RAZUMIKHIN-TYPE THEOREMS ON EXPONENTIAL STABILITY OF NEUTRAL STOCHASTIC FUNCTIONAL-DIFFERENTIAL EQUATIONS

Recently, we initiated in [Systems Control Lett., 26 (1995), pp. 245-2 51] the study of exponential stability of neutral stochastic functiona l differential equations, and in this paper, we shall further our stud y in this area. We should emphasize that the main technique employed i n this paper is the well-known Rasumikhin argument and is completely d ifferent from those used in our previous paper [Systems Control Lett., 26 (1995), pp. 245-251]. The results obtained in [Systems Control Let t., 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 w ith neutral stochastic differential delay equations. Moreover, in [Sys tems Control Lett., 26 (1995), pp. 245-251], we only studied the expon ential stability in mean square, but in this paper, we shall also stud y 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 disc ussed in [Systems Control Lett., 26 (1995), pp. 245-251], the results there are sharper.