Non-exponential Stability and Decay Rates in Nonlinear Stochastic Difference Equation with Unbounded Noises

We consider stochastic difference equation xn+1 = xn ( 1− hf(xn) + √ hg(xn)ξn+1 ) , n = 0, 1, . . . , x0 ∈ R, where functions f and g are nonlinear and bounded, random variables ξi are independent and h > 0 is a nonrandom parameter. We establish results on asymptotic stability and instability of the trivial solution xn ≡ 0. We also show, that for some natural choices of the nonlinearities f and g, the rate of decay of xn is approximately polynomial: we find α > 0 such that xn decays 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, xnn → 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.