Nualart, David
Stochastic calculus with respect to fractional Brownian motion
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6 : Tome 15 (2006) no. 1 , p. 63-78
MR 2225747 | Zbl pre05208249
doi : 10.5802/afst.1113
URL stable : http://www.numdam.org/item?id=AFST_2006_6_15_1_63_0

Le mouvement brownien fractionnaire (MBF) est un processus gaussien centré auto-similaire à accroissements stationnaires qui dépend d’un paramètre H(0,1), appelé paramètre de Hurst. Dans cette conférence, nous donnerons une synthèse des résultats nouveaux en calcul stochastique par rapport à un MBF. Dans le cas particulier H=1/2, ce processus est le mouvement brownien classique, sinon, ce n’est pas une semi-martingale et on ne peut pas utiliser le calcul d’Itô. Différentes approches ont été utilisées pour construire des intégrales stochastiques par rapport à un MBF : techniques trajectorielles, calcul de Malliavin, approximation par des sommes de Riemann. Nous décrivons ces méthodes et présentons les formules de changement de variables associées. Plusieurs applications seront présentées.
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.

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