On the estimation of the jump activity index in the case of random observation times

Abstract

We propose a nonparametric estimator of the jump activity index beta of a pure-jump semimartingale X driven by a beta -stable process when the underlying observations are coming from a high-frequency setting at irregular times. The proposed estimator is based on an empirical characteristic function using rescaled increments of X, with a limit that depends in a complicated way on beta and the distribution of the sampling scheme. Utilising an asymptotic expansion we derive a consistent estimator for beta and prove an associated central limit theorem.