1687-1847-2004-513569 1687-1847 Research Article <p>Global asymptotic stability of solutions of cubic stochastic difference equations</p> RodkinaAlexandraalexandra.rodkina@uwimona.edu.jm SchurzHenrihschurz@math.siu.edu

Department of Mathematics and Computer Science, University of the West Indies at Mona, Kingston 7, Jamaica

Department of Mathematics, Southern Illinois University, 1245 Lincoln Drive, Carbondale, IL 62901-4408, USA

 Advances in Difference Equations 1687-1847 2004 2004 3 513569 http://www.advancesindifferenceequations.com/content/2004/3/513569 10.1155/S1687183904309015 1892003221220031272004 2004Rodkina and Schurz

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.