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

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

2006-10-13

arxiv.org/abs/math/0610425v2

Stochastics and Stochastics Reports, 81, 99-127 (2009)

Mathematics

Probability

Major revision: 24 pages, fixed iterated logarithm mistake

Scientific paper

10.1080/17442500802088541

We consider stochastic difference equation x_{n+1} = x_n (1 - h f(x_n) + \sqrt{h} g(x_n) \xi_{n+1}), where functions f and g are nonlinear and bounded, random variables \xi_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 \alpha>0 such that x_n decay faster than n^{-\alpha+\epsilon} but slower than n^{-\alpha-\epsilon} for any \epsilon>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^\alpha -> 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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