On the practical global uniform asymptotic stability of stochastic differential equations

Abstract

The method of Lyapunov functions is one of the most effective ones for the investigation of stability of dynamical systems, in particular, of stochastic differential systems. The main purpose of the paper is the analysis of the stability of stochastic differential equations (SDEs) by using Lyapunov functions when the origin is not necessarily an equilibrium point. The global uniform boundedness and the global practical uniform exponential stability of solutions of SDEs based on Lyapunov techniques are investigated. Furthermore, an example is given to illustrate the applicability of the main result.

Keywords:

• stochastic differential equations
• Lyapunov techniques
• practical asymptotic stability

Acknowledgement

We would like to thank the referee for the interesting comments and suggestions which allowed us to improve the presentation of this paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The research of T. Caraballo has been partially supported by FEDER and the Ministerio de Economíay Competitividad (Spain) [grant number MTM2011-22411], and Junta de Andalucía (Spain) under Proyecto de Excelencia [grant number P12-FQM-1492] and the Ayudas de consolidación for the research group [grant number FQM314].