Optimal estimation of the supremum and occupation times of a self-similar Lévy process

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In this paper we present new theoretical results on optimal estimation of certain random quantities based on high frequency observations of a Lévy process. More specifically, we investigate the asymptotic theory for the conditional mean and conditional median estimators of the supremum/infimum of a linear Brownian motion and a strictly stable Lévy process. Another contribution of our article is the conditional mean estimation of the local time and the occupation time of a linear Brownian motion. We demonstrate that the new estimators are considerably more efficient compared to the classical estimators studied in e.g. [6, 14, 29, 30, 38]. Furthermore, we discuss pre-estimation of the parameters of the underly-ing models, which is required for practical implementation of the proposed statistics.

Original languageEnglish
JournalElectronic Journal of Statistics
Pages (from-to)892-934
Number of pages43
Publication statusPublished - Jul 2022


  • Conditioning to stay positive
  • local time
  • Lévy processes
  • occupation time
  • optimal estimation
  • self-similarity
  • supremum
  • weak limit theorems


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