On the oscillation of solutions of stochastic difference equations with state-independent perturbations*

This paper considers the pathwise oscillatory behaviour of the scalar nonlinear stochastic difference equation 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 nonhyperbolic. 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 nonoscillation 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.