# American Institute of Mathematical Sciences

September  2017, 22(7): 2923-2938. doi: 10.3934/dcdsb.2017157

## 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

Received  July 2016 Revised  April 2017 Published  May 2017

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.

Citation: Min Zhu, Panpan Ren, Junping Li. Exponential stability of solutions for retarded stochastic differential equations without dissipativity. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2923-2938. doi: 10.3934/dcdsb.2017157
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