Compound Poisson approximation to estimate the Lévy density

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 of the L\'evy density, we circumvent the use of spectral techniques and in particular of the L\'evy-Khintchine formula. A linear wavelet estimators is built and its performance is studied in terms of $L_p$ loss functions, $p\geq 1$, over Besov balls. The resulting rates are minimax-optimal for a large class of L\'evy processes. We discuss the robustness of the procedure to the presence of a Brownian part and to the estimation set getting close to the critical value 0.