Identifying the successive blumenthal-getoor indices of a discretely observed process

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Abstract

This paper studies the identification of the Ĺvy 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.

Original languageEnglish (US)
Pages (from-to)1430-1464
Number of pages35
JournalAnnals of Statistics
Volume40
Issue number3
DOIs
StatePublished - Jun 2012

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

Keywords

  • Brownian motion
  • Discrete sampling
  • Finite activity
  • High frequency
  • Infinite activity
  • Jumps
  • Semimartingale

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