# Spectral-free estimation of L\'evy densities in high-frequency regime

## TL;DR

This paper introduces a pathwise, spectral-free method for estimating the Le9vy density of pure jump Le9vy processes from high-frequency data, avoiding spectral techniques and the Le9vy--Khintchine formula.

## Contribution

It proposes a novel direct, pathwise estimation procedure for Le9vy densities that works for infinite variation processes, bypassing spectral methods and using wavelet estimators.

## Key findings

- Achieves classical nonparametric rates for finite variation processes
- Demonstrates robustness near the critical estimation boundary
- Shows effectiveness even with a Brownian component present

## Abstract

We construct an estimator of the L\'evy density of a pure jump L\'evy process, possibly of infinite variation, from the discrete observation of one trajectory at high frequency. The novelty of our procedure is that we directly estimate the L\'evy density relying on a pathwise strategy, whereas existing procedures rely on spectral techniques. By taking advantage of a compound Poisson approximation, we circumvent the use of spectral techniques and in particular of the L\'evy--Khintchine formula. A linear wavelet estimator is built and its performance is studied in terms of $L_p$ loss functions, $p\geq 1$, over Besov balls. We recover classical nonparametric rates for finite variation L\'evy processes and for a large nonparametric class of symmetric infinite variation L\'evy processes. We show that the procedure is robust when the estimation set gets close to the critical value 0 and also discuss its robustness to the presence of a Brownian part.