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

We consider the models Yi,n = ∫ i/n 0 σ(s)dWs + τ(i/n) i,n, and Ỹi,n = σ(i/n)Wi/n + τ(i/n) i,n, i = 1, . . . , n, where Wt denotes a standard Brownian motion and i,n are centered i.i.d. random variables with E ( i,n ) = 1 and finite fourth moment. Furthermore, σ and τ are unknown deterministic functions and Wt and ( 1,n, . . . , n,n) are assumed to be independent processes. Based on a spectral decomposition of the covariance structures we derive series estimators for σ and τ 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 i,n introduces an additionally degree of ill-posedness of 1/2; irrespectively of the tail behavior of i,n. The method is illustrated by a small numerical study. AMS 2000 Subject Classification: Primary 62M09, 62M10; secondary 62G08.