Mean square estimates of solutions of stochastic differential equations with delay and Poisson switching (Q2761525)

The authors consider the stochastic differential equation with delay NEWLINE\[NEWLINE\begin{multlined} dx(t)=[A_0x(t)+A_1x(t-\tau)] dt+ [B_0x(t)+ B_1x(t-\tau)] dw(t)+\\ \int_{U}[C_0(u)x(t)+C_1(u)x(t-\tau)]\widetilde\nu(du,dt)\end{multlined}NEWLINE\]NEWLINE with the initial condition \(x(t)=\psi(t),\;t\in[t_0-\tau,t_0],\;\tau>0\), where \(A_0,A_1,B_0,B_1\) are real matrices; \(\psi\in C([t_0-\tau,t_0])\); \(w(t)\) is a one-dimensional Wiener process; \(\widetilde\nu(du,dt)=\nu(du,dt)-\Pi(du) dt\) is a centered Poisson measure; \(C_0(u), C_1(u)\) are matrix-valued functions. Under some condition on the solution \(H\) of the Silvester equation NEWLINE\[NEWLINEA^{\text{T}}H+HA+B^{\text{T}}HB+\int_{U}C^{\text{T}}(u)HC(u)\Pi(du)= -G,NEWLINE\]NEWLINE NEWLINE\[NEWLINEA\equiv A_0+A_1,\quad B\equiv B_0+B_1, \quad C(u)\equiv C_0(u)+C_1(u),NEWLINE\]NEWLINE where \(G\) is an arbitrary symmetric positive definite matrix, the following estimate is obtained: For arbitrary \(\varepsilon>0\): \(M|x(t)|^2<\varepsilon,\;\forall t>t_0\) if \(\sup_{-\tau\leq s\leq 0}|\psi(t_0+s)|<\varepsilon/\varphi(H)\), where \(\varphi(H)=\lambda_{\max}/ \lambda_{\min}\), \(\lambda_{\max}\) and \(\lambda_{\min}\) are maximal and minimal positive eigenvalues of the matrix \(H\), respectively.