Sup-norm adaptive simultaneous drift estimation for ergodic diffusions

We consider the question of estimating the drift and the invariant density for a large class of scalar ergodic diffusion processes, based on continuous observations, in $\sup$-norm loss. The unknown drift $b$ is supposed to belong to a nonparametric class of smooth functions of unknown order. We suggest an adaptive approach which allows to construct drift estimators attaining minimax optimal $\sup$-norm rates of convergence. In addition, we prove a Donsker theorem for the classical kernel estimator of the invariant density and establish its semiparametric efficiency. Finally, we combine both results and propose a fully data-driven bandwidth selection procedure which simultaneously yields both a rate-optimal drift estimator and an asymptotically efficient estimator of the invariant density of the diffusion. Crucial tool for our investigation are uniform exponential inequalities for empirical processes of diffusions.