Second moment boundedness of linear stochastic delay differential equations

Zhen Wang; Xiong Li; Jinzhi Lei

doi:10.3934/dcdsb.2014.19.2963

Article Contents

2014, Volume 19, Issue 9: 2963-2991. Doi: 10.3934/dcdsb.2014.19.2963

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Second moment boundedness of linear stochastic delay differential equations

1.
School of Mathematics, Hefei University of Technology, Hefei 230009
2.
School of Mathematical Sciences, Beijing Normal University, Beijing 100875
3.
Zhou Pei-Yuan Center for Applied Mathematics, MOE Key Laboratory of Bioinformatics, Tsinghua University, Beijing, 100084

Received: June 30, 2013

Revised: February 28, 2014

Published: October 2014

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Abstract

This paper studies the second moment boundedness of solutions of linear stochastic delay differential equations. First, we give a framework--for general $\mathrm{N}$-dimensional linear stochastic differential equations with a single discrete delay--of calculating the characteristic function for the second moment boundedness. Next, we apply the proposed framework to a specific case of a type of $2$-dimensional equation that the stochastic terms are decoupled. For the $2$-dimensional equation, we obtain the characteristic function that is explicitly given by equation coefficients, and the characteristic function gives sufficient conditions for the second moment to be bounded or unbounded.

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Mathematics Subject Classification: Primary: 34K06, 34K50.

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