Stability of Nonlinear Stochastic Volterra Difference Equations with Respect to a Fading Perturbation

The paper concerns studies the stochastic stability and stochastic asymptotic stability of the equilibrium solution of a nonlinear Volterra difference equation which is subject to stochastic state independent disturbances. It is shown that if the linearized deterministic equation has summable solutions, then the nonlinear stochastic equation will be stable or asymptotically stable, provided that the initial condition, and the intensity of the stochastic disturbances are sufficiently small. The smallness of the intensity follows closely the conditions required for the sta-bility of the stochastically perturbed linear Volterra difference equation