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Nonparametric inference for discretely sampled Lévy processes. (English. French summary) Zbl 1235.62121

Summary: Given a sample from a discretely observed Lévy process \(X = (X_{t})_{t\geq 0}\) of afinite jump activity, the problem of nonparametric estimation of the Lévy density \(\rho\) corresponding to the process \(X\) is studied. An estimator of \(\rho\) is proposed that is based on a suitable inversion of the Lévy-Khintchine formula and a plug-in device. The main results of the paper deal with upper risk bounds for estimation of \(\rho\) over suitable classes of Lévy triplets. The corresponding lower bounds are also discussed.

MSC:

62M09 Non-Markovian processes: estimation
62G07 Density estimation
60G51 Processes with independent increments; Lévy processes
62M05 Markov processes: estimation; hidden Markov models
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