Qualitative study of Caputo Erdélyi-Kober stochastic fractional delay differential equations

We present new results on the well-posedness and time regularity of solutions to stochastic fractional delay differential equations (SFDDEs) using the Caputo-Erdélyi-Kober fractional derivative. Additionally, we prove the averaging principle. We establish all results in the pth moment, which generalizes the case p = 2. First, by applying fixed-point theory (FPT), we prove that the solution exists, is unique, and continuously depends on the initial values as well as the fractional derivative. Second, we establish a smoothness theorem for the solution and demonstrate that the solution of the original system converges to the averaged system in the pth moment. Finally, to support our theoretical findings, we present illustrative examples.