Stochastic calculus with respect to fractional Brownian motion

David Nualart[1]

  • [1] Facultat de Matemàtiques, Universitat de Barcelona, Gran Via 585, 08007 Barcelona (Spain).

Annales de la faculté des sciences de Toulouse Mathématiques (2006)

  • Volume: 15, Issue: 1, page 63-78
  • ISSN: 0240-2963

Abstract

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Fractional Brownian motion (fBm) is a centered self-similar Gaussian process with stationary increments, which depends on a parameter H ∈ ( 0 , 1 ) called the Hurst index. In this conference we will survey some recent advances in the stochastic calculus with respect to fBm. In the particular case H = 1 / 2 , the process is an ordinary Brownian motion, but otherwise it is not a semimartingale and Itô calculus cannot be used. Different approaches have been introduced to construct stochastic integrals with respect to fBm: pathwise techniques, Malliavin calculus, approximation by Riemann sums. We will describe these methods and present the corresponding change of variable formulas. Some applications will be discussed.

How to cite

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Nualart, David. "Stochastic calculus with respect to fractional Brownian motion." Annales de la faculté des sciences de Toulouse Mathématiques 15.1 (2006): 63-78. <http://eudml.org/doc/10039>.

@article{Nualart2006,
abstract = {Fractional Brownian motion (fBm) is a centered self-similar Gaussian process with stationary increments, which depends on a parameter $H\in (0,1)$ called the Hurst index. In this conference we will survey some recent advances in the stochastic calculus with respect to fBm. In the particular case $H=1/2$, the process is an ordinary Brownian motion, but otherwise it is not a semimartingale and Itô calculus cannot be used. Different approaches have been introduced to construct stochastic integrals with respect to fBm: pathwise techniques, Malliavin calculus, approximation by Riemann sums. We will describe these methods and present the corresponding change of variable formulas. Some applications will be discussed.},
affiliation = {Facultat de Matemàtiques, Universitat de Barcelona, Gran Via 585, 08007 Barcelona (Spain).},
author = {Nualart, David},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {stochastic integrals; Malliavin calculus; change of variable formulas},
language = {eng},
number = {1},
pages = {63-78},
publisher = {Université Paul Sabatier, Toulouse},
title = {Stochastic calculus with respect to fractional Brownian motion},
url = {http://eudml.org/doc/10039},
volume = {15},
year = {2006},
}

TY - JOUR
AU - Nualart, David
TI - Stochastic calculus with respect to fractional Brownian motion
JO - Annales de la faculté des sciences de Toulouse Mathématiques
PY - 2006
PB - Université Paul Sabatier, Toulouse
VL - 15
IS - 1
SP - 63
EP - 78
AB - Fractional Brownian motion (fBm) is a centered self-similar Gaussian process with stationary increments, which depends on a parameter $H\in (0,1)$ called the Hurst index. In this conference we will survey some recent advances in the stochastic calculus with respect to fBm. In the particular case $H=1/2$, the process is an ordinary Brownian motion, but otherwise it is not a semimartingale and Itô calculus cannot be used. Different approaches have been introduced to construct stochastic integrals with respect to fBm: pathwise techniques, Malliavin calculus, approximation by Riemann sums. We will describe these methods and present the corresponding change of variable formulas. Some applications will be discussed.
LA - eng
KW - stochastic integrals; Malliavin calculus; change of variable formulas
UR - http://eudml.org/doc/10039
ER -

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