Razumikhin Type Theorems on Exponential Stability of Neutral Stochastic Functional Diierential Equations

Recently we initiated in 11] the study of exponential stability of neutral stochastic functional diierential equations and in this paper we shall further our study in this area. We should emphasize that the main technique employed in this paper is the well-known Razumikhin argument and is completely diierent from those used in our previous paper 11]. The results obtained in 11] can only be applied to a certain class of neutral stochastic functional diierential equations excluding neutral stochastic diierential delay equations, but the results obtained in this paper are more general, especially can be used to deal with neutral stochastic diierential delay equations. Moreover, in 11] we only studied the exponential stability in mean square but in this paper we shall also study the almost sure exponential stability. It should be pointed out that although the results established in this paper are applicable to more general neutral type equations, for a particular type of equations discussed in 11] the results there are sharper.