Qualitative Analysis of a Neutral Fractional Stochastic Differential Equation with G-L$\acute{e}$vy Noise

Abstract

This paper investigates the existence and uniqueness of solutions to Caputo-type neutral fractional stochastic differential equations driven by multiplicative and fractional noises within the framework of the G-L$\acute{e}$vy process, where the Hurst index satisfies $H \in \left( \frac{1}{2}, 1 \right)$. The analysis employs Cauchy's inequality and Gronwall's inequality as essential mathematical tools to obtain rigorous estimates and establish the well-posedness of the system. To validate the theoretical findings, a detailed comparison is carried out between the exact solution and its approximation obtained via the Picard iterative method, with particular emphasis on evaluating the associated error bounds. Furthermore, an exponential estimation for the solutions is derived, providing deeper insight into their long-term behavior. Finally, two carefully designed illustrative examples are presented to demonstrate the applicability and effectiveness of the proposed theoretical framework.