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Approximation and existence of periodic solutions for controlled diffusion equations. (English) Zbl 0703.60052
The author proved the existence of bounded T-periodic controls $$u_ i(t,x)$$, $$i=1,...,m$$, acting in the same directions as the Wiener process, such that the equation with T-periodic coefficients $dx=[f(t,x)+\sum^{m}_{i=1}u_ i(t,x)g_ i(t,x)]dt+\sum^{m}_{i=1}g_ i(t,x)dw_ i(t),$ $$t\geq 0$$, $$x\in {\mathbb{R}}^ n$$, has a solution T-periodic in distribution. The main assumption is a nondegeneracy condition on the diffusion part: There exist $$h_ 1,...,h_ n\in {\mathcal L}(g_ 1,...,g_ m)$$ and $$K>0$$ such that $\lambda^*H(t,x)H^*(t,x)\lambda \geq K| \lambda |^ 2\text{ for } every\quad \lambda \in {\mathbb{R}}^ n,\quad t\in [0,T]\text{ and } x\in {\mathbb{R}}^ n,$ where $$H(t,x)=(h_ 1(t,x),...,h_ n(t,x))$$ and $${\mathcal L}(g_ 1,...,g_ m)$$ is the Lie algebra generated by $$g_ 1,...,g_ m$$. The method of the author is based on an approximation theorem for solutions of SDEs.
Reviewer: Tran Van Nhung

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