Adaptive nonparametric estimation for L\'evy processes observed at low frequency

TL;DR

This paper develops adaptive nonparametric kernel estimators for Le9vy processes observed at low frequency, focusing on bandwidth selection via model selection techniques and addressing unknown variance issues.

Contribution

It introduces a novel adaptive bandwidth selection method for nonparametric estimation of Le9vy processes, with potential extensions to density deconvolution.

Findings

Established upper risk bounds for the estimators

Derived convergence rates under regularity assumptions

Proposed a new approach to model selection with unknown variance

Abstract

This article deals with adaptive nonparametric estimation for L\'evy processes observed at low frequency. For general linear functionals of the L\'evy measure, we construct kernel estimators, provide upper risk bounds and derive rates of convergence under regularity assumptions. Our focus lies on the adaptive choice of the bandwidth, using model selection techniques. We face here a non-standard problem of model selection with unknown variance. A new approach towards this problem is proposed, which also allows a straightforward generalization to a classical density deconvolution framework.