On the existence of periodic solutions of stochastic differential equations with small parameter (Q2740442)

This paper deals with the system of stochastic differential equations NEWLINE\[NEWLINEdx(t)=\varepsilon b(t,x(t))dt+\sqrt\varepsilon\sum_{r=1}^{N}\sigma^{r}(t,x(t))dw_{r}(t),NEWLINE\]NEWLINE where \(x(t),\) \(b(t,x)\) and \(\sigma^{r}(t,x)\) are \(n\)-dimensional vectors, \(r=1,\dots,N,\) \(w^{r}(t)\) are independent Wiener processes, \(\varepsilon\in (0,\varepsilon_{0}]\) is a small positive parameter. The author applies averaging by time method and considers averaged linear system. If the linear averaged system is stable with probability \(1\), then the Lyapunov function is constructed which provides the existence of periodical solutions of the initial system. This result is based on a previous paper by the author [Random Oper. Stoch. Equ. 1, No. 1, 47-55 (1993; Zbl 0842.60056)]. Such approach gives a possibility to take into account a diffusion influence and to construct a linear autonomous system stability of which provides the existence of periodic solutions.