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.