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Discrete and Continuous Dynamical Systems - Series B (DCDS-B)
 

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

Pages: 1521 - 1531, Volume 18, Issue 6, August 2013      doi:10.3934/dcdsb.2013.18.1521

 
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Evelyn Buckwar - Johannes Kepler University, Institute for Stochastics, Altenbergerstra├če 69, 4040 Linz, Austria (email)
Girolama Notarangelo - Johannes Kepler University, Institute for Stochastics, Altenbergerstra├če 69, 4040 Linz, Austria (email)

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.

Received: December 2011;      Revised: April 2012;      Available Online: March 2013.

 References