BEGIN:VCALENDAR VERSION:2.0 PRODID:-//Talks.cam//talks.cam.ac.uk// X-WR-CALNAME:Talks.cam BEGIN:VEVENT SUMMARY:About small jumps of Lévy processes: approximations and estimatio n - Ester Mariucci (Université de Versailles Saint-Quentin-en-Yvelines) DTSTART:20240424T084500Z DTEND:20240424T093000Z UID:TALK214156@talks.cam.ac.uk DESCRIPTION:Abstract: We consider the problem of estimating the density of the process associated with the small jumps of a pure jump Lé\;vy p rocess\, possibly of infinite variation\, from discrete observations of on e trajectory. The interest of such a question lies on the observation that even when the Lé\;vy measure is known\, the density of the incremen ts of the small jumps of the process cannot be computed. We discuss result s both from low and high frequency observations. In a low frequency settin g\, assuming the Lé\;vy density associated with the jumps larger tha n $\\eps\\in (0\,1]$ in absolute value is known\, \;a spectral \; estimator relying on the convolution structure of the problem achieves&nbs p\; minimax parametric rates of convergence with respect to the integrated $L_2$ loss\, up to a logarithmic factor. In a high frequency setting\, we remove the assumption on the knowledge of the Lé\;vy measure of the large jumps and show that the rate of convergence depends both on the sam pling scheme and on the behaviour of the Lé\;vy measure in a neighbo rhood of zero. We show that the rate we find is minimax up to a log-factor . \; An adaptive penalized procedure is studied to select the cutoff p arameter. These results are extended to encompass the case where a Brownia n component is present in the Lé\;vy process. Furthermore\, we illus trate the performances of our procedures through an extensive simulation s tudy.\nThis is a joint work with Taher Jalal and Cé\;line Duval. LOCATION:External END:VEVENT END:VCALENDAR