Preface

Summary

The aim of this book is to provide a straightforward and accessible introduction to stochastic integrals and stochastic differential equations driven by Lévy processes.

Lévy processes are essentially stochastic processes with stationary and independent increments. Their importance in probability theory stems from the following facts:

• they are analogues of random walks in continuous time;

• they form special subclasses of both semimartingales and Markov processes for which the analysis is on the one hand much simpler and on the other hand provides valuable guidance for the general case;

• they are the simplest examples of random motion whose sample paths are right-continuous and have a number (at most countable) of random jump discontinuities occurring at random times, on each finite time interval.

• they include a number of very important processes as special cases, including Brownian motion, the Poisson process, stable and self-decomposable processes and subordinators.

Although much of the basic theory was established in the 1930s, recent years have seen a great deal of new theoretical development as well as novel applications in such diverse areas as mathematical finance and quantum field theory. Recent texts that have given systematic expositions of the theory have been Bertoin and Sato. Samorodnitsky and Taqqu is a bible for stable processes and related ideas of self-similarity, while a more applications-oriented view of the stable world can be found in Uchaikin and Zolotarev.