Non-exponential stability and decay rates in nonlinear stochastic difference equation with unbounded noises
Abstract
We consider stochastic difference equation xn+1 = xn (1 - h f(xn) + h g(xn) n+1), where functions f and g are nonlinear and bounded, random variables i 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 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, xn nα -> 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.
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