Asymptotic stability of Itô differential systems with retarded argument (Q5926553)

For a given number \(p\), \(1\leq p<\infty\), and a positive scalar function \(\gamma(t)\), \(t\in [0, +\infty)\), introduce the following space: \[ M_p^\gamma=\left\{x:x\in D^n,\quad\|x\|_{M_p^\gamma}\overset\text{def}= \left(\sup_{t\geq 0}E|\gamma(t)x(t)|^p\right)^{1/p}<\infty\right\}, \] where \(D^n\) is the linear space of \(n\)-dimensional progressively measurable random processes on \([0, +\infty)\) whose trajectories are continuous almost surely and \(E\) is the expectation. The author investigates the \(M_p^\gamma\)-stability for linear Itô differential equations with lumped delay. The case of unbounded delay is investigated in detail. \(M_p^\gamma\)-stability conditions for the studied equations in terms of their parameters are given.