Qualitative Analysis and Numerical Simulations of γ-Hilfer Fractional Stochastic Dynamical Systems

This study presents findings on the existence, uniqueness, averaging principle, and numerical solutions for fractional stochastic systems influenced by both Brownian motion and Poisson jumps within the pth-moment framework. While most existing results for fractional stochastic differential equations are derived using the mean-square approach, this research offers results in the more general pth-moment framework, enhancing applicability. The results are derived using the γ-Hilfer fractional derivative, a generalized operator defined in relation to another function. This operator enables the memory effect to vary according to a nonlinear time scale. The main motivation behind this work is that there is no research work on γ-Hilfer fractional stochastic systems with standard Brownian motion and Poisson jumps regarding existence, uniqueness, and averaging principles in the pth-moment.