Compound Poisson approximation to estimate the Lévy density

Céline Duval, Ester Mariucci

Published 2017-02-28Version 1

We construct an estimator of the L\'evy density, with respect to the Lebesgue measure, of a pure jump L\'evy process from high frequency observations: we observe one trajectory of the L\'evy process over [0, T] at the sampling rate $\Delta$, where $\Delta$ $\rightarrow$ 0 as T $\rightarrow$ $\infty$. The main novelty of our result is that we directly estimate the L\'evy 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 $\le$ p \textless{} $\infty$, whereas existing results only focus on p $\in$ {2, $\infty$}. 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 L\'evy 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 $\ge$ 1, over Besov balls. The results are illustrated on several examples.