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We consider the models Y_{i,n}=\int_0^{i/n}\sigma(s)dW_s+\tau(i/n)\epsilon_{i,n}, and \tildeY_{i,n}=\sigma(i/n)W_{i/n}+\tau(i/n)\epsilon_{i,n}, i=1,...,n, where W_tdenotes a standard Brownian motion and \epsilon_{i,n} are centered i.i.d.random variables with E(\epsilon_{i,n}^2)=1 and finite fourth moment.Furthermore, \sigma and \tau are unknown deterministic functions and W_t and(\epsilon_{1,n},...,\epsilon_{n,n}) are assumed to be independent processes.Based on a spectral decomposition of the covariance structures we derive seriesestimators for \sigma^2 and \tau^2 and investigate their rate of convergence ofthe MISE in dependence of their smoothness. To this end specific basisfunctions and their corresponding Sobolev ellipsoids are introduced and we showthat our estimators are optimal in minimax sense. Our work is motivated bymicrostructure noise models. Our major finding is that the microstructure noise\epsilon_{i,n} introduces an additionally degree of ill-posedness of 1/2;irrespectively of the tail behavior of \epsilon_{i,n}. The method isillustrated by a small numerical study.

Nonparametric estimation of the volatility function in a high-frequency model corrupted by noise