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

TL;DR

This paper investigates the limits of identifying jump activity indices of a Lévy process from discrete data, establishing which indices are identifiable and proposing an estimation method with convergence analysis.

Contribution

It introduces the concept of successive Blumenthal-Getoor indices, clarifies their identifiability boundaries, and develops an estimation procedure with convergence rate analysis.

Findings

Leading index is always identifiable.

Higher order indices are only identifiable if close to previous ones.

Proposed estimators achieve near-optimal convergence rates in certain cases.

Abstract

This paper studies the identification of the L\'{e}vy 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.