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

We consider the models Y-i,Y-n = integral(i/n)(0) sigma(s)dW(s) + tau(i/n)epsilon(i,n), and (Y) over tilde (i,n) = sigma(i/n)W-i/n + tau(i/n)epsilon(i,n), i = 1,..., n, where (W-t)(t is an element of[0,1]) denotes a standard Brownian motion and epsilon(i,n) are centered i.i.d. random variables with E(epsilon(2)(i,n)) = 1 and finite fourth moment. Furthermore, sigma and tau are unknown deterministic functions and (W-t)(t is an element of[0,1]) 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. A 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 performance of the estimates is illustrated by a small numerical study.