Robustness of Exponential Stability of Stochastic Differential Delay Equations

Regard the stochastic differential delay equation dx(t)=[(A+Ā(t))x(t)+(B+B̄(t− τ)) x(t−τ)]dt+g(t,x(t),x(t−τ))dw(t) as the result of the effects of uncertainty, stochastic perturbation and time lag to a linear ordinary differential equation ẋ(t)=(A+B)x(t). Assume the linear system is exponentially stable. In this paper we shall characterize how much the uncertainty, stochastic perturbation and time lag the linear system can bear such that the stochastic delay system remains exponentially stable. The result will also be extended to non-linear systems.