Boundedness analysis of stochastic delay differential equations with Lévy noise

• Both time delay and Lévy noise are taken into considered in the stochastic differential equations. • Mean square asymptotic boundedness criteria of the solutions are established by the method of reduction and the Itô formula. • Based on the Chebyshev inequality and the Borel–Cantelli lemma, the almost surely boundedness criteria are obtained. The present paper is concerned with the mean square asymptotic boundedness and twice power almost surely asymptotic boundedness of stochastic delay differential equations driven by Lévy noise. First, mean square asymptotic boundedness criteria of the solutions are established by the method of reduction and the generalized Itô formula. Then, based on the Chebyshev inequality and the Borel–Cantelli lemma, the twice power almost surely asymptotic boundedness criteria are also derived for the addressed equations. Finally, a example is provided to demonstrate the validity of the proposed results.