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.
General
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