Razumikhin-type Theorems on Exponential Stability of Neutral Stochastic Functional Differential Equations

Recently, we initiated in [Systems Control Lett., 26 (1995), pp. 245–251] the study of exponential stability of neutral stochastic functional differential 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 different from those used in our previous paper [Systems Control Lett., 26 (1995), pp. 245–251]. The results obtained in [Systems Control Lett., 26 (1995), pp. 245–251] can only be applied to a certain class of neutral stochastic functional differential equations excluding neutral stochastic differential delay equations, but the results obtained in this paper are more general, and they especially can be used to deal with neutral stochastic differential delay equations. Moreover, in [Systems Control Lett., 26 (1995), pp. 245–251], 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 equation, for a particular type of equation discussed in [Systems Control Lett., 26 (1995), pp. 245–251], the results there are sharper.