Asymptotic behavior of solutions for stochastic fractional differential equations involving bounded delay

Abstract

In this paper, we investigate the asymptotic mean-square behavior of solutions for a class of stochastic fractional delay differential equations (SFDDEs) when the fractional derivative is in the Caputo sense. To this end, we first establish a variation of constants formula for the considered system. Subsequently, by employing the Jordan normal form of matrices together with a Djrbashian-type summation formula, we obtain precise estimates for the matrix-form Mittag-Leffler function. We then construct a suitable weighted norm and utilize its favorable properties to establish the desired asymptotic results. Crucially, this tailored weighted-norm approach isolates the asymptotic tail behavior of the solution, allowing us to completely remove the restrictive 2 η scaling condition typically imposed on delay Lipschitz constants by standard fractional Halanay inequalities, where η is fractional order. These findings contribute to the qualitative analysis of stochastic fractional differential systems with delays, providing sharper stability criteria and new insights into their long-term stability in the mean-square sense. Finally, we present some numerical simulations to support the theoretical results.