Selected Aspects of Fractional Brownian Motion

By Ivan Nourdin. 2013

287 citations recorded by the source corpus.

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.

Subfields: Finance

View Paper: https://www.semanticscholar.org/paper/b2eb32a3021549471010947044b0c6bf80b144fb

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Citations in the corpus

Fractional Calculus: Quo Vadimus? (Where are we Going?) (2015, result): “We have taken the present occasion, as a compliment to our previous survey [107], to cite books published during the years 2011 [92, 80, 44, 26, 65, 47, 5, 106, 117], 2012 [7, 3, 77, 1, 51, 38, 102, 56, 122, 15, 112, 110, 118], 2013 [14, 18, 89, 29, 36], and 2014 [57, 4, 121, 39, 105, 40, 66, 86, 2,”
Generalized Hermite processes, discrete chaos and limit theorems (2013, methodology): “The first-order Hermite process, namely fractional Brownian motion, has been studied intensively by numerous researchers since its popularization by Mandelbrot and Van Ness [19], and we refer the reader to a recent monograph Nourdin [21] and the references Key words Long memory; Discrete chaos; Wien”
Science metrics on fractional calculus development since 1966 (2013, background): “Figure 2 (left) depicts the evolution of Da [19, 12, 56, 75, 57, 99, 60, 87, 100, 100, 77, 68, 85, 95, 69, 70, 71, 72, 78, 23, 102, 13, 96, 64, 114, 36, 79, 73, 91, 63, 84, 80, 105, 109, 4, 111, 82, 22, 27, 28, 110, 81, 67, 32, 29, 86, 52, 33, 117, 106, 55, 58, 112, 113, 108, 6, 15, 2, 41, 59, 10, 4”
The optimal fourth moment theorem (2013, methodology): “19) is a difficult problem, that has generated a large amount of research (see [10, 15] for an overview of the available literature, as well as [1, 2, 11] for recent developments).”
New Berry-Esseen bounds for non-linear functionals of Poisson random measures (2013, background): “Applications of findings dealing with the Gaussian case have found notable applications in the theory and statistics of Gaussian random processes [3, 4] (most prominently the fractional Brownian motion [15]), spherical random fields [1, 14], random matrix theory [17] and universality questions [19],”
Short-time at-the-money skew and rough fractional volatility (2015, background): “2 of Nourdin (2012), sup0≤s≤1 |Ŷ 1 s | has exponential moments of any order.”