Bounded and periodic solutions of linear and weakly nonlinear stochastic Itô’s system.(Ukrainian, English)Zbl 1050.60063

Teor. Jmovirn. Mat. Stat. 68, 133-141 (2003); translation in Theory Probab. Math. Stat. 68, 147-156 (2004).
Let us consider the system of Itô’s equations
$dx(t)=[A(t)x(t)+\alpha(t)]\,dt+ \beta(t)\,dw(t), \qquad t\in R^{1},$
where $$A(t)$$ is a bounded continuous matrix; $$\alpha(t), \beta(t)$$ are continuous $${\mathcal F}_{t}$$-measurable functions, $$\sup_{t\in R}E| \alpha(t)| ^2<\infty$$, $$\sup_{t\in R}E| \beta(t)| ^2<\infty$$. The author proves that if the deterministic system $$dx/dt=A(t)x$$ is exponentially stable, then the considered stochastic system has a unique mean-square bounded solution which is mean-square exponentially stable. If the matrix $$A(t)$$ is $$T$$-periodic and the process $$\eta(t)=(\alpha(t),\beta(t))$$ has $$T$$-periodic finite-dimensional distributions, then solutions of the considered stochastic system have $$T$$-periodic finite-dimensional distributions. The same problems are investigated for the weakly nonlinear stochastic system
$dx=[A(t)x+f(t,x)]\,dt+ g(t,x)\,dw(t).$

MSC:

 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 34F05 Ordinary differential equations and systems with randomness