Algebraic criteria and sufficient conditions for asymptotic stability and boundedness with probability 1 for the solutions of a system of linear stochastic difference equations

Abstract

In this article, using the method of stochastic Lyapunov functions (SLF), we obtain new algebraic criteria and sufficient conditions for the asymptotic stability with probability 1 for the solutions of a system of linear stationary difference equations with random (of the type of a vector "white" sequence of random variables) coefficients, which are discrete analogs of the conditions established earlier by several authors for stochastic Ito equations with continuous time. We assume that in the absence of parametric random perturbations, an unperturbed determined system of difference equations is asymptotically Lyapunov stable (the matrix A of the system is convergent). The conditions are formulated either in terms of the negative definiteness of certain matrix relations, involving the matrix A and the matrices of the parametric perturbations Bi, i = i, 2 .... , r (sufficient conditions, which in many cases are nearly necessary), or in terms of the existence of a positive-definite solution H of the Sylvester matrix algebraic equation e