Non-exponential stability and decay rates in nonlinear stochastic difference equations with unbounded noise

We consider the stochastic difference equation x(n+1) = x(n)(1 + hf(x(n)) + root hg(x(n))xi(n+1)), n = 0, 1, ... , x(0) is an element of R-1, where f and g are nonlinear, bounded functions, {ji} is a sequence of independent random variables, and h>0 is a nonrandom parameter. We establish results on asymptotic stability and instability of the trivial solution x(n) equivalent to 0. We also show that, for some natural choices of f and g, the rate of decay of x(n) is approximately polynomial: there exists alpha>0 such that x(n) decays faster than n(-alpha+epsilon) but slower than n(-alpha-epsilon), for any epsilon>0. I t turns out that, if g(x) decays faster than f(x) as x -> 0, the polynomial rate of decay can be established precisely: x(n)n(alpha) tends to a constant limit. On the other hand, if g does not decay quickly enough, the approximate decay rate is the best possible result.