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

Abstract

We consider the stochastic difference equation where f and g are nonlinear, bounded functions, 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 . We also show that, for some natural choices of f and g, the rate of decay of is approximately polynomial: there exists such that decays faster than but slower than , for any . It turns out that, if decays faster than as , the polynomial rate of decay can be established precisely: 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.