On properties of the second moment of solutions of stochastic differential functional equations with varying coefficients. (Ukrainian, English) Zbl 1074.60070

Yasyns’kyj, V. K.; Antonyuk, S. V.

Teor. Jmovirn. Mat. Stat. 70, 157-163 (2004); translation in Theory Probab. Math. Stat. 70, 177-184 (2005).

The authors prove a theorem on the mean square stability of solutions of the stochastic differential functional equations with Poisson switching \[ dx(t) = a\left( {t,x_{t}} \right)dt + b\left( {t,x_{t}} \right)dw\left( {t} \right) + \int\limits_{\mathbb R} g(t,x_{t},u)\bar\nu(dt,du),\;x(t)=\varphi(t),\;t\in(-\infty,0], \] where \(x_{t}=\{x(t+\theta), -\infty<\theta\leq0\}\), \(w(t)\) is a standard Wiener process, \(\bar\nu(t,A)\) is a centered Poisson measure independent of \(w(t)\).

Reviewer: Mikhail P. Moklyachuk (Kyïv)

MSC:

60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
60F15	Strong limit theorems
60G42	Martingales with discrete parameter
62J05	Linear regression; mixed models

Keywords:

asymptotic stability