Theoretical analysis and numerical simulation of solutions to fractional Itô-Doob stochastic systems

Fractional stochastic equations are gaining more and more importance for the modelling of real systems where memory and randomness are involved. In this paper, we study a class of Itô-Doob fractional stochastic systems with proportional delays and fractional derivative of Caputo-Katugampola type. The main difference between our work and the mean-square analysis is that we use the more general p th moment framework, with \(p≥ 2\) . By choosing this way, it is not only more robust but also more reliable. First, we prove that the system is well-posed by showing existence, uniqueness, and continuous dependence of the solution on the initial data under reasonable assumptions. The derivation employs a combination of fixed-point arguments and stochastic calculus. In the following step, we establish the Ulam–Hyers stability in the p th moment sense, demonstrating that the solution remains close to the exact solution in instances where the system experiences a small perturbation. We also prove the averaging principle. This principle allows for the substitution of a complex and rapidly oscillating system with a more straightforward, averaged version. This is particularly useful for systems with multiple time-scale dynamics. We give a rigorous proof of the closeness of the solutions of the averaged and the original systems. We demonstrate our theoretical results with some numerical examples using a suitable adaptation of the Euler–Maruyama scheme to the fractional case.