  We study nonparametric estimation of the diffusion coefficient from discrete
data, when the observations are blurred by additional noise. Such issues have
been developed over the last 10 years in several application fields and in
particular in high frequency financial data modelling, however mainly from a
parametric and semiparametric point of view. This paper addresses the
nonparametric estimation of the path of the (possibly stochastic) diffusion
coefficient in a relatively general setting. By developing pre-averaging
techniques combined with wavelet thresholding, we construct adaptive estimators
that achieve a nearly optimal rate within a large scale of smoothness
constraints of Besov type. Since the diffusion coefficient is usually genuinely
random, we propose a new criterion to assess the quality of estimation; we
retrieve the usual minimax theory when this approach is restricted to a
deterministic diffusion coefficient. In particular, we take advantage of recent
results of Reiss [33] of asymptotic equivalence between a Gaussian diffusion
with additive noise and Gaussian white noise model, in order to prove a sharp
lower bound.
