Some Prerequisites

This second preliminary chapter is devoted to recalling some basic properties of semimartingales, and it is about the convergence of processes. Both of these topics are prerequisites for the rest of the book.

In Sect. 2.1 the main properties of semimartingales are recalled, with a special emphasis on a description of the so-called

Itô semimartingales

. We also recall basic features of the characteristics of a semimartingale. Most of the results established in this book hold only for Itô semimartingales. Almost all the properties of semimartingales we review here can be found in various books already, so proofs are omitted. A few results are new in book form, and these are proved in the Appendix: this mainly concerns a number of estimates on the increments of a semimartingale, or an Itô semimartingale. These results are scattered in various papers but are essential to our aims, so it is prudent to collect them here.

Section 2.2 is devoted to recalling facts about the convergence of processes, starting with a quick reminder about the Skorokhod topology. Here, special emphasis is put on

stable convergence in law

, which is central for almost all statistical applications of the results of this book. Again, the proofs of most results, which can be found in various books, is omitted, whereas the proofs of the few results which are new in book form is given in the Appendix.