Compound Poisson approximation to estimate the Lévy density

We construct an estimator of the Levy density, with respect to the Lebesgue measure, of a pure jump Levy process from high frequency observations: we observe one trajectory of the Levy process over [0, T] at the sampling rate ∆, where ∆ → 0 as T → ∞. The main novelty of our result is that we directly estimate the Levy density in cases where the process may present infinite activity. Moreover, we study the risk of the estimator with respect to L_p loss functions, 1 ≤ p < ∞, whereas existing results only focus on p ∈ {2, ∞}. The main idea behind the estimation procedure that we propose is to use that "every infinitely divisible distribution is the limit of a sequence of compound Poisson distributions" (see e.g. Corollary 8.8 in Sato (1999)) and to take advantage of the fact that it is well known how to estimate the Levy density of a compound Poisson process in the high frequency setting. We consider linear wavelet estimators and the performance of our procedure is studied in term of L_p loss functions, p ≥ 1, over Besov balls. The results are illustrated on several examples.