Ulam–Hyers Stability of Caputo-Type Fractional Stochastic Differential Equations with Time Delays

Abstract

In this paper, we study a class of Caputo-type fractional stochastic differential equations (FSDEs) with time delays. Under some new criteria, we get the existence and uniqueness of solutions to FSDEs by Carath $\acute{e}$ odory approximation. Furthermore, with the help of H $\ddot{o}$ lder’s inequality, Jensen’s inequality, It $\widehat{o}$ isometry, and Gronwall’s inequality, the Ulam–Hyers stability of the considered system is investigated by using Lipschitz condition and non-Lipschitz condition, respectively. As an application, we give two representative examples to show the validity of our theories.