Stability Analysis and Numerical Simulations of Fractional Stochastic Systems

This paper considers a class of fractional stochastic differential systems driven jointly by the Rosenblatt process and a compensated Poisson random measure. The Rosenblatt process captures non-Gaussian fluctuations with memory effects; the Poisson jumps model sudden random shocks. By employing the mild solution formulation associated with fractional resolvent operators, we establish the existence of solutions via Krasnoselskii’s fixed point theorem (KFPT) and prove uniqueness through the Banach contraction under suitable Lipschitz conditions on the drift, diffusion, and jump coefficients together with contraction assumptions. Furthermore, Ulam–Hyers stability is established, ensuring that approximate solutions remain close to exact solutions in the mean-square sense. We also establish Mittag–Leffler-type continuous dependence on initial data, demonstrating that solutions depend continuously on their initial histories in the mean-square sense. In addition, open-loop approximate trajectory realization is established through the construction of an explicit open-loop control law that steers the stochastic system along any prescribed admissible trajectory under suitable invertibility and regularity assumptions. An example validates the theoretical results, demonstrating fractional memory, Rosenblatt noise, and Poisson jumps.