Báo Cáo Báo cáo hóa học GLOBAL ASYMPTOTIC STABILITY OF SOLUTIONS OF CUBIC STOCHASTIC DIFFERENCE EQUATIONS AL

GLOBAL ASYMPTOTIC STABILITY OF SOLUTIONS OF CUBIC STOCHASTIC DIFFERENCE EQUATIONS ALEXANDRA RODKINA AND HENRI SCHURZ

Received 18 September 2003 and in revised form 22 December 2003

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.

1. Introduction Suppose that a filtered probability space (Ω, , { n }n∈N , P) is given as a stochastic basis with filtrations { n }n∈N . Let {ξn }n∈N be a one-dimensional real-valued { n }n∈N martingale difference (for details, see [2, 14]) and let (S) denote the set of all Borel