Sup-norm adaptive drift estimation for multivariate nonreversible diffusions

We consider the question of estimating the drift for a large class of ergodic multivariate and possibly nonreversible diffusion processes, based on continuous observations, in sup -norm loss. Nonparametric classes of smooth functions of unknown order are considered, and we suggest an adaptive approach which allows to construct drift estimators attaining optimal sup -norm rates of convergence. Reversibility structures and related functional inequalities are known to be key tools for these estimation problems. We can discard such restrictions by making use of mixing properties which are satisﬁed for the very general class of processes under consideration. Analysing diffusions, the scalar case is very distinct from the general multivariate setting. Therefore, we treat scalar and multivariate processes separately which leads to in several aspects improved univariate results. While we consider drift estimation on bounded domains for exponentially β -mixing multivariate processes, for scalar diffusion processes we work under minimal assumptions that allow estimation of unbounded drift terms over the entire real line, and we provide classical minimax results (including lower bounds) which cannot be obtained under state-of-the-art conditions in the multivariate case. In addition, we prove a Donsker theorem for the classical kernel estimator of the invariant density in the scalar setting and establish its semiparametric efﬁciency.