Integral Contraction Approach to Coupled Integro-Fractional Stochastic Differential Systems

We consider a coupled system of Caputo–Katugampola integro–fractional stochastic differential equations with Volterra memory, nonlocal integral boundary conditions, Wiener perturbations, and compensated Poisson jumps. The main purpose is to give a mathematically explicit fixed-point treatment in which the two components are solved simultaneously on a product space. A Green representation is derived directly from the two boundary conditions; this derivation fixes the boundary denominator and the complete Green kernel used in all subsequent estimates. Integral contractors are formulated with a correction operator and a residual modulus. In particular, a cubic nonlinearity is used to show that a coefficient need not be globally Lipschitz on the whole state space in order to satisfy the contractor hypothesis on an invariant ball. The moment order p≥2 is fixed throughout each theorem, so all contractor and contraction constants are explicitly allowed to depend on p. The coupled fixed-point problem is represented by a nonnegative 2×2 matrix Qp and the condition ρ(Qp)<1 is used as the actual contraction condition. Ulam–Hyers stability is obtained from the same matrix by an explicit Neumann-series estimate. For trajectory control we distinguish ordinary predictable finite-energy controls from generalized semimartingale controls; this distinction is necessary because a two-point stochastic boundary-value problem is generally noncausal and an ordinary drift control cannot cancel independent martingale terms pathwise. A numerical experiment based on the same Green representation verifies the contraction criterion, the non-Lipschitz contractor construction, the Ulam–Hyers estimate, and the generalized trajectory-control mechanism. The presentation also records the precise scope of the adaptedness assumption for the stochastic integrals.