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

## TL;DR

This paper introduces a robust two-scales power variation method for estimating a noisy subordinated Brownian motion, outperforming previous estimators even without microstructure noise, with practical guidelines for implementation.

## Contribution

It develops a novel two-scales power variation estimator for semiparametric jump processes that is noise-robust and achieves improved convergence rates over existing methods.

## Key findings

- Estimators are robust against microstructure noise.
- Proposed method attains better convergence rates than traditional estimators.
- Data-driven procedure effectively selects optimal parameters.

## Abstract

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 Ito 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.