Ultimate boundedness of impulsive stochastic delay differential equations with delayed impulses

Abstract

This paper is devoted to studying the ultimate boundedness of impulsive stochastic delay differential equations (SDDEs) with delayed impulses. Using suitable Lyapunov functionals and Itoˆ’s formula, some necessary criteria are met to guarantee the system’s pth moment global exponential ultimate boundedness (p-MGEUB). The results show that the unbounded stochastic delay differential equation can be transformed into a bounded system by impulses. Moreover, our results generalize and improve some results in the literature. Finally, we provide an illustration to clarify our findings.