Nonparametric Estimation of the Volatility Function in a High-Frequency Model corrupted by Noise

We consider the models Yi;n = R i=n 0 (s)dWs + (i=n) i;n, and ~ Yi;n = (i=n)Wi=n + (i=n) i;n, i = 1;:::;n, where (Wt) t2[0;1] denotes a standard Brownian motion and i;n are centered i.i.d. random variables with E 2 = 1 and nite fourth moment. Furthermore, and are unknown deterministic functions and (Wt) t2[0;1] 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 2 and 2 and investigate their rate of convergence of the MISE in dependence of their smoothness. To this end specic 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 nding 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 performance of the estimates is illustrated by a small numerical study.

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