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Sufficient stability conditions for stochastic systems with aftereffect. (English. Russian original) Zbl 0824.93069
Differ. Equations 30, No. 4, 509-517 (1994); translation from Differ. Uravn. 30, No. 4, 555-564 (1994).
The stochastic functional differential equation \(dx(t) = [(Vx) (t) + f(t)] dZ(t)\), \(t \geq 0\) is under study. Here \(x(t)\) is an \(n\)- dimensional stochastic process on \([0,\infty)\), \(Z\) is an \(m\)-dimensional semimartingale, \(f(t)\) is an \(n \times m\)-matrix function on \([0, \infty)\), and \(V\) is a related linear operator. The \(p\)-stability (including asymptotical and exponential) of the trivial solution of the above equation is defined by the behavior of the mean value of \(| x(t)|^ p\) when \(t\) tends to infinity. The author finds necessary and sufficient conditions for the \(p\)-stability of the stochastic system. As a special case the stability of a linear stochastic system with distributed delay is studied.

93E15 Stochastic stability in control theory
93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory
34K50 Stochastic functional-differential equations