About small jumps of Lévy processes : approximations and estimation

Ester Mariucci (Versailles)

In the first part of the talk, we discuss how to quantify the goodness of a Gaussian approximation of the small jumps of a Lévy process according to a given metric. We will focus on two metrics: the Wasserstein distance of order p and the total variation distance. Upper bounds for these metrics are discussed. In particular, sharp and non asymptotic bounds for a Gaussian approximation for jumps of infinite activity are presented. 

In the second part of the talk, we consider the problem of estimating the density of the small jumps of a pure jump Lévy process, from discrete observations of one trajectory. We discuss results both from low and high frequency observations, for Lévy processes possibly of infinite variation. We propose an estimator obtained via a spectral approach which achieves the minimax rate with respect to the L2 loss in the low frequency case. In a high frequency setting, the rate we obtain depends on the sampling scheme and on the Blumenthal-Getoor index of the process. Finally, we discuss the optimality of these results.