Estimation of a noisy subordinated Brownian Motion via two-scale power variations

Abstract: High frequency based estimation methods for a pure-jump subordinated Brownian motion exposed to a small additive microstructure noise are developed building on the two-scale realized variations approach developed by Zhang et al. (2005) for the estimation of a continuous Itô process. The proposed estimators are shown to be robust against the noise and to attain better rates of convergence than those of standard method of moment estimators even in the absence of noise. Our main results give approximate optimal values for the number K of regular sparse subsamples to be used, which is a crucial tune-up parameter of the method. Finally, a two-step data-driven procedure is devised to implement the proposed estimators with the optimal K-value. The superior finite-sample performance and empirical robustness of the resulting estimators are illustrated by Monte Carlo simulations and a real high-frequency data application.