Algebraic coefficient conditions for absolute (not depending on delay) asymptotic stability with probability 1, for solutions of a system of linear stochastic Itô equations with contagion (Q1081209)

Sufficient conditions are derived for the equation \[ (*)\quad dx(t)=[Ax(t)+A_ 1x(t-\tau)]dt+[Bx(t)+B_ 1x(t-\tau)]dw(t), \] \[ x(t_ 0-\theta)=x_ 0\neq 0\quad for\quad 0\leq \theta \leq \tau; \] to be asymptotically stable with probability one for any \(\tau >0\) (constant). More precisely, the Lyapunov-Krasovskij quadratic functional, defined with respect to the supposed stability of (*) with \(B=B_ 1=0\), is differentiated along the solution of (*) using Itô's formula, and the requirement for the mean value of the differential to be negative yields the algebraic sufficient condition.