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New criteria on exponential stability of neutral stochastic differential delay equations. (English) Zbl 1100.93048
Summary: Neutral stochastic differential delay equations (NSDDEs) have recently been studied intensively (see e.g. V.B. Kolmanovskii, V.R. Nosov [Stability and periodic regimes of control systems with aftereffect, Nauka, Moscow (1981; Zbl 0457.93002)] and X. Mao [Syst. Control Lett. 26, 245–251 (1995; Zbl 0877.93133 ); SIAM J. Math. Anal. 28, No. 2, 389–401 (1997; Zbl 0876.60047); Stochastic differential equations and their applications, Horwood Publishing, Chichester (1997; Zbl 0892.60057)]). More recently, X. Mao [Asymptotic properties of neutral stochastic differential delay equations, Stochastics Stochastics Rep. 68, No. 3-4, 273–295 (2000; Zbl 0964.60066)] 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.
MSC:
93E15Stochastic stability