Existence theory and stability analysis of neutral $\psi$–Hilfer fractional stochastic differential system with fractional noises and non-instantaneous impulses

In this article, with the help of Laplace transform, the existence of solution was established in a finite dimensional setting for nonlinear $\psi$-Hilfer fractional stochastic equation with both retarded and advanced arguments driven by multiplicative and fractional noises, with Hurst index $H\in (\frac{1}{2},1)$. At first, we obtained the existence and uniqueness results by using the Banach fixed point theorem (FPT). Second, the existence result was also obtained by applying Schaefer's fixed point theorem with less conservative conditions. Furthermore, we investigated the Hyers Ulam Rasisas stability for the aforementioned system. At the end, an example was illustrated to validate the obtained theoretical results.