Algebraic criteria and sufficient conditions for asymptotic stability and boundedness with probability 1 for the solutions of a system of linear stochastic difference equations (Q1095497)

A criterion and sufficient conditions are given for asymptotic stability and boundedness with probability 1 of solutions of a system of linear stochastic difference equations of the form \[ x(k+1)=Ax(k)+\sum^{r}_{i=1}B_ i(\epsilon)x(k)z_ i(k),\quad k=0,1,2,..., \] \[ z_ i(k)=w_ i(k+1)-w_ i(k),\quad i=1,2,...,r, \] where A and \(B_ i(\epsilon)\) are constant \(n\times n\) matrices, w is an \(R^ r\)-valued Wiener process and \(\epsilon\) is a parameter.