Estimation of the activity of jumps in time-changed Lévy models

Abstract

In this paper we consider a class of time-changed Lévy processes that can be represented in the form $Y_{s}=X_{\mathcal{T}(s)}$, where $X$ is a Lévy process and $\mathcal{T}$ 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 Lévy 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.