Abstract
We construct an estimator of the Lévy density of a pure jump Lévy 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 Lévy density relying on a pathwise strategy, whereas existing procedures rely on spectral techniques. By taking advantage of a compound Poisson approximation, we circumvent the use of spectral techniques and in particular of the Lévy–Khintchine formula. A linear wavelet estimator is built and its performance is studied in terms of loss functions, , over Besov balls. We recover classical nonparametric rates for finite variation Lévy processes and for a nonparametric class of symmetric infinite variation Lévy processes. We show that the procedure is robust when the estimation set gets close to the critical value 0 and also discuss its robustness to the presence of a Brownian part.
Funding Statement
The work of E. Mariucci has been partially funded by the Federal Ministry for Education and Research through the Sponsorship provided by the Alexander von Humboldt Foundation, by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – 314838170, GRK 2297 MathCoRe, and by Deutsche Forschungsgemeinschaft (DFG) through grant CRC 1294 ‘Data Assimilation’.
Acknowledgements
The authors are very grateful to two anonymous Referees whose valuable comments helped making the statements of the results clearer.
Citation
Céline Duval. Ester Mariucci. "Spectral-free estimation of Lévy densities in high-frequency regime." Bernoulli 27 (4) 2649 - 2674, November 2021. https://doi.org/10.3150/21-BEJ1326
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