Estimation of the activity of jumps for time-changed Lévy processes

Abstract

In this paper we consider a class of processes that can be represented in the form Ys = XT (s), where X is a Levy process and T is a non-negative and non-decreasing stochastic process independent of X. The aim of this work is to infer on the Blumenthal-Getoor index of the process X from low-frequency observations of the time-changed Levy process Y . We propose a consistent estimator for this index, derive the minimax rates of convergence and show that these rates can not be improved in general. The performance of the estimator is illustrated by numerical examples.