Abstract
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 underlying models, which is required for practical implementation of the proposed statistics.
Highlights
During the past decades the increasing availability of high frequency data in economics and finance has led to an immense progress in high frequency statistics
High frequency functionals of Itosemimartingales have received a great deal of attention in the statistical and probabilistic literature, where the focus has been on estimation of quadratic variation, realised jumps and related quantities
In the case of local/occupation time we only work with the class (i) of linear Brownian motions and focus on the conditional mean estimators exclusively, which is dictated by the structure of the problem and the tools currently available
Summary
During the past decades the increasing availability of high frequency data in economics and finance has led to an immense progress in high frequency statistics. The aim of our paper is to study optimal estimation of extrema, local time and occupation time of certain Levy processes Accurate estimation of these random functionals is important for numerous applications. In the case of local/occupation time we only work with the class (i) of linear Brownian motions and focus on the conditional mean estimators exclusively, which is dictated by the structure of the problem and the tools currently available. The proofs are collected in Appendix A and Appendix B for the supremum and local/occupation time, respectively The former requires some additional theory for Levy processes conditioned to stay positive which is given in Appendix C
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