Serge Cohen Livres

This is the second volume in a subseries of the Lecture Notes in Mathematics called Lévy Matters, which is published at irregular intervals over the years. Each volume examines a number of key topics in the theory or applications of Lévy processes and pays tribute to the state of the art of this rapidly evolving subject with special emphasis on the non-Brownian world. The expository articles in this second volume cover two important topics in the area of Lévy processes. The first article by Serge Cohen reviews the most important findings on fractional Lévy fields to date in a self-contained piece, offering a theoretical introduction as well as possible applications and simulation techniques. The second article, by Alexey Kuznetsov, Andreas E. Kyprianou, and Victor Rivero, presents an up to date account of the theory and application of scale functions for spectrally negative Lévy processes, including an extensive numerical overview.

This book focuses on fractional Brownian fields and their extensions, serving as a teaching resource for graduate students at Grenoble and Toulouse's Universities. It is designed to be self-contained and includes numerous exercises with solutions in an appendix. Following a foreword by Stéphane Jaffard, the first chapter covers classical results from stochastic fields and fractal analysis. A key concept is self-similarity, explored in the second chapter with an emphasis on Gaussian self-similar fields, known as fractional Brownian fields, stemming from Mandelbrot and Van Ness's work. Fundamental properties of these fields are established and proven. The book also introduces local asymptotic self-similarity (lass) in the third chapter, focusing on lass fields with finite variance, which include both Gaussian and non-Gaussian Lévy fields that connect fractional Brownian fields to stable self-similar fields. Another significant topic is the identification of fractional parameters, addressed in the statistics chapter through generalized quadratic variations methods for estimation. Finally, the simulation of fractional fields is discussed in the last chapter, highlighting ongoing research in this area. The algorithms presented are efficient but do not claim to resolve all issues.