International Journal of Difference Equations ISSN 0973-6069, Volume 4, Number 2, pp. 165–184 (2009) http://campus.mst.edu/ijde Stability of Nonlinear Stochastic Volterra Difference Equations with Respect to a Fading Perturbation John A. D. Appleby Dublin City University School of Mathematical Sciences Dublin 9, Ireland john.appleby@dcu.ie Alexandra Rodkina The University of the West Indies, Mona Campus Department of Mathematics and Computer Science Mona, Kingston 7, Jamaica alexandra.rodkina@uwimona.edu.jm Abstract 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. AMS Subject Classifications: 39A11, 60F15, 60F10. Keywords: Volterra difference equation, nonlinear stochastic Volterra difference equa- tion, resolvent, almost sure asymptotic stability, stochastic asymptotic stability. Received September 24, 2008; Accepted June 4, 2009 Communicated by Martin Bohner