On asymptotic stability and instability with respect to a fading stochastic perturbation

Abstract

We develop necessary and sufficient conditions for the a.s. asymptotic stability of solutions of a scalar, non-linear stochastic equation with state-independent stochastic perturbations that fade in intensity. These conditions are formulated in terms of the intensity function: roughly speaking, we show that as long as the perturbations fade quicker than some identifiable critical rate, the stability of the underlying deterministic equation is unaffected. These results improve on those of Chan and Williams; for example, we remove the monotonicity requirement on the drift coefficient and relax it on the intensity of the stochastic perturbation. We also employ different analytic techniques.

Keywords:

• almost sure asymptotic stability
• stochastic differential equation
• fading stochastic perturbations

AMS Subject Classifications::

• 60H10
• 93E15
• 93D09
• 93D20

Acknowledgement

The authors are grateful to Dr Cónall Kelly for his suggestions regarding the exposition, structure and layout of the article. This research is supported under the Enterprise Ireland International Collaboration Programme (grant number IC/2004/003). J.A.D. Appleby was also partially supported by an Albert College Fellowship awarded by Dublin City University's Research Advisory Panel. J.P. Glesson was partially supported by Science Foundation Ireland grant 06/IN.1/I366 and by the Boole Centre for Research in Informatics, University College Cork. A. Rodkina was supported by a Mona Research Fellowship awarded by University of the West Indies, Mona, and by the Boole Centre for Research in Informatics, University College Cork.