Finite-Time Stability and Fixed-Time Stability of Stochastic Delay System with Lévy Noise

This paper investigates the finite-time and fixed-time stability of stochastic delay differential equations (SDDEs) driven by Lévy noise, employing Itô’s formula and the stopping-time technique. First, a novel inequality inspired by the Bihari inequality is derived, and a sufficient condition for finite-time stability of SDDEs under Lévy noise is established. Second, by integrating the Razumikhin method with Lyapunov function theory, we establish Razumikhin-type theorems that guarantee fixed-time stability and provide an explicit upper bound for the settling time. Finally, an example is given to verify the correctness of the theoretical results.