The Existence and Averaging Principle for Caputo Fractional Stochastic Delay Differential Systems with Poisson Jumps

Abstract

In this paper, we obtain the existence and uniqueness theorem for solutions of Caputo-type fractional stochastic delay differential systems(FSDDSs) with Poisson jumps by utilizing the delayed perturbation of the Mittag–Leffler function. Moreover, by using the Burkholder–Davis–Gundy inequality, Doob’s martingale inequality, and Hölder inequality, we prove that the solution of the averaged FSDDSs converges to that of the standard FSDDSs in the sense of Lp. Some known results in the literature are extended.

Keywords

• stochastic fractional delay differential systems
• delayed Mittag–Leffler-type matrix function
• existence and uniqueness
• averaging principle
• <i>L^p</i> convergence