equations has drawn increasing attentions recently. For example, Ahmed and Zhu 8 discussed the averaging principle for delay stochastic Hilfer‐type fractional systems; then, the results were generalized to the non‐Lipschitz conditions by Luo et al 9 ; Xiao and Wang 10 derived the stability of Caputo‐type fractional SDEs by stopping time technique. For infinite‐dimensional fractional SDEs, Li and Wang 11 established the existence, uniqueness, and continuous dependence of mild solutions to stochastic delay evolution equations with a Caputo fractional derivative; Sakthivel et al 12 presented the issue of existence of mild solutions for a class of fractional SDEs with impulses, and Zhang et al 13 gave the regularity of the trajectories of mild solutions for the fractional evolution equations in the Hilbert space.…”