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.

Keywords:

• nonlinear stochastic difference equations
• almost sure stability
• decay rates
• martingale convergence theorem

AMS Subject Classification:

• 39A10
• 39A11
• 37H10
• 34F05
• 93E15

Acknowledgements

John Appleby was partially supported by an Albert College Fellowship awarded by Dublin City University's Research Advisory Panel. Alexandra Rodkina was supported by the Mona Research Fellowship Programme awarded by University of the West Indies, Mona.