Razumikhin-type theorems on exponential stability of stochastic functional differential equations

Although the Razumikhin-type theorems have been well developed for the stability of functional differential equations and they are very useful in applications, so far there is almost no result of Razumikhin type on the stability of stochastic functional differential equations. The main aim of this paper is to close this gap by establishing several Razumikhin-type theorems on the exponential stability for stochastic functional differential equations. By applying these new results to stochastic differential delay equations and stochastically perturbed equations we improve or generalize several known results, and this shows the powerfulness of our new results clearly.