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 $\mathfrak{p}$th moment, which generalizes the case $\mathfrak{p}=2$. 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 $\mathfrak{p}$th moment. Finally, to support our theoretical findings, we present illustrative examples.