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

Stanzhyts’kyj, O. M.

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
\[ dx=[A(t)x+f(t,x)]\,dt+ g(t,x)\,dw(t). \]

Reviewer: A. D. Borisenko (Kyïv)

MSC:

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

Keywords:

bounded and periodic solutions; linear stochastic Itô’s system; weakly nonlinear stochastic Itô’s system