Khasminskii-type theorems for stochastic functional differential equations

Song, Minghui and Hu, Liangjian and Mao, Xuerong and Zhang, Liguo (2013) Khasminskii-type theorems for stochastic functional differential equations. Discrete and Continuous Dynamical Systems - Series B, 18 (6). pp. 1697-1714. ISSN 1531-3492

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Abstract

For a stochastic functional differential equation (SFDE) to have a unique global solution it is in general required that the coefficients of the SFDE obey the local Lipschitz condition and the linear growth condition. However, there are many SFDEs in practice which do not obey the linear growth condition. The main aim of this paper is to establish existence-and-uniqueness theorems for SFDEs where the linear growth condition is replaced by more general Khasminskii-type conditions in terms of a pair of Laypunov-type functions.