International Journal of Difference Equations
Open Access
  • Year: 2006
  • Volume: 1
  • Issue: 1

On Global Asymptotic Stability of Nonlinear Stochastic Difference Equations with Delays

  • Author:
  • Yoshihiro Hamaya1, Alexandra Rodkina2
  • Total Page Count: 18
  • Page Number: 101 to 118

1Department of Information Science, Okayama University of Science, Japan.

2Department of Math/CSci, University of the West Indies, Jamaica.

AMS subject classification: 39A11, 34K20, 34F05, 93E15, 60H10, 92B20.

Abstract

We consider the stochastic difference equation with arbitrary initial conditions , non-linear continuous functions F and g, and independent zero mean random variables . Equation (0.1) describes the dynamics of a neural network under stochastic perturbations.

We prove results on global a.s. asymptotic stability of the trivial solution Xn of equation (0.1). We show that (0.1) is a good discrete model for a corresponding stochastic continuous Itô equation, since under the same conditions on the functions F and g, their solutions have similar asymptotic behavior.

Keywords

Stochastic difference and differential equations, a.s. asymptotic stability, martingale convergence theorems, Lyapunov–Krasovkii functionals, neural networks