On cubic difference equations with variable coefficients and fading stochastic perturbations
Abstract
We consider the stochastically perturbed cubic difference equation with variable coefficients \[ xn+1=xn(1-hnxn2)+ρn+1ξn+1, n∈ N, x0∈ R. \] Here (ξn)n∈ N is a sequence of independent random variables, and (ρn)n∈ N and (hn)n∈ N are sequences of nonnegative real numbers. We can stop the sequence (hn)n∈ N after some random time N so it becomes a constant sequence, where the common value is an FN-measurable random variable. We derive conditions on the sequences (hn)n∈ N, (ρn)n∈ N and (ξn)n∈ N, which guarantee that n ∞ xn exists almost surely (a.s.), and that the limit is equal to zero a.s. for any initial value x0∈ R.
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