Stochastic integration with respect to fractional brownian motion
Annales de l'I.H.P. Probabilités et statistiques, Volume 39 (2003) no. 1, p. 27-68
@article{AIHPB_2003__39_1_27_0,
     author = {Carmona, Philippe and Coutin, Laure and Montseny, G\'erard},
     title = {Stochastic integration with respect to fractional brownian motion},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Elsevier},
     volume = {39},
     number = {1},
     year = {2003},
     pages = {27-68},
     zbl = {1016.60043},
     mrnumber = {1959841},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2003__39_1_27_0}
}
Carmona, Philippe; Coutin, Laure; Montseny, Gérard. Stochastic integration with respect to fractional brownian motion. Annales de l'I.H.P. Probabilités et statistiques, Volume 39 (2003) no. 1, pp. 27-68. http://www.numdam.org/item/AIHPB_2003__39_1_27_0/

[1] E. Alos, O. Mazet, D. Nualart, Stochastic calculus with respect to fractional Brownian motion with Hurst parameter less than 1/2, Stochastic Process. Appl. 86 (2000) 121-139. | MR 1741199 | Zbl 1028.60047

[2] A. Benassi, S. Jaffard, D. Roux, Elliptic Gaussian random processes, Rev. Mat. Iberoamericana 13 (1997) 19-90. | MR 1462329 | Zbl 0880.60053

[3] J. Beran, N. Terrin, Testing for a change of the long-memory parameter, Biometrika 83 (1996) 627-638. | MR 1423879 | Zbl 0866.62055

[4] Z. Ciesielski, G. Kerkyacharian, B. Roynette, Quelques espaces fonctionnels associés à des processus gaussiens. (Some function spaces associated with gaussian processes), Stud. Math. 107 (1993) 171-204. | MR 1244574 | Zbl 0809.60004

[5] F. Comte, E. Renault, Long memory continuous time models, J. Econometrics 73 (1996) 101-150. | MR 1410003 | Zbl 0856.62104

[6] L. Coutin, Z. Qian, Stochastic analysis, rough path analysis and fractional Brownian motions, To be published in PTRF, 2000. | Zbl 1047.60029

[7] L. Coutin, Z. Qian, Stochastic differential equations for fractional Brownian motions, C. R. Acad. Sci. Paris Sér. I Math. 331 (2000) 75-80. | MR 1780221 | Zbl 0981.60040

[8] W. Dai, C. Heyde, Ito's formula with respect to fractional Brownian motion and its application, J. Appl. Math. Stochastic Anal. 9 (1996) 439-448. | MR 1429266 | Zbl 0867.60029

[9] L. Decreusefond, A. Üstunel, Stochastic analysis of the fractional Brownian motion, Potential Anal. 10 (1997) 177-214. | MR 1677455 | Zbl 0924.60034

[10] C. Dellacherie, B. Maisonneuve, P. Meyer, Probabilités et potentiel. Chapitres XVII à XXIV : Processus de Markov (fin) Compléments de calcul stochastique, Hermann, Paris, 1992.

[11] R. Dudley, R. Norvaisa, An introduction to p-variation and Young integrals, Tech. Rep. 1, Maphysto, Centre for Mathematical Physics and Stochastics, University of Aarhus, Ny Munkegade, DK-8000 Aarhus C, Denmark, 1998, Concentrated advanced course. | Zbl 0937.28001

[12] T. Duncan, Y. Hu, B. Pasik-Duncan, Stochastic calculus for fractional Brownian motion I. Theory, SIAM J. Control Optim. 38 (2000) 582-612. | MR 1741154 | Zbl 0947.60061

[13] D. Feyel, A.De La Pradelle, On the approximate solution of the Stratonovitch equation, Electron. J. Probab. 3 (1998). | MR 1624858 | Zbl 0901.60028

[14] A. Kolmogorov, Wienersche spiralen und einige andere interessante kurven im Hilbertschen raum, S. R. (Dokl.) Acad. Sci. USSR (N.S.) 26 (1940) 115-118. | JFM 66.0552.03 | MR 3441

[15] N. Lebedev, Special Functions and their Applications, Dover Publications, New York, 1972, (Translated and edited by Richard A. Silverman). | MR 350075 | Zbl 0271.33001

[16] W. Leland, M. Taqqu, W. Willinger, D. Wilson, On the self-similar nature of Ethernet traffic, IEEE/ACM Trans. Networking 2 (1994) 1-15.

[17] J. Leon, Fubini theorem for anticipating stochastic integrals in Hilbert space, Appl. Math. Optimization 27 (1993) 313-327. | MR 1201627 | Zbl 0771.60039

[18] S. Lin, Stochastic analysis of fractional Brownian motions, Stochastics Stochastics Rep. 55 (1995) 121-140. | MR 1382288 | Zbl 0886.60076

[19] R. Liptser, A. Shyriaev, Theory of Martingales, Mathematics and its Applications, Kluwer Academic Publishers, 1989. | MR 1022664 | Zbl 0728.60048

[20] T. Lyons, Differential equations driven by rough signals, Rev. Mat. Iberoamericana 14 (1998) 215-310. | MR 1654527 | Zbl 0923.34056

[21] I. Norros, E. Valkeila, J. Virtamo, An elementary approach to a Girsanov formula and other analytical results on fractional Brownian motions, Bernoulli 5 (1999) 571-588. | MR 1704556 | Zbl 0955.60034

[22] D. Nualart, The Malliavin Calculus and Related Topics, Probability and its Applications, Springer-Verlag, New York, NY, 1995. | MR 1344217 | Zbl 0837.60050

[23] V. Pipiras, M.S. Taqqu, Integration questions related to fractional Brownian motion, Probab. Theory Related Fields (2000) 251-291. | MR 1790083 | Zbl 0970.60058

[24] N. Privault, Skorokhod stochastic integration with respect to non-adapted processes on Wiener space, Stochastics Stochastics Rep. 65 (1998) 13-39. | MR 1708428 | Zbl 0918.60038

[25] P. Protter, Stochastic Integration and Differential Equations, Applications of Mathematics, 21, Springer-Verlag, 1992. | Zbl 0694.60047

[26] L. Rogers, Arbitrage with fractional Brownian motion, Math. Finance 7 (1997) 95-105. | MR 1434408 | Zbl 0884.90045

[27] F. Russo, P. Vallois, Forward, backward and symmetric stochastic integration, Probab. Theory Related Fields 97 (1993) 403-421. | MR 1245252 | Zbl 0792.60046

[28] F. Russo, P. Vallois, The generalized covariation process and Itô formula, Stochastic Process. Appl. 59 (1995) 81-104. | MR 1350257 | Zbl 0840.60052

[29] A.A. Ruzmaikina, Stieltjes integrals of Hölder continuous functions with applications to fractional Brownian motion, J. Statist. Phys. 100 (2000) 1049-1069. | MR 1798553 | Zbl 0970.60045

[30] S. Samko, A. Kilbas, O. Marichev, Fractional Integrals and Derivatives, Gordon & Breach Science, 1993. | MR 1347689 | Zbl 0818.26003

[31] D. Stroock, A Concise Introduction to the Theory of Integration, Birkhauser, 1994. | MR 1267228 | Zbl 0912.28001

[32] L. Young, An inequality of Hölder type, connected with Stieltjes integration, Acta Math. 67 (1936) 251-282. | MR 1555421 | Zbl 0016.10404

[33] M. Zähle, Integration with respect to fractal functions and Stochastic Calculus, Probab. Theory Related Fields 111 (1998) 333-374. | MR 1640795 | Zbl 0918.60037

[34] M. Zähle, On the link between fractional and stochastic calculus, in: Crauel H. (Ed.), Stochastic Dynamics, Conference on Random Dynamical Systems, Bremen, Germany, April 28-May 2, 1997, Springer, 1999, pp. 305-325, Dedicated to Ludwig Arnold on the occasion of his 60th birthday. | Zbl 0947.60060