Selected Aspects of Fractional Brownian Motion

This book explores several aspects of fractional Brownian motion, including the stochastic integration, the study of its supremum and its appearance as limit of partial sums involving stationary sequences.

Fractional Brownian motion (fBm) is a stochastic process which deviates significantly from Brownian motion and semimartingales, and others classically used in probability theory. As a centered Gaussian process, it is characterized by the stationarity of its increments and a medium- or long-memory property which is in sharp contrast with martingales and Markov processes. FBm has become a popular choice for applications where classical processes cannot model these non-trivial properties; for instance long memory, which is also known as persistence, is of fundamental importance for financial data and in internet traffic. The mathematical theory of fBm is currently being developed vigorously by a number of stochastic analysts, in various directions, using complementary and sometimes competing tools. This book is concerned with several aspects of fBm, including the stochastic integration with respect to it, the study of its supremum and its appearance as limit of partial sums involving stationary sequences, to name but a few. The book is addressed to researchers and graduate students in probability and mathematical statistics. With very few exceptions (where precise references are given), every stated result is proved.

Except for very few exception, every result stated in this book is proved in details: the book is then perfectly tailored for self-learning My guiding thread was to develop only the most aesthetic topics related to fractional Brownian motion: the book will appeal to readers who are not necessarily familiar with fractional Brownian motion and who like beautiful mathematics A special chapter on a recent link between fractional Brownian motion and free probability introduces the reader to a new and promising line of research Includes supplementary material: sn.pub/extras Includes supplementary material: sn.pub/extras

Auteur

Ivan Nourdin is full professor in Mathematics at Université de Lorraine (France). His research interests include Malliavin calculus, Stein's method and free probability.

Contenu

1. Preliminaries.- 2. Fractional Brownian motion.- 3. Integration with respect to fractional Brownian motion.- 4. Supremum of the fractional Brownian motion.- 5. Malliavin calculus in a nutshell.- 6. Central limit theorem on the Wiener space.- 7. Weak convergence of partial sums of stationary sequences.- 8. Non-commutative fractional Brownian motion.