Averaging Principle for Itô–Doob Fractional Stochastic Systems in mth Moment

This study presents results on the well-posedness, Ulam–Hyers stability, and mth moment averaging principle for the Itô–Doob fractional stochastic system within the framework of η-Caputo fractional derivatives. We demonstrate well-posedness using the fixed-point approach. A generalized Grönwall inequality is employed to establish sufficient conditions for Ulam–Hyers stability. Furthermore, we establish the averaging principle that facilitates obtaining a simplified averaged system from the original complex, multiple time-scale system. Finally, numerical simulations using the Euler–Maruyama method are provided to support the theoretical findings.