Keywords:Brownian motion Neutral stochastic functional differential equation Razumikhin-type theorems pth moment exponential stability Almost sure exponential stability The paper discusses both pth moment and almost sure exponential stability of solutions to neutral stochastic functional differential equations and neutral stochastic differential delay equations, by using the Razumikhin-type technique. The main goal is to find sufficient stability conditions that could be verified more easily then by using the usual method with Lyapunov functionals. The analysis is based on paper [X. Mao, Razumikhin-type theorems on exponential stability of neutral stochastic functional differential equations, SIAM J. Math. Anal. 28 (2) (1997) 389-401], referring to mean square and almost sure exponential stability.

The paper discusses the pth moment exponential stability for a general class of neutral stochastic functional differential equations of the Ito type. This investigation can be very complicated, even in many special cases, by using usual methods based on Lyapunov functionals. In this paper we present criteria which are relatively easy to verify the pth moment exponential stability of the solutions of such equations.

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Since neutral stochastic functional differential equations are essentially complex, it is not easy to study stability problems of their solutions by applying usual procedures based on Lyapunov functionals. In this paper we present criteria on the bases of which it is relatively easy to verify the p-th moment exponential stability.

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