tailieunhanh - GLOBAL ASYMPTOTIC STABILITY OF SOLUTIONS OF CUBIC STOCHASTIC DIFFERENCE EQUATIONS ALEXANDRA RODKINA

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. | 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 ớ-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 O Í XeK P is given as a stochastic basis with filtrations nlneN. Let iidneN be a one-dimensional real-valued n neN martingale difference for details see 2 14 and let S denote the set of all Borel sets of the set S. Furthermore let a an neN be a nonincreasing sequence of strictly positive real numbers an and let K Kn neN be a sequence of real numbers Kn. We use . as the abbreviation for wordings P-almost sure or P-almost surely . In this paper we consider discrete-time stochastic difference equations DSDEs V IS 7 3 7 7 3 111 1 111 1 xn 1 xn Knxn anxn 1 fn xl 0 l n On xl 0 l n Sn 1 with cubic-type main part of their drift in R1 real parameters an Kn e R1 driven by the square-integrable martingale difference S n 1 new of independent random variables Sn 1 with E Sn 1 0 and E Sn 1 2 TO. We are especially interested in conditions ensuring the almost sure global asymptotic stability of solutions of these DSDEs . The main result should be such that it can be applied to numerical methods for related continuous-time stochastic differential equations CSDEs as its potential limits. For example consider dXt a1 t Xt a2 t Xt dt b t Xt dWt Copyright 2004 Hindawi Publishing Corporation Advances in Difference Equations 2004 3 .