A Lepski\u{i}-type stopping rule for the covariance estimation of multi-dimensional L\'evy processes

TL;DR

This paper develops a data-driven, adaptive method for estimating the covariance of multi-dimensional Lévy processes observed discretely, using a Lepskii-type stopping rule to achieve near-optimal estimation rates.

Contribution

It introduces a Lepskii-type stopping rule for adaptive covariance estimation of Lévy processes, improving upon existing methods with a data-driven parameter choice.

Findings

The adaptive estimator attains near-optimal convergence rates.

Numerical experiments demonstrate the effectiveness of the proposed selection rule.

The method provides a practical approach for covariance estimation in Lévy processes.

Abstract

We suppose that a L\'evy process is observed at discrete time points. Starting from an asymptotically minimax family of estimators for the continuous part of the L\'evy Khinchine characteristics, i.e., the covariance, we derive a data-driven parameter choice for the frequency of estimating the covariance. We investigate a Lepski\u{i}-type stopping rule for the adaptive procedure. Consequently, we use a balancing principle for the best possible data-driven parameter. The adaptive estimator achieves almost the optimal rate. Numerical experiments with the proposed selection rule are also presented.