IDENTIFYING THE SUCCESSIVE BLUMENTHAL-GETOOR INDICES OF A DISCRETELY OBSERVED PROCESS

• 成果类型: Article
• 署名作者: Ait-Sahalia, Yacine; Jacod, Jean
• 署名单位: Princeton University; National Bureau of Economic Research; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Universite Paris Cite; Sorbonne Universite
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/12-AOS976
• 发表日期: 2012
• 页码: 1430-1464
• 关键词: High-frequency data; nonparametric-estimation; Levy processes; independent increments; jumps

摘要

This paper studies the identification of the Levy 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.