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.

Well-posedness is crucial in studying fractional stochastic differential equations, as it ensures that solutions are mathematically sound and applicable to practical situations. A well-formulated model satisfies the essential requirements for solutions, such as existence, uniqueness, and stability concerning various parameters. Using fixed-point theory, we prove that the solution to stochastic fractional delay differential equations with the Hilfer fractional operator exists, is unique, and continuously depends on the initial values and the fractional derivative. Additionally, we establish a smoothness theorem for the solution and demonstrate that the solution of the original system converges to the averaged system in the $\mathrm{p}$th moment. Last, to support our theoretical findings, we provide examples and graphical illustrations. The primary tools used in our proofs include the Burkholder-Davis-Gundy inequality, Jensen's inequality, and Hölder's inequality.

Most existing research on well-posedness and stability has focused on fractional stochastic differential equations, with relatively fewer studies addressing fractional stochastic integro-differential equations (FSIDEs). In this work, we address this gap by establishing theoretical results on the well-posedness of FSIDEs. In particular, we derive a generalized Grönwall inequality and present results on Ulam-Hyers stability (UHS). Moreover, we extend existing findings by incorporating both the $\mathrm{\Phi }$-Caputo fractional derivative and the $\mathrm{p}$th moment, thereby unifying and generalizing current results in the literature.

Fractional stochastic differential equations (FSDEs) driven by fractional Brownian motion (fBm) have attracted growing attention due to their ability to model systems exhibiting non-Markovian dynamics and long-range dependence, which naturally arise in many real-world phenomena characterized by hereditary and persistent randomness. In this work, we establish the existence and uniqueness of mild solutions using the Picard iteration technique for the case where the Hurst parameter satisfies $H\in \left(\frac{1}{2},1\right)$. Moreover, we establish the approximate controllability of the systems under suitable conditions. To generalize the theoretical framework, we employ the Caputo–Katugampola fractional derivative (CKFD), thereby extending the analysis to a broader class of fractional stochastic systems.