Stochastic calculus with respect to fractional Brownian motion

David Nualart

doi:10.5802/afst.1113

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, Série 6, Tome 15 (2006) no. 1, pp. 63-78.

Résumé
Abstract

Le mouvement brownien fractionnaire (MBF) est un processus gaussien centré auto-similaire à accroissements stationnaires qui dépend d’un paramètre $H \in (0, 1)$ , 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 $H \in (0, 1)$ 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.

MR

DOI : 10.5802/afst.1113

Affiliations des auteurs :

David Nualart ¹

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

@article{AFST_2006_6_15_1_63_0,
     author = {David Nualart},
     title = {Stochastic calculus with respect to fractional {Brownian} motion},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {63--78},
     publisher = {Universit\'e Paul Sabatier, Institut de Math\'ematiques},
     address = {Toulouse},
     volume = {Ser. 6, 15},
     number = {1},
     year = {2006},
     doi = {10.5802/afst.1113},
     mrnumber = {2225747},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1113/}
}

TY  - JOUR
AU  - David Nualart
TI  - Stochastic calculus with respect to fractional Brownian motion
JO  - Annales de la Faculté des sciences de Toulouse : Mathématiques
PY  - 2006
SP  - 63
EP  - 78
VL  - 15
IS  - 1
PB  - Université Paul Sabatier, Institut de Mathématiques
PP  - Toulouse
UR  - https://afst.centre-mersenne.org/articles/10.5802/afst.1113/
DO  - 10.5802/afst.1113
LA  - en
ID  - AFST_2006_6_15_1_63_0
ER  -

%0 Journal Article
%A David Nualart
%T Stochastic calculus with respect to fractional Brownian motion
%J Annales de la Faculté des sciences de Toulouse : Mathématiques
%D 2006
%P 63-78
%V 15
%N 1
%I Université Paul Sabatier, Institut de Mathématiques
%C Toulouse
%U https://afst.centre-mersenne.org/articles/10.5802/afst.1113/
%R 10.5802/afst.1113
%G en
%F AFST_2006_6_15_1_63_0

David Nualart. Stochastic calculus with respect to fractional Brownian motion. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 15 (2006) no. 1, pp. 63-78. doi : 10.5802/afst.1113. https://afst.centre-mersenne.org/articles/10.5802/afst.1113/

Bibliographie
Cité par

[1] E. Alòs; J. A. León; D. Nualart Stratonovich stochastic calculus with respect to fractional Brownian motion with Hurst parameter less than 1/2, Taiwanesse Journal of Mathematics, Volume 5 (2001), pp. 609-632 | MR | Zbl

[2] E. Alòs; O. Mazet; D. Nualart Stochastic calculus with respect to fractional Brownian motion with Hurst parameter lesser than $\frac{1}{2}$ , Stoch. Proc. Appl., Volume 86 (1999), pp. 121-139 | MR | Zbl

[3] E. Alòs; O. Mazet; D. Nualart Stochastic calculus with respect to Gaussian processes, Annals of Probability, Volume 29 (2001), pp. 766-801 | MR | Zbl

[4] E. Alòs; D. Nualart Stochastic integration with respect to the fractional Brownian motion, Stochastics and Stochastics Reports, Volume 75 (2003), pp. 129-152 | MR | Zbl

[5] S. Berman Local nondeterminism and local times of Gaussian processes, Indiana Univ. Math. J., Volume 23 (1973), pp. 69-94 | MR | Zbl

[6] P. Carmona; L. Coutin Stochastic integration with respect to fractional Brownian motion, Ann. Institut Henri Poincaré, Volume 39 (2003), pp. 27-68 | Numdam | MR | Zbl

[7] Z. Ciesielski; G. Kerkyacharian; B. Roynette Quelques espaces fonctionnels associés à des processus gaussiens, Studia Math., Volume 107 (1993), pp. 171-204 | MR | Zbl

[8] P. Cheridito Mixed fractional Brownian motion, Bernoulli, Volume 7 (2001), pp. 913-934 | MR | Zbl

[9] P. Cheridito; D. Nualart Stochastic integral of divergence type with respect to the fractional Brownian motion with Hurst parameter $H < \frac{1}{2}$ , Ann. Institut Henri Poincaré, Volume 41 (2005), pp. 1049-1081 | Numdam | MR | Zbl

[10] A. Chorin Vorticity and Turbulence, Springer-Verlag, 1994 | MR | Zbl

[11] L. Coutin; D. Nualart; C. A. Tudor Tanaka formula for the fractional Brownian motion, Stochastic Processes Appl., Volume 94 (2001), pp. 301-315 | MR | Zbl

[12] L. Coutin; Z. Qian Stochastic analysis, rough paths analysis and fractional Brownian motions, Probab. Theory Rel. Fields, Volume 122 (2002), pp. 108-140 | MR | Zbl

[13] L. Decreusefond; A. S. Üstünel Stochastic analysis of the fractional Brownian motion, Potential Analysis, Volume 10 (1998), pp. 177-214 | MR | Zbl

[14] N. Eisenbaum; C. A. Tudor On squared fractional Brownian motions, Lecture Notes in Math., Volume 1857 (2005), pp. 282-289 | MR | Zbl

[15] F. Flandoli On a probabilistic description of small scale structures in 3D fluids, Ann. Inst. Henri Poincaré, Volume 38 (2002), pp. 207-228 | Numdam | MR | Zbl

[16] F. Flandoli; M. Gubinelli The Gibbs ensemble of a vortex filament, Probab. Theory Relat. Fields, Volume 122 (2001), pp. 317-340 | MR | Zbl

[17] J. Guerra; D. Nualart The $1 / H$ -variation of the divergence integral with respect to the fractional Brownian motion for $H > 1 / 2$ and fractional Bessel processes, Stoch. Proc. Applications, Volume 115 (2005), p. 289-289 | MR | Zbl

[18] Y. Hu Integral transformations and anticipative calculus for fractional Brownian motions, Mem. Amer. Math. Soc., Volume 175 (2005) no. 825, pp. viii+127 | MR | Zbl

[19] Y. Hu; D. Nualart Some Processes Associated with Fractional Bessel Processes, J. Theoretical Probability, Volume 18 (2005), pp. 377-397 | MR | Zbl

[20] Y. Hu; B. Øksendal Fractional white noise calculus and applications to finance, Infin. Dimens. Anal. Quantum Probab. Relat. Top., Volume 6 (2003), pp. 1-32 | MR | Zbl

[21] A. N. Kolmogorov Wienersche Spiralen und einige andere interessante Kurven im Hilbertschen Raum., C. R. (Doklady) Acad. URSS (N.S.), Volume 26 (1940), pp. 115-118 | MR | Zbl

[22] T. Lyons Differential equations driven by rough signals (I): An extension of an inequality of L. C. Young, Mathematical Research Letters, Volume 1 (1994), pp. 451-464 | MR | Zbl

[23] T. Lyons; Z. Qian System control and rough paths, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2002 (Oxford Science Publications) | MR | Zbl

[24] B. B. Mandelbrot; J. W. Van Ness Fractional Brownian motions, fractional noises and applications, SIAM Review, Volume 10 (1968), pp. 422-437 | MR | Zbl

[25] A. Millet; M. Sanz-Solé Large deviations for rough paths of the fractional Brownian motion (Preprint)

[26] Y. Nourdin Calcul stochastique généralisé et applications au mouvement brownien fractionnaire; Estimation non-paramétrique de la volatilité et test d’adéquation, Université de Nancy I, juin (2004) (doctorat)

[27] D. Nualart; E. Pardoux Stochastic calculus with anticipating integrands, Prob. Th. Rel. Fields, Volume 78 (1988), pp. 535-581 | MR | Zbl

[28] D. Nualart; A. Rascanu Differential equations driven by fractional Brownian motion, Collectanea Mathematica, Volume 53 (2002), pp. 55-81 | MR | Zbl

[29] D. Nualart; C. Rovira; S. Tindel Probabilistic models for vortex filaments based on fractional Brownian motion, Annals of Probability, Volume 31 (2003), pp. 1862-1899 | MR | Zbl

[30] V. Pipiras; M. S. Taqqu Integration questions related to fractional Brownian motion, Probab. Theory Rel. Fields, Volume 118 (2000), pp. 121-291 | MR | Zbl

[31] V. Pipiras; M. S. Taqqu Are classes of deterministic integrands for fractional Brownian motion on a interval complete?, Bernoulli, Volume 7 (2001), pp. 873-897 | MR | Zbl

[32] L. C. G. Rogers Arbitrage with fractional Brownian motion, Math. Finance, Volume 7 (1997), pp. 95-105 | MR | Zbl

[33] F. Russo; P. Vallois Forward, backward and symmetric stochastic integration, Probab. Theory Rel. Fields, Volume 97 (1993), pp. 403-421 | MR | Zbl

[34] A. V. Skorohod On a generalization of a stochastic integral, Theory Probab. Appl., Volume 20 (1975), pp. 219-233 | MR | Zbl

[35] H. J. Sussmann On the gap between deterministic and stochastic ordinary differential equations, Ann. Probability, Volume 6 (1978), pp. 19-41 | MR | Zbl

[36] L. C. Young An inequality of the Hölder type connected with Stieltjes integration, Acta Math., Volume 67 (1936), pp. 251-282 | Zbl

[37] M. Zähle Integration with respect to fractal functions and stochastic calculus. I., Probab. Theory Related Fields, Volume 111 (1998), pp. 333-374 | MR | Zbl

Cité par Sources :