Khasminskii-type theorems for stochastic functional differential equations

Minghui Song; Liangjian Hu; Xuerong Mao; Liguo Zhang

doi:10.3934/dcdsb.2013.18.1697

Article Contents

2013, Volume 18, Issue 6: 1697-1714. Doi: 10.3934/dcdsb.2013.18.1697

This issue Previous Article Exponential growth rate for a singular linear stochastic delay differential equation Next Article Mean-square random attractors of stochastic delay differential equations with random delay

Khasminskii-type theorems for stochastic functional differential equations

1.
Department of Mathematics, Harbin Institute of Technology, Harbin 150001
2.
Department of Applied Mathematics, Donghua Univerisity, Shanghai 201600
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

Received: September 30, 2011

Revised: November 30, 2011

Published: July 2013

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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.

Keywords:

Brownian motion,
Itô's formula,
Khasminskii-test,
Khasminskii-type condition,
stochastic functional differential equations.

Mathematics Subject Classification: 34K50, 60H10, 34F05, 93E03.

Citation:

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References

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