On the existence of periodic solutions of stochastic differential equations with small parameter. (English. Ukrainian original) Zbl 0986.60049

Il’chenko, O. V.

Theory Probab. Math. Stat. 61, 33-39 (2000); translation from Teor. Jmovirn. Mat. Stat. 61, 33-38 (2000).

This paper deals with the system of stochastic differential equations \[ dx(t)=\varepsilon b(t,x(t))dt+\sqrt\varepsilon\sum_{r=1}^{N}\sigma^{r}(t,x(t))dw_{r}(t), \] 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.

Reviewer: A.V.Swishchuk (Kyïv)

MSC:

60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
93E15	Stochastic stability in control theory
34D20	Stability of solutions to ordinary differential equations
60F05	Central limit and other weak theorems

Keywords:

stochastic differential equation; periodic solution; stochastic stability; averaging system; Lyapunov function

Citations:

Zbl 0842.60056