Exponential stability of solutions for retarded stochastic differential equations without dissipativity

Min Zhu; Panpan Ren; Junping Li

doi:10.3934/dcdsb.2017157

Article Contents

2017, Volume 22, Issue 7: 2923-2938. Doi: 10.3934/dcdsb.2017157

This issue Previous Article LaSalle type stationary oscillation theorems for Affine-Periodic Systems Next Article Minimization of carbon abatement cost: Modeling, analysis and simulation

Exponential stability of solutions for retarded stochastic differential equations without dissipativity

1.
College of Traffic Engineering, Hunan University of Technology, Zhuzhou, Hunan 412007, China
2.
School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, China
3.
Department of Mathematics, Swansea University, Singleton Park, SA2, 8PP, UK

* Corresponding author: Min Zhu
* Corresponding author: Min Zhu

Received: July 2016

Revised: April 2017

Published: October 2017

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Abstract

This work focuses on a class of retarded stochastic differential equations that need not satisfy dissipative conditions. The principle technique of our investigation is to use variation-of-constants formula to overcome the difficulties due to the lack of the information at the current time. By using variation-of-constants formula and estimating the diffusion coefficients we give sufficient conditions for $p$-th moment exponential stability, almost sure exponential stability and convergence of solutions from different initial value. Finally, we provide two examples to illustrate the effectiveness of the theoretical results.

Keywords:

Retarded stochastic differential equations,
variation-of-constants formula,
exponential stability,
almost sure exponential stability,
dissipativity.

Mathematics Subject Classification: 60H10, 39B82, 60H30, 37H10.

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