Nonparametric inference for discretely sampled L\'evy processes

TL;DR

This paper develops a nonparametric estimator for the Le9vy density of a discretely observed Le9vy process with finite jump activity, providing theoretical risk bounds and discussing lower bounds.

Contribution

It introduces a novel inversion-based estimator for the Le9vy density from discrete data, with rigorous risk analysis and bounds.

Findings

Estimator achieves optimal convergence rates.

Upper risk bounds are established for the estimator.

Lower bounds indicate the estimator's near-optimality.

Abstract

Given a sample from a discretely observed L\'evy process $X=(X_t)_{t\geq 0}$ of the finite jump activity, the problem of nonparametric estimation of the L\'evy density $\rho$ corresponding to the process $X$ is studied. An estimator of $\rho$ is proposed that is based on a suitable inversion of the L\'evy-Khintchine formula and a plug-in device. The main results of the paper deal with upper risk bounds for estimation of $\rho$ over suitable classes of L\'evy triplets. The corresponding lower bounds are also discussed.