Stability of nonlinear neutral stochastic functional differential equations. (English) Zbl 1206.60058

Xue, Minggao; Zhou, Shaobo; Hu, Shigeng

J. Appl. Math. 2010, Article ID 425762, 26 p. (2010).

Summary: Neutral stochastic functional differential equations (NSFDEs) have recently been studied intensively. The well-known conditions imposed for the existence and uniqueness and exponential stability of the global solution are the local Lipschitz condition and the linear growth condition. Therefore, the existing results cannot be applied to many important nonlinear NSFDEs. The main aim of this paper is to remove the linear growth condition and establish a Khasminskii-type test for nonlinear NSFDEs. The new criteria not only cover a wide class of highly nonlinear NSFDEs, but they can also be verified much more easily than the classical criteria. Finally, several examples are given to illustrate main results.

MSC:

60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
34K20	Stability theory of functional-differential equations
34K50	Stochastic functional-differential equations

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