Finer Analysis of Jumps: The Degree of Jump Activity

Abstract

The notions of Blumenthal–Getoor (BG) indices BG indices can be extended to Itô (or even, general) semimartingale, and successive BG indices also make sense for semimartingales. This chapter is concerned with estimating those quantities, together with the related intensities. For general Itô semimartingales the problem seems utterly intractable, so we need some assumptions on the process, which may be considered as restrictive from a theoretical point of view. However, in practice all models used by practitioners fulfill these assumptions. Our objective in doing so is to provide specification tools for financial models, where the presence or at least the possibility of large jumps is generally accepted. There is much less consensus in the literature regarding the nature or even the need for small jumps, and this is where knowing the BG index might prove very useful for modeling.