Adaptive pointwise estimation for pure jump L\'evy processes

TL;DR

This paper develops an adaptive kernel estimation method for the L\'evy density of pure-jump processes, achieving optimal convergence rates and demonstrating effectiveness through simulations and irregular sampling analysis.

Contribution

It introduces a novel adaptive bandwidth selection technique for kernel estimation of L\'evy densities in high-frequency data, with proven optimal convergence rates.

Findings

Achieves minimax optimal convergence rates.

Provides effective adaptive estimation method.

Demonstrates robustness with irregular sampling.

Abstract

This paper is concerned with adaptive kernel estimation of the L\'evy density N(x) for bounded-variation pure-jump L\'evy processes. The sample path is observed at n discrete instants in the "high frequency" context (\Delta = \Delta(n) tends to zero while n\Delta tends to infinity). We construct a collection of kernel estimators of the function g(x)=xN(x) and propose a method of local adaptive selection of the bandwidth. We provide an oracle inequality and a rate of convergence for the quadratic pointwise risk. This rate is proved to be the optimal minimax rate. We give examples and simulation results for processes fitting in our framework. We also consider the case of irregular sampling.