Non-exponential stability and decay rates in nonlinear stochastic difference equation with unbounded noises

Abstract

We consider stochastic difference equation x_{n+1} = x_n (1 - h f(x_n) + \sqrt{h} g(x_n) ξ_{n+1}), where functions f and g are nonlinear and bounded, random variables ξ_i are independent and h>0 is a nonrandom parameter. We establish results on asymptotic stability and instability of the trivial solution x_n=0. We also show, that for some natural choices of the nonlinearities f and g, the rate of decay of x_n is approximately polynomial: we find α>0 such that x_n decay faster than n^{-α+ε} but slower than n^{-α-ε} for any ε>0. It also turns out that if g(x) decays faster than f(x) as x->0, the polynomial rate of decay can be established exactly, x_n n^α-> const. On the other hand, if the coefficient by the noise does not decay fast enough, the approximate decay rate is the best possible result.