On the stability of the solutions of stochastic differential equations with delay which non solved on derivatives (Q2703348)

The author obtains necessary and sufficient conditions for the asymptotic Lyapunov stability in the mean square of the trivial solution of stochastic differential equations NEWLINE\[NEWLINED dx(t)= [Ax(t)+A_1x(t-\tau)] dt+ [Bx(t)+B_1x(t-\tau)] dw(t)+ \int_{U}[C(u)x(t)+C_1x(t-\tau)]\tilde\nu(du,dt),NEWLINE\]NEWLINE with condition \(x(\theta)=x_0\neq 0\), \(0\leq t_0\leq t\); \(\tau>0\); \(t_0-\tau\leq \theta\leq t_0\), where \(w(t)\) is the standard one-dimensional Wiener process, \(\widetilde\nu(du,dt)=\nu(du,dt)-\Pi(du) dt\) is the centered Poisson measure; \(A,B,A_1,B_1,D\) are constant matrices, \(C(u), C_1(u)\) are matrix-valued functions such that \(\int_{U}\|C(u)\|^2\Pi(du)<\infty\), \(\int_{U}\|C_1(u)\|^2\Pi(du)<\infty\).