Exponential stability of solutions for retarded stochastic differential equations without dissipativity

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.