# American Institute of Mathematical Sciences

August  2013, 18(6): 1697-1714. doi: 10.3934/dcdsb.2013.18.1697

## Khasminskii-type theorems for stochastic functional differential equations

 1 Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China 2 Department of Applied Mathematics, Donghua Univerisity, Shanghai 201600, China 3 Department of Mathematics and Statistics, University of Strathclyde, Glasgow G1 1XH 4 School of Electronic Information and Control Engineering, Beijing University of Technology, Beijing 100124, China

Received  October 2011 Revised  December 2011 Published  March 2013

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.
Citation: Minghui Song, Liangjian Hu, Xuerong Mao, Liguo Zhang. Khasminskii-type theorems for stochastic functional differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1697-1714. doi: 10.3934/dcdsb.2013.18.1697
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