Nonparametric adaptive estimation for pure jump L\'evy processes

TL;DR

This paper develops a nonparametric adaptive method for estimating the Le9vy density of pure jump Le9vy processes from discrete observations, providing risk bounds and convergence rates.

Contribution

It introduces a new adaptive estimation procedure using deconvolution techniques and characteristic function derivatives for pure jump Le9vy processes.

Findings

Provides risk bounds for the estimators.

Establishes convergence rates under certain conditions.

Demonstrates applicability to various models.

Abstract

This paper is concerned with nonparametric estimation of the L\'evy density of a pure jump L\'evy process. The sample path is observed at $n$ discrete instants with fixed sampling interval. We construct a collection of estimators obtained by deconvolution methods and deduced from appropriate estimators of the characteristic function and its first derivative. We obtain a bound for the ${\mathbb L}^2$-risk, under general assumptions on the model. Then we propose a penalty function that allows to build an adaptive estimator. The risk bound for the adaptive estimator is obtained under additional assumptions on the L\'evy density. Examples of models fitting in our framework are described and rates of convergence of the estimator are discussed.