Olga Aryasova (Inst. of Geophysics, Nat. Acad. of Sciences of Ukraine / Friedrich–Schiller–Univ. Jena)
Céline Duval (Université de Lille)
We consider the problem of estimating the Lévy density of a pure jump Lévy process, possibly of infinite variation, from the high frequency observation of one trajectory. To directly construct an estimator of the Lévy density, we use a compound Poisson approximation and we build a linear wavelet estimator. Its performance is studied in terms of \(L_p\) loss functions, \(p\geq1\), over Besov balls. To show that the resulting rates are minimax-optimal for a large class of Lévy processes, we propose new non-asymptotic bounds of the cumulative distribution function of Lévy processes with Lévy density bounded from above by the density of an alpha-stable type Lévy process in a neighbourhood of the origin. It is a joint work with Ester Mariucci.
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