Stochastic calculus with respect to fractional Brownian motion
Nualart, David1
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.

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.

DOI : 10.5802/afst.1113
Nualart, David 1

1 Facultat de Matemàtiques, Universitat de Barcelona, Gran Via 585, 08007 Barcelona (Spain). 
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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, pp. 63-78. doi : 10.5802/afst.1113. https://www.numdam.org/articles/10.5802/afst.1113/

[1] Alòs, E.; León, J. A.; Nualart, D. 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] Alòs, E.; Mazet, O.; Nualart, D. Stochastic calculus with respect to fractional Brownian motion with Hurst parameter lesser than 12, Stoch. Proc. Appl., Volume 86 (1999), pp. 121-139 | MR | Zbl

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

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

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

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

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

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

[9] Cheridito, P.; Nualart, D. Stochastic integral of divergence type with respect to the fractional Brownian motion with Hurst parameter H<12, Ann. Institut Henri Poincaré, Volume 41 (2005), pp. 1049-1081 | Numdam | MR | Zbl

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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