Identifying the successive Blumenthal-Getoor indices of a discretely observed process

Abstract

This paper studies the identification of the Lévy jump measure of a discretely-sampled semimartingale. We define successive Blumenthal-Getoor indices of jump activity, and show that the leading index can always be identified, but that higher order indices are only identifiable if they are sufficiently close to the previous one, even if the path is fully observed. This result establishes a clear boundary on which aspects of the jump measure can be identified on the basis of discrete observations, and which cannot. We then propose an estimation procedure for the identifiable indices and compare the rates of convergence of these estimators with the optimal rates in a special parametric case, which we can compute explicitly.

arXiv commentPublished in at http://dx.doi.org/10.1214/12-AOS976 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (6)

• Testing for jumps in a discretely observed process
• Estimating the degree of activity of jumps in high frequency data
• Nonparametric tests for pathwise properties of semimartingales
• Testing whether jumps have finite or infinite activity
• Spectral estimation of the fractional order of a Lévy process
• Risk bounds for the non-parametric estimation of Lévy processes

Cited by in corpus (3)

• Efficient maximum likelihood estimation for Lévy-driven Ornstein-Uhlenbeck processes
• Near-optimal estimation of jump activity in semimartingales
• Estimation of state-dependent jump activity and drift for Markovian semimartingales