Fractional Stochastic Systems Driven by Fractional Brownian Motion: Existence, Uniqueness, and Approximate Controllability with Generalized Memory Effects

In this research work, we present findings on fractional stochastic systems characterized by fractional Brownian motion, which is defined by a Hurst parameter $\mathbb{H}\in \left(\frac{1}{2},1\right)$. These systems are crucial for modeling complex phenomena that diverge from Markovian behavior and exhibit long-range dependence, particularly in areas such as financial engineering and statistical physics. We utilize the fixed-point iteration method to demonstrate the existence and uniqueness (Ex-Un) of mild solutions. Additionally, we investigate the approximate controllability of the system. We establish all results within the framework of the $\mu$-Caputo fractional derivative. This study makes a meaningful contribution to the existing body of literature by rigorously establishing the existence, uniqueness, and approximate controllability of mild solutions to generalized Caputo fractional stochastic differential equations driven by fractional Brownian motion.