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

Abstract

The stability of equilibrium solutions of a deterministic linear system of delay differential equationscan 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 onthe 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, providingsufficient 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 andthen compare them with the same equilibrium in the deterministic case. Finally the theoretical results are illustrated by numerical simulations.