On estimating a dynamic function of a stochastic system with averaging

We consider a two-scaled diffusion system, when drift and diffusion parameters of the 'slow' component are contaminated by the ' fast' unobserved component. The goal is to estimate the dynamic function which is defined by averaging the drift coefficient of the 'slow' component w.r.t. the stationary distribution of the 'fast' one. We apply a locally linear smoother with a datadriven bandwidth choice. The procedure is fully adaptive and nearly optimal up to a log log factor.