Stability with probability 1 of solutions of systems of linear stochastic differential-difference Itô equations (Q806164)

Let \[ (1)\quad dx(t)=[Ax(t)+\sum^{k}_{i=1}B_ ix(t-\tau_ i)]dt+\sum^{r}_{j=1}C_ jx(t)dw_ j \] be a system of differential- difference equations of Itô type, where \(0\leq \tau_ 1\leq \tau_ 2\leq...\leq \tau_ k<\infty\), A, \(B_ i\), \(i=1,...,k\), \(C_ j\), \(j=1,...,r\), are matrices and \(w_ j\), \(j=1,...,r\), are independent standard Wiener processes. It is shown that if there exist \(\gamma_ i>0\) such that \[ Re \lambda_ i\{A_{[2]}+\sum^{k}_{i=1}[\gamma_ iI+(1/\gamma_ i)B_ i^{[2]}]+\sum^{r}_{j=1}C_ j^{[2]}\}<0,\quad i=1,...,m \] (where \(A_{[2]}\), \(B_ i^{[2]}\), \(C_ j^{[2]}\) are certain transformations of matrices), then the trivial solution of (1) is asymptotic stable with probability one.