Stability in the Sense of Hyers-Ulam of Proportional Fractional Stochastic Integral Equations

This work investigates the existence, uniqueness, and stability in the sense of Hyers-Ulam for a class of proportional fractional Itô-Doob stochastic integral equations (PFIDSIE). To establish these properties, we employ the Banach fixed point theorem (BFPT) in combination with several fundamental mathematical inequalities that provide insight into the structure of PFIDSIEs. The approach is structured to demonstrate not only the theoretical foundation of the existence and uniqueness of solutions but also the stability of these solutions in the Hyers-Ulam sense, which ensures that approximate solutions remain close to the exact solution under small perturbations. The results contribute to the broader field of fractional stochastic differential equations, particularly in situations where fractional dynamics and stochastic processes intersect. Furthermore, the findings are illustrated through three examples, showcasing the applicability and utility of the developed theory in practical settings.OPEN ACCESS Received: 29/01/2025 Accepted: 14/03/2025 Published: 20/04/2025