We consider the models Y_{i,n}=\int_0^{i/n}
\sigma(s)dW_s+\tau(i/n)\epsilon_{i,n}, and \tilde
Y_{i,n}=\sigma(i/n)W_{i/n}+\tau(i/n)\epsilon_{i,n}, i=1,...,n, where W_t
denotes 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 series
estimators for \sigma^2 and \tau^2 and investigate their rate of convergence of
the MISE in dependence of their smoothness. To this end specific basis
functions and their corresponding Sobolev ellipsoids are introduced and we show
that our estimators are optimal in minimax sense. Our work is motivated by
microstructure 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 is
illustrated by a small numerical study.