Estimation of a noisy subordinated Brownian motion via two-scales power variations

- High-frequency estimators for a Lévy model with microstructure noise are developed.
- Besides a volatility parameter, the model is equipped with a kurtosis parameter.
- The latter is important to model extremes in finance, insurance, and many other fields.
- The estimators exhibit better performance than others, even in the absence of noise.
- This is due to an optimal tuning procedure of the estimator's parameters.
- Data driven implementation of the optimal formulas are devised and numerically studied.

High frequency based estimation methods for a semiparametric pure-jump subordinated Brownian motion exposed to a small additive microstructure noise are developed building on the two-scales realized variations approach originally developed by Zhang et al. (2005) for the estimation of the integrated variance of a continuous Itô process. The proposed estimators are shown to be robust against the noise and, surprisingly, to attain better rates of convergence than their precursors, method of moment estimators, even in the absence of microstructure noise. Our main results give approximate optimal values for the number K of regular sparse subsamples to be used, which is an important tune-up parameter of the method. Finally, a data-driven plug-in procedure is devised to implement the proposed estimators with the optimal K-value. The developed estimators exhibit superior performance as illustrated by Monte Carlo simulations and a real high-frequency data application.