cnrs   Université de Toulouse
With cedram.org

Table of contents for this issue | Previous article | Next article
David Nualart
Stochastic calculus with respect to fractional Brownian motion
Annales de la faculté des sciences de Toulouse Sér. 6, 15 no. 1 (2006), p. 63-78, doi: 10.5802/afst.1113
Article PDF | Reviews MR 2225747 | Zbl pre05208249

Résumé - 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.

Bibliography

[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 5 (2001), p. 609-632  MR 1849782 |  Zbl 0989.60054
[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. 86 (1999), p. 121-139  MR 1741199 |  Zbl 1028.60047
[3] E. Alòs, O. Mazet & D. Nualart, “Stochastic calculus with respect to Gaussian processes”, Annals of Probability 29 (2001), p. 766-801 Article |  MR 1849177 |  Zbl 1015.60047
[4] E. Alòs & D. Nualart, “Stochastic integration with respect to the fractional Brownian motion”, Stochastics and Stochastics Reports 75 (2003), p. 129-152  MR 1978896 |  Zbl 1028.60048
[5] S. Berman, “Local nondeterminism and local times of Gaussian processes”, Indiana Univ. Math. J. 23 (1973), p. 69-94  MR 317397 |  Zbl 0264.60024
[6] P. Carmona & L. Coutin, “Stochastic integration with respect to fractional Brownian motion”, Ann. Institut Henri Poincaré 39 (2003), p. 27-68 Numdam |  MR 1959841 |  Zbl 1016.60043
[7] Z. Ciesielski, G. Kerkyacharian & B. Roynette, “Quelques espaces fonctionnels associés à des processus gaussiens”, Studia Math. 107 (1993), p. 171-204 Article |  MR 1244574 |  Zbl 0809.60004
[8] P. Cheridito, “Mixed fractional Brownian motion”, Bernoulli 7 (2001), p. 913-934 Article |  MR 1873835 |  Zbl 1005.60053
[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 41 (2005), p. 1049-1081 Numdam |  MR 2172209 |  Zbl 02231407
[10] A. Chorin, Vorticity and Turbulence, Springer-Verlag, 1994  MR 1281384 |  Zbl 0795.76002
[11] L. Coutin, D. Nualart & C. A. Tudor, “Tanaka formula for the fractional Brownian motion”, Stochastic Processes Appl. 94 (2001), p. 301-315  MR 1840834 |  Zbl 1053.60055
[12] L. Coutin & Z. Qian, “Stochastic analysis, rough paths analysis and fractional Brownian motions”, Probab. Theory Rel. Fields 122 (2002), p. 108-140  MR 1883719 |  Zbl 1047.60029
[13] L. Decreusefond & A. S. Üstünel, “Stochastic analysis of the fractional Brownian motion”, Potential Analysis 10 (1998), p. 177-214  MR 1677455 |  Zbl 0924.60034
[14] N. Eisenbaum & C. A. Tudor, “On squared fractional Brownian motions”, Lecture Notes in Math. 1857 (2005), p. 282-289  MR 2126980 |  Zbl 1071.60023
[15] F. Flandoli, “On a probabilistic description of small scale structures in 3D fluids”, Ann. Inst. Henri Poincaré 38 (2002), p. 207-228 Numdam |  MR 1899111 |  Zbl 1017.76074
[16] F. Flandoli & M. Gubinelli, “The Gibbs ensemble of a vortex filament”, Probab. Theory Relat. Fields 122 (2001), p. 317-340  MR 1892850 |  Zbl 0992.60058
[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 115 (2005), p. 289-289  MR 2105371 |  Zbl 1075.60056
[18] Y. Hu, “Integral transformations and anticipative calculus for fractional Brownian motions”, Mem. Amer. Math. Soc. 175 (2005) no. 825  MR 2130224 |  Zbl 1072.60044
[19] Y. Hu & D. Nualart, “Some Processes Associated with Fractional Bessel Processes”, J. Theoretical Probability 18 (2005), p. 377-397  MR 2137449 |  Zbl 1074.60050
[20] Y. Hu & B. Øksendal, “Fractional white noise calculus and applications to finance”, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 6 (2003), p. 1-32  MR 1976868 |  Zbl 1045.60072
[21] A. N. Kolmogorov, “Wienersche Spiralen und einige andere interessante Kurven im Hilbertschen Raum.”, C. R. (Doklady) Acad. URSS (N.S.) 26 (1940), p. 115-118  MR 3441 |  Zbl 0022.36001
[22] T. Lyons, “Differential equations driven by rough signals (I): An extension of an inequality of L. C. Young”, Mathematical Research Letters 1 (1994), p. 451-464  MR 1302388 |  Zbl 0835.34004
[23] T. Lyons & Z. Qian, System control and rough paths, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2002, Oxford Science Publications  MR 2036784 |  Zbl 1029.93001
[24] B. B. Mandelbrot & J. W. Van Ness, “Fractional Brownian motions, fractional noises and applications”, SIAM Review 10 (1968), p. 422-437  MR 242239 |  Zbl 0179.47801
[25] A. Millet & M. Sanz-Solé, “Large deviations for rough paths of the fractional Brownian motion”, Preprint arXiv
[26] Y. Nourdin, Calcul stochastique généralisé et applications au mouvement brownien fractionnaire; Estimation non-paramétrique de la volatilité et test d’adéquation, doctorat, Université de Nancy I, 2004
[27] D. Nualart & E. Pardoux, “Stochastic calculus with anticipating integrands”, Prob. Th. Rel. Fields 78 (1988), p. 535-581  MR 950346 |  Zbl 0629.60061
[28] D. Nualart & A. Rascanu, “Differential equations driven by fractional Brownian motion”, Collectanea Mathematica 53 (2002), p. 55-81  MR 1893308 |  Zbl 1018.60057
[29] D. Nualart, C. Rovira & S. Tindel, “Probabilistic models for vortex filaments based on fractional Brownian motion”, Annals of Probability 31 (2003), p. 1862-1899 Article |  MR 2016603 |  Zbl 1047.76013
[30] V. Pipiras & M. S. Taqqu, “Integration questions related to fractional Brownian motion”, Probab. Theory Rel. Fields 118 (2000), p. 121-291  MR 1790083 |  Zbl 0970.60058
[31] V. Pipiras & M. S. Taqqu, “Are classes of deterministic integrands for fractional Brownian motion on a interval complete?”, Bernoulli 7 (2001), p. 873-897 Article |  MR 1873833 |  Zbl 1003.60055
[32] L. C. G. Rogers, “Arbitrage with fractional Brownian motion”, Math. Finance 7 (1997), p. 95-105  MR 1434408 |  Zbl 0884.90045
[33] F. Russo & P. Vallois, “Forward, backward and symmetric stochastic integration”, Probab. Theory Rel. Fields 97 (1993), p. 403-421  MR 1245252 |  Zbl 0792.60046
[34] A. V. Skorohod, “On a generalization of a stochastic integral”, Theory Probab. Appl. 20 (1975), p. 219-233  MR 391258 |  Zbl 0333.60060
[35] H. J. Sussmann, “On the gap between deterministic and stochastic ordinary differential equations”, Ann. Probability 6 (1978), p. 19-41 Article |  MR 461664 |  Zbl 0391.60056
[36] L. C. Young, “An inequality of the Hölder type connected with Stieltjes integration”, Acta Math. 67 (1936), p. 251-282  Zbl 0016.10404 |  JFM 62.0250.02
[37] M. Zähle, “Integration with respect to fractal functions and stochastic calculus. I.”, Probab. Theory Related Fields 111 (1998), p. 333-374  MR 1640795 |  Zbl 0918.60037
Search for an article
Search within the site
top