A note on the analysis of asymptotic mean-square stability properties for systems of linear stochastic delay differential equations

Evelyn Buckwar; Girolama Notarangelo

doi:10.3934/dcdsb.2013.18.1521

Article Contents

2013, Volume 18, Issue 6: 1521-1531. Doi: 10.3934/dcdsb.2013.18.1521

This issue Previous Article Preface Next Article Invariance and monotonicity for stochastic delay differential equations

A note on the analysis of asymptotic mean-square stability properties for systems of linear stochastic delay differential equations

Evelyn Buckwar^1, and
Girolama Notarangelo^1,

1.
Johannes Kepler University, Institute for Stochastics, Altenbergerstraße 69, 4040 Linz

Received: December 2011

Revised: April 2012

Published: August 2013

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Abstract

The stability of equilibrium solutions of a deterministic linear system of delay differential equations can be investigated by studying the characteristic equation. For stochastic delay differential equations stability analysis is usually based on Lyapunov functional or Razumikhin type results, or Linear Matrix Inequality techniques. In [7] the authors proposed a technique based on the vectorisation of matrices and the Kronecker product to transform the mean-square stability problem of a system of linear stochastic differential equations into a stability problem for a system of deterministic linear differential equations. In this paper we extend this method to the case of stochastic delay differential equations, providing sufficient and necessary conditions for the stability of the equilibrium. We apply our results to a neuron model perturbed by multiplicative noise. We study the stochastic stability properties of the equilibrium of this system and then compare them with the same equilibrium in the deterministic case. Finally the theoretical results are illustrated by numerical simulations.

Keywords:

Mean-square stability analysis,
linear stochastic delay differential systems,
Kronecker product,
neuron model.

Mathematics Subject Classification: Primary: 34K50, 60H20; Secondary: 34K20, 93E15.

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References

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