Non-parametric kernel estimation of the coefficient of a diffusion

In this work we exhibit a non-parametric estimator of kernel type, for the diffusion coefficient when one observes a one-dimensional diffusion process at times i/n for i = ,..., n and study its asymptotics as n --> infinity. When the diffusion coefficient has regularity r greater than or equal to 1, we obtain a rate 1/n(r/(1+2r)), both for pointwise estimation and for esti mation on a compact subset of R: this is the same rate as for non-parametri c estimation of a density with i.i.d. observations.