ON THE OSCILLATION OF SOLUTIONS OF STOCHASTIC DIFFERENCE EQUATIONS WITH STATE-INDEPENDENT PERTURBATIONS JOHN A. D. APPLEBY, GREGORY BERKOLAIKO, AND ALEXANDRA RODKINA Abstract. This paper considers the pathwise oscillatory behaviour of the scalar nonlinear stochastic difference equation (1) X(n + 1) = X(n) − f (X(n)) + σ(n)ξ(n + 1), n =0, 1,..., where (ξ(n)) n≥0 is a sequence of independent random variables with zero mean and unit variance. The real–valued function f : R → R is presumed to be continuous with f (0) = 0 and xf (x) > 0 for x = 0. It is shown that when the stochastic sequence is identically distributed, oscillation occurs if the noise intensity is not square summable, or if the mean reversion is relatively strong sufficiently far from the equilibrium, even in the case when the equilibrium is non–hyperbolic. If the noise intensity is square summable, it can be shown for both linear equations and for equations with a hyperbolic equilibrium that oscillation as well as non–oscillation can occur. This depends on the relation between the rates of decay of the noise intensity and of the solution of the underlying unperturbed deterministic equation. 1. Introduction The oscillation of the solutions of deterministic difference equation has been discussed in many papers; a comprehensive survey of this literature is contained in [2]. In this paper we concentrate on the oscillation of solutions to scalar stochastic non-linear difference equations. Aside from results concerning the preservation of oscillation and non–oscillation in solutions of discretised linear stochastic delay differential equations in [3, 4] there is little known about the oscillation of the solution of stochastic non-linear difference equations. Global a.s. asymptotic stability of the solutions to stochastic non-linear differ- ence equation was discussed in many papers, see the most relevant publications: [2], [5], [6], [7], [10], [11]. In this paper we consider the oscillatory behaviour of sample paths of the sto- chastic difference equation (2) X(n + 1) = X(n) − f (X(n)) + σ(n)ξ (n + 1), n =0, 1, .... Date : April 28, 2007. 1991 Mathematics Subject Classification. Primary: 37H10, 39A11; Secondary: 60H10, 34F05, 65C20. Key words and phrases. Stochastic difference equation; oscillation of solution; state- independent perturbation. The first author was partially supported by an Albert College Fellowship awarded by Dublin City University’s Research Advisory Panel. The third author was supported by a New Initiative Fellowship awarded by University of the West Indies, Mona. 1