Stability of a class of stochastic differential systems (Q1081208)

The authors consider on a complete probability space (\(\Omega\),\({\mathcal F},P)\) the system \[ dX_ t=A(\omega)X_ t dt+B_ 1(\omega)f_ 1(t,\sigma,\omega)dw_ t+B_ 2(\omega)f_ 2(t,\sigma,\omega)dz_ t, \] \[ \sigma \equiv \sigma (t,\omega):=C^ T(\omega)X_ t,\quad X(0)=X_ 0,\quad t\in [0,\infty). \] Here A, \(B_ 1\), \(B_ 2\) and C are random n-vectors, \(f_ i: {\mathbb{R}}^+\times {\mathbb{R}}\times \Omega \to {\mathbb{R}}\), \(i=1,2\), \(w_ t\) is a standard Wiener process and \(z_ t\) a Poisson process independent of \(w_ t.\) Using results of \textit{A. C. H. Lee} and the second author [Nonlinear Anal. Theory, Methods Appl. 1, 175-185 (1977; Zbl 0347.60045)] and the authors [J. Integral Equations 4, 145-162 (1982; Zbl 0506.60058)] they obtain under a lot of technical assumptions mainly the following results: Theorem 2.1: There exists a unique solution of the above system. Theorem 3.2: There is a constant \(c>0\) such that \(\sup_{t>0} E(\sigma^ 2(t,\omega))\leq cE(h^ 2(t,\omega))\) where \(h(t,\omega)=C^ T(\omega)\exp (A(\omega)t)X_ 0\).