Optimal estimation of some random quantities of a L\'evy process

Jevgenijs Ivanovs; Mark Podolskij

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arXiv · 2001.02517

Optimal estimation of some random quantities of a L\'evy process

Jevgenijs Ivanovs·Mark Podolskij

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Abstract

In this paper we present new theoretical results on optimal estimation of certain random quantities based on high frequency observations of a L\'evy 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 stable L\'evy process. Another contribution of our article is the conditional mean estimation of the local time and the occupation time measure of a linear Brownian motion. We demonstrate that the new estimators are considerably more efficient compared to the classical estimators. Furthermore, we discuss pre-estimation of the parameters of the underlying models, which is required for practical implementation of the proposed statistics.

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Jevgenijs Ivanovs, Mark Podolskij. 2020-01-08. Optimal estimation of some random quantities of a L\'evy process. https://arxiv.org/abs/2001.02517

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