Advances in Difference Equations 
Volume 2004 (2004), Issue 3, Pages 249-260
doi:10.1155/S1687183904309015

Global asymptotic stability of solutions of cubic stochastic difference equations

Alexandra Rodkina1 and Henri Schurz2

 1Department of Mathematics and Computer Science, University of the West Indies at Mona, Kingston 7, Jamaica
2Department of Mathematics, Southern Illinois University, 1245 Lincoln Drive, Carbondale 62901-4408, IL, USA
Received 18 September 2003; Revised 22 December 2003

Abstract

Global almost sure asymptotic stability of solutions of some nonlinear stochastic difference equations with cubic-type main part in their drift and diffusive part driven by square-integrable martingale differences is proven under appropriate conditions in ℝ1. As an application of this result, the asymptotic stability of stochastic numerical methods, such as partially drift-implicit θ-methods with variable step sizes for ordinary stochastic differential equations driven by standard Wiener processes, is discussed.