New criteria on exponential stability of neutral stochastic differential delay equations

Neutral stochastic differential delay equations (NSDDEs) have recently been studied intensively (see e.g. [V.B. Kolmanovskii, V.R. Nosov, Stability and Periodic Modes of Control Systems with Aftereffect, Nauka, Moscow, 1981; X. Mao, Exponential stability in mean square of neutral stochastic differential functional equations, Systems Control Lett. 26 (1995) 245–251; X. Mao, Razumikhin type theorems on exponential stability of neutral stochastic functional differential equations, SIAM J. Math. Anal. 28(2) (1997) 389–401; X. Mao, Stochastic Differential Equations and Their Applications, Horwood Publishing, Chichester, 1997]). More recently, Mao [Asymptotic properties of neutral stochastic differential delay equations, Stochastics and Stochastics Rep. 68 (2000) 273–295] provided with some useful criteria on the exponential stability for NSDDEs. However, the criteria there require not only the coefficients of the NSDDEs to obey the linear growth condition but also the time delay to be a constant. One of our aims in this paper is to remove these two restrictive conditions. Moreover, the key condition on the diffusion operator associated with the underlying NSDDE will take a much more general form. Our new stability criteria not only cover many highly non-linear NSDDEs with variable time delays but they can also be verified much more easily than the known criteria.