Abstract
This paper is concerned with two classes of retarded stochastic differential equations. Sufficient conditions are derived to guarantee the pth moment exponential stability and almost sure exponential stability. Moreover, we construct some examples to demonstrate the theory derived.
Highlights
1 Introduction The theory of retarded stochastic differential equations (SDEs) has received a great deal of attention since it is academically challenging and of practical importance, and it has played an important role in many ways such as in life insurance, risk management, wireless communication, and optimal control of multiagent systems
In the past few decades, the theory of neutral stochastic differential equations has received a great deal of attention
There have been few results presented on the exponential asymptotic behavior of solutions of semi-linear retarded stochastic differential equations
Summary
The theory of retarded stochastic differential equations (SDEs) has received a great deal of attention since it is academically challenging and of practical importance, and it has played an important role in many ways such as in life insurance, risk management, wireless communication, and optimal control of multiagent systems. Under a non-Lipschitz condition and a weakened linear growth condition, Bao et al [ ] investigated the existence and uniqueness of mild solutions to stochastic neutral partial functional differential equations. Milošević [ ] considered global almost sure asymptotic exponential stability of the equilibrium solution for a class of neutral stochastic differential equations with time-dependent delay under nonlinear growth conditions and established moment estimates for solutions of equations of this type. There have been few results presented on the exponential asymptotic behavior of solutions of semi-linear retarded stochastic differential equations. Section is devoted to the study of asymptotic behavior of the solution for semi-linear retarded SDEs and derived sufficient conditions to guarantee the pth moment exponential stability and almost sure exponential stability. The following two theorems provide the pth moment exponential stability and almost sure exponential stability of the solution of equation ( ).
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