m-order integrals and generalized Itô’s formula ; the case of a fractional brownian motion with any Hurst index
Gradinaru, Mihai ; Nourdin, Ivan ; Russo, Francesco ; Vallois, Pierre
Annales de l'I.H.P. Probabilités et statistiques, Tome 41 (2005), p. 781-806 / Harvested from Numdam
@article{AIHPB_2005__41_4_781_0,
     author = {Gradinaru, Mihai and Nourdin, Ivan and Russo, Francesco and Vallois, Pierre},
     title = {$m$-order integrals and generalized It\^o's formula ; the case of a fractional brownian motion with any Hurst index},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     volume = {41},
     year = {2005},
     pages = {781-806},
     doi = {10.1016/j.anihpb.2004.06.002},
     zbl = {1083.60045},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPB_2005__41_4_781_0}
}
Gradinaru, Mihai; Nourdin, Ivan; Russo, Francesco; Vallois, Pierre. $m$-order integrals and generalized Itô’s formula ; the case of a fractional brownian motion with any Hurst index. Annales de l'I.H.P. Probabilités et statistiques, Tome 41 (2005) pp. 781-806. doi : 10.1016/j.anihpb.2004.06.002. http://gdmltest.u-ga.fr/item/AIHPB_2005__41_4_781_0/

[1] E. Alos, J.L. Léon, D. Nualart, Stratonovich calculus for fractional Brownian motion with Hurst parameter less than 1 2, Taiwanese J. Math. 5 (2001) 609-632. | MR 1849782 | Zbl 0989.60054

[2] C. Bender, An Itô formula for generalized functionals of a fractional Brownian motion with arbitrary Hurst parameter, Stochastic Process. Appl. 124 (1) (2003) 81-106. | MR 1956473 | Zbl 1075.60530

[3] P. Carmona, L. Coutin, G. Monseny, Stochastic integration with respect to fractional Brownian motion, Ann. Inst. H. Poincaré Probab. Statist. 39 (1) (2003) 27-68. | Numdam | MR 1959841 | Zbl 1016.60043

[4] P. Cheridito, D. Nualart, Stochastic integral of divergence type with respect to fractional Brownian motion with Hurst parameter H(0,1 2), Preprint, Barcelona, 2002.

[5] L. Coutin, Z. Qian, Stochastic analysis, rough path analysis and fractional Brownian motions, Probab. Theory Related Fields 122 (1) (2002) 108-140. | MR 1883719 | Zbl 1047.60029

[6] L. Decreusefond, A.S. Ustunel, Stochastic analysis of the fractional Brownian motion, Potential Anal. 10 (1998) 177-214. | MR 1677455 | Zbl 0924.60034

[7] R.M. Dudley, R. Norvaisa, Differentiability of Six Operators on Nonsmooth Functions and p-variation, Lecture Notes in Math., vol. 1703, Springer-Verlag, 1999. | MR 1705318 | Zbl 0973.46033

[8] M. Errami, F. Russo, Covariation de convolutions de martingales, C. R. Acad. Sci. Sér. 1 326 (1998) 601-609. | MR 1649341 | Zbl 0917.60054

[9] M. Errami, F. Russo, n-covariation and symmetric SDEs driven by finite cubic variation process, Stoch. Process. Appl. 104 (2) (2000) 259-299. | MR 1961622 | Zbl 1075.60531

[10] D. Feyel, A. De La Pradelle, On fractional Brownian processes, Potential Anal. 10 (3) (1999) 273-288. | MR 1696137 | Zbl 0944.60045

[11] H. Föllmer, Calcul d'Itô sans probabilités, in: Séminaire de Probabilités XV 1979/80, Lecture Notes in Math., vol. 850, Springer-Verlag, 1981, pp. 143-150. | Numdam | MR 622559 | Zbl 0461.60074

[12] H. Föllmer, P. Protter, A.N. Shiryaev, Quadratic covariation and an extension of Itô's formula, Bernoulli 1 (1995) 149-169. | MR 1354459 | Zbl 0851.60048

[13] M. Fukushima, T.T. Oshima, Dirichlet Forms and Symmetric Markov Processes, Walter de Gruyter, 1994. | MR 1303354 | Zbl 0838.31001

[14] M. Gradinaru, I. Nourdin, Approximation at first and second order of the m-variation of the fractional Brownian motion, Electron. J. Probab. 8 (2003) 1-26, Paper 18. | MR 2041819 | Zbl 1063.60079

[15] M. Gradinaru, F. Russo, P. Vallois, Generalized covariations, local time and Stratonovich Itô’s formula for fractional Brownian motion with Hurst index H1 4, Ann. Probab. 31 (2003) 1772-1820. | Zbl 1059.60067

[16] T.J. Lyons, Differential equations driven by rough signals, Rev. Mat. Iberoamericana 14 (2) (1998) 215-310. | MR 1654527 | Zbl 0923.34056

[17] T.J. Lyons, W. Zheng, A crossing estimate for the canonical process on a Dirichlet space and tightness result, Astérisque 157-158 (1998) 249-271. | MR 976222 | Zbl 0654.60059

[18] S. Nakao, Stochastic calculus for continuous additive functionals of zero energy, Wahrs. Verw. Geb. 68 (1995) 557-578, (1985). | MR 772199 | Zbl 0604.60068

[19] I. Nourdin, PhD Thesis, Nancy, 2004.

[20] D. Nualart, The Malliavin Calculus and Related Topics, Springer-Verlag, 1995. | MR 1344217 | Zbl 0837.60050

[21] N. Privault, C. Tudor, Skorohod and pathwise stochastic calculus with respect to a L 2 - process, Random Operators and Stochastic Equations 8 (3) (2000) 1-24. | MR 1796673 | Zbl 0973.60062

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

[23] F. Russo, P. Vallois, Itô formula for C 1 -functions of semimartingales, Probab. Theory Related. Fields 104 (1996) 27-41. | MR 1367665 | Zbl 0838.60045

[24] F. Russo, P. Vallois, Stochastic calculus with respect to a finite quadratic variation process, Stochastics and Stochastics Reports 70 (2000) 1-40. | MR 1785063 | Zbl 0981.60053

[25] J. Stoer, R. Bulirsch, R. Bartels, W. Gautschi, C. Witzgall, Introduction to Numerical Analysis, Springer-Verlag, 1983. | Zbl 0423.65002

[26] G. Trutnau, Stochastic calculus of generalized Dirichlet forms and applications to stochastic differential equations in infinite dimensions, Osaka J. Math. 37 (2) (2000) 315-343. | MR 1772835 | Zbl 0963.60071

[27] J. Wolf, An Itô Formula for local Dirichlet processes, Stochastics and Stochastics Reports 62 (2) (1997) 103-115. | MR 1489183 | Zbl 0890.60044

[28] M. Yor, Sur quelques approximations d'intégrales stochastiques, in: Séminaire de Probabilités XI 1975/76, Lecture Notes in Math., vol. 581, Springer-Verlag, 1975, pp. 518-528. | Numdam | MR 448556 | Zbl 0367.60058

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