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.
Liaqat et al. (Thu,) studied this question.