Dear all, On Thursday, 12th February, at 12:00 in room A304b, the Department of AI, Data and Decision Sciences of Luiss University of Rome will host a seminar by Ester Mariucci (Université Paris-Saclay) Title: "Understanding the Small Jumps of Lévy Processes: Approximations and Nonparametric Estimation” Abstract: There is a rich literature on approximations of the small jumps of a Lévy process, with a particular focus on Gaussian approximations. These approximations are often used to simplify analysis, but their validity depends critically on the observation scheme, in particular whether the data are collected at high or low frequency. In many situations, the Gaussian approximation may not be appropriate, and a more precise understanding of the law of small jumps is necessary. In this talk, we will discuss two complementary approaches. First, we explore approximations of the small jumps and provide quantitative control of their accuracy using distances such as the Wasserstein distance and total variation. Second, we investigate nonparametric density estimation of the small jumps based on discrete observations of a single trajectory. Estimating this density allows us to better understand the underlying distribution of small jumps. We present results for both low-frequency and high-frequency observation settings. In the low-frequency case, assuming that the Lévy density associated with large jumps is known, a spectral estimator exploiting the deconvolution structure achieves minimax parametric rates of convergence with respect to the integrated L2 loss, up to a logarithmic factor. In the high-frequency case, the assumption on the knowledge of the Lévy measure of large jumps can be relaxed; the rate of convergence then depends on the sampling scheme and on the behavior of the Lévy measure near zero, and is minimax up to a logarithmic factor. We also propose an adaptive penalized procedure to select the cutoff parameter. These results are further extended to include the case where a Brownian component is present. Finally, we illustrate our findings through extensive simulations. Anyone interested in attending is welcome; just email me for logistical purposes. Hoping to meet you there! Best, Matteo Iacopini La presente e-mail proviene da Luiss Guido Carli e s'intende inviata per scopi lavorativi. Tutte le informazioni ivi contenute, compresi eventuali allegati, sono da ritenersi esclusivamente confidenziali e riservati secondo i termini del vigente D.Lgs. 196/2003 in materia di privacy e del Regolamento europeo 679/2016 - GDPR. È vietato qualsiasi ulteriore utilizzo non autorizzato. Qualora la stessa Le fosse pervenuta per errore, La preghiamo di eliminarla immediatamente e di darcene tempestiva comunicazione. Grazie. This e-mail message is sent by Luiss Guido Carli for business purposes. All informations contained therein, including any attachments, are for the sole use of the intended recipient and may contain confidential and privileged information pursuant to Legislative Decree 196/2003 and the European General Data Protection Regulation 679/2016 - GDPR -. Any unauthorized review, use, disclosure or distribution is prohibited. If you are not the intended recipient, please contact the sender by soon reply this e-mail and destroy all copies of the original message. Thanks