Approximations of the brownian rough path with applications to stochastic analysis
Annales de l'I.H.P. Probabilités et statistiques, Tome 41 (2005) no. 4, pp. 703-724.
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     title = {Approximations of the brownian rough path with applications to stochastic analysis},
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Friz, Peter; Victoir, Nicolas. Approximations of the brownian rough path with applications to stochastic analysis. Annales de l'I.H.P. Probabilités et statistiques, Tome 41 (2005) no. 4, pp. 703-724. doi : 10.1016/j.anihpb.2004.05.003. http://www.numdam.org/articles/10.1016/j.anihpb.2004.05.003/

[1] S. Aida, S. Kusuoka, D. Stroock, On the support of Wiener functionals, in: Asymptotic Problems in Probability Theory: Wiener Functionals and Asymptotics (Sanda/Kyoto, 1990), Pitman Res. Notes Math. Ser., vol. 284, Longman Sci. Tech., Harlow, 1993, pp. 3-34. | MR | Zbl

[2] P. Baldi, M. Chaleyat-Maurel, Sur l'équivalent du module de continuité des processus de diffusion, in: Séminaire de Probabilités, XXI, Lecture Notes in Math., vol. 1247, Springer, Berlin, 1987, pp. 404-427. | Numdam | MR | Zbl

[3] P. Baldi, M. Sanz-Solé, Modulus of continuity for stochastic flows, in: Barcelona Seminar on Stochastic Analysis (St. Feliu de Guíxols, 1991), Progr. Probab., vol. 32, Birkhäuser, Basel, 1993, pp. 1-20. | MR | Zbl

[4] P. Baldi, G. Ben Arous, G. Kerkyacharian, Large deviations and the Strassen theorem in Hölder norm, Stochastic Process. Appl. 42 (1) (1992) 171-180. | MR | Zbl

[5] G. Ben Arous, M. Ledoux, Grandes déviations de Freidlin-Wentzell en norme hölderienne, in: Séminaire de Probabilités, XXVIII, Lecture Notes in Math., vol. 1583, Springer, Berlin, 1994, pp. 293-299. | Numdam | MR | Zbl

[6] G. Ben Arous, M. Gruadinaru, Normes hölderiennes et support des diffusions, C. R. Acad. Sci. Paris Sér. I Math. 316 (3) (1993) 283-286. | MR | Zbl

[7] G. Ben Arous, M. Gruadinaru, M. Ledoux, Hölder norms and the support theorem for diffusions, Ann. Inst. H. Poincaré Probab. Statist. 30 (3) (1994) 415-436. | Numdam | MR | Zbl

[8] J.-D. Deuschel, D. Stroock, Large Deviations, Academic Press, New York, 1989. | MR | Zbl

[9] A. Dembo, O. Zeitouni, Large Deviations Techniques and Applications, Jones and Bartlett, Boston, MA, 1993. | MR | Zbl

[10] M. Eddahbi, M. N'Zi, Y. Ouknine, Grandes déviations des diffusions sur les espaces de Besov-Orlicz et application, Stochastics Stochastics Rep. 65 (3-4) (1999) 299-315. | MR | Zbl

[11] G.B. Folland, E.M. Stein, Hardy Spaces on Homogeneous Groups, Princeton University Press, 1982. | MR | Zbl

[12] P.K. Friz, Continuity of the Itô-map for Hölder rough path with applications to the Support Theorem in Hölder norm, preprint, 2003. | MR

[13] P.K. Friz, N. Victoir, A note on the notion of geometric rough paths, preprint, 2004.

[14] B. Gaveau, Principe de moindre action, propagation de la chaleur et estimées sous-elliptiques sur certains groupes nilpotents, Acta Math. 139 (1-2) (1977) 95-153. | MR | Zbl

[15] M. Gromov, Carnot-Carathéodory spaces seen from within, in: Sub-Riemannian Geometry, Progr. Math., vol. 144, Birkhäuser, Basel, 1996, pp. 79-323. | MR | Zbl

[16] I. Gyöngy, D. Nualart, M. Sanz-Solé, Approximation and support theorems in modulus spaces, Probab. Theory Related Fields 101 (4) (1995) 495-509. | MR | Zbl

[17] B.M. Hambly, T.J. Lyons, Stochastic area for Brownian motion on the Sierpinski gasket, Ann. Probab. 26 (1) (1998) 132-148. | MR | Zbl

[18] W. Hebisch, A. Sikora, A smooth subadditive homogeneous norm on a homogeneous group, Studia Math. 96 (1990) 231-236. | MR | Zbl

[19] I. Karatzas, E. Shreve, Brownian Motion and Stochastic Calculus, 1991. | Zbl

[20] T. Lyons, Differential equations driven by rough signals, Math. Res. Lett. 1 (1994) 451-464. | MR | Zbl

[21] A. Lejay, Introduction to Rough Paths, Séminaire de probabilités, Lecture Notes in Math., vol. XXXVII, Springer-Verlag, 2003. | MR | Zbl

[22] P. Levy, Processus Stochastiques et Mouvement Brownien, Gauthier-Villars, 1948. | MR | Zbl

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

[24] T. Lyons, Z. Qian, System Control and Rough Paths, Oxford University Press, 2002. | MR | Zbl

[25] M. Ledoux, Z. Qian, T. Zhang, Large deviations and support theorem for diffusion processes via rough paths, Stochastic Process. Appl. 102 (2) (2002) 265-283. | MR | Zbl

[26] P. Malliavin, Stochastic Analysis, Grundlehren der Mathematischen Wissenschaften, vol. 313, Springer-Verlag, Berlin, 1997. | MR | Zbl

[27] M. Mellouk, Support des diffusions dans les espaces de Besov-Orlicz, C. R. Acad. Sci. Paris Sér. I Math. 319 (3) (1994) 261-266. | MR | Zbl

[28] A. Millet, M. Sanz-Solé, A simple proof of the support theorem for diffusion processes, in: Séminaire de Probabilités, XXVIII, Lecture Notes in Math., vol. 1583, Springer, Berlin, 1994, pp. 36-48. | Numdam | MR | Zbl

[29] R. Montgomery, A Tour of Subriemannian Geometries, their Geodesics and Applications, Mathematical Surveys and Monographs, vol. 91, American Mathematical Society, Providence, RI, 2002. | MR | Zbl

[30] D. Neuenschwander, Probability on the Heisenberg Group, Lecture Notes in Math., Springer, 1996. | Zbl

[31] C. Reutenauer, Free Lie Algebras, London Mathematical Society Monographs. New Series, vol. 7, Oxford Science Publications, 1993. | MR | Zbl

[32] D. Revuz, N. Yor, Continuous Martingales and Brownian Motion, Springer, 2001. | Zbl

[33] E.M. Sipilainen, A pathwise view of solutions of stochastic differential equations, PhD Thesis, University of Edinburgh, 1993.

[34] E.M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, vol. 30, 1970. | MR | Zbl

[35] D. Stroock, S. Taniguchi, Diffusions as integral curves, or Stratonovich without Itô, in: The Dynkin Festschrift, Progr. Probab., vol. 34, 1994, pp. 333-369. | MR | Zbl

[36] D. Stroock, S.R. Varadhan, Multidimensional Diffusion Processes, Springer, 1979. | MR | Zbl

[37] D. Stroock, S.R. Varadhan, On the support of diffusion processes with applications to the strong maximum principle, in: Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, vol. III: Probability Theory, 1972, pp. 333-359. | MR | Zbl

[38] S.R.S. Varadhan, Brownian Motion. Lecture notes.

[39] V.S. Varadarajan, Lie Groups, Lie Algebras and their Presentations, Springer, 1984. | MR | Zbl

[40] N. Victoir, An extension theorem to rough path, preprint, 2003.

[41] F. Warner, Foundations of Differentiable Manifolds and Lie Groups, Springer, 1983. | MR | Zbl

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

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