Selected Aspects of Fractional Brownian Motion

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.

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.