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 data-driven bandwidth choice. The procedure is fully adaptive and nearly optimal up to a log log factor. This revised version was published online in June 2006 with corrections to the Cover Date.