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 ←∞. When the diffusion coefficient has regularity r ≥ 1, we obtain a rate 1/ n ^{r /(1+2 r )}, both for pointwise estimation and for estimation on a compact subset of R: this is the same rate as for non-parametric estimation of a density with i.i.d. observations.