Advances in Difference Equations
Volume 2004 (2004), Issue 3, Pages 249-260
Global asymptotic stability of solutions of cubic stochastic difference equations
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
Copyright © 2004 Alexandra Rodkina and Henri Schurz. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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
. As an application of this result, the asymptotic
stability of stochastic numerical methods, such as partially
drift-implicit -methods with variable step sizes for
ordinary stochastic differential equations driven by standard
Wiener processes, is discussed.