Mean square absolute stability of solutions of the stochastic differential equations unsolvable with respect to derivatives (Q2703333)

The author considers the system of stochastic differential equations NEWLINE\[NEWLINED dx(t)=Ax(t) dt+ Bx(t) dw(t)+\int_{U}C(u)x(t)\widetilde\nu(du,dt), \quad x(t_{0})=x_{0},\;t_{0}\leq t,NEWLINE\]NEWLINE where \(w(t)\) is a Wiener process; \(\widetilde\nu(du,dt)=\nu(du,dt)-\Pi(du)dt\) is a centered Poisson measure; \(A,B,D\) are non-random \(n\times n\) matrices; \(C(u)\) is a matrix-valued function such that \(\int_{U}|C(u)|^{2}\Pi(du)<+\infty\). One of the results is the following. Let the matrix bundle \(A-\lambda D\) be a Hurwitz bundle. The existence of a positive definite solution \(H\) of the matrix generalized Silvester equation NEWLINE\[NEWLINEA^{\text{T}}HD+D^{\text{T}}HA+B^{\text{T}}HB+ \int_{U}C^{\text{T}}(u)HC(u)\Pi(du)=-G,NEWLINE\]NEWLINE where \(G=G^{\text{T}}>0\), is a necessary and sufficient condition for the asymptotic mean square stability of the trivial solution of the considered system. Sufficient conditions for the asymptotic mean square stability and a necessary and sufficient condition for the exponential mean square stability of the trivial solution of the considered system are presented.