Fractional Stochastic Difference Systems: Well-Posedness, Stability, Perturbation Analysis, and Macroeconomic Application

We investigate nonlinear Caputo–nabla fractional stochastic difference systems driven by independent Gaussian increments. The fractional summation identity yields an exact pathwise Volterra representation whose strictly causal structure establishes finite-horizon existence and pathwise uniqueness under measurability and linear-growth assumptions, without requiring a contraction condition. Under global Lipschitz assumptions, separate weighted ℓ1 and ℓ2 kernel estimates provide quantitative mean-square bounds for perturbations of initial data, drift and diffusion coefficients, and the fractional order. The dependence of these estimates on the exponential weight and the observation horizon is made explicit. We also establish residual-based Ulam-type stability estimates, finite-horizon moment bounds, and finite-step exactness of Picard iteration. For long-time analysis, we prove discrete Caputo–nabla comparison principles, establish decay of the associated scalar resolvent, and derive an exact quadratic identity for the corresponding dissipativity condition. The Mittag–Leffler stability result remains conditional on a history-dependent dissipativity inequality. Independently, a directly verifiable coefficient-level criterion establishes asymptotic mean-square stability and is applied to a three-dimensional illustrative macroeconomic model at fractional order ϑ=0.75. Numerical experiments, including a nonlinear Monte Carlo simulation with sampling uncertainty estimates, illustrate the theoretical results.