×

zbMATH — the first resource for mathematics

A note on the notion of geometric rough paths. (English) Zbl 1108.34052
The paper clarifies two definitions of Lyons’ geometric rough paths, and proves a conjecture by Ledoux concerning fractional Brownian motions.

MSC:
34F05 Ordinary differential equations and systems with randomness
60G17 Sample path properties
34A26 Geometric methods in ordinary differential equations
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Ciesielski, Z.: On the isomorphisms of the spaces H \(\alpha\) and m. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 8, 217–222 (1960) · Zbl 0093.12301
[2] Coutin, L., Qian Z.: Stochastic analysis, rough path analysis and fractional Brownian motions Probab. Theory Relat. Fields 122, 108–140 (2002) · Zbl 1047.60029 · doi:10.1007/s004400100158
[3] Decreusefond, L., Üstünel, A.S.: Stochastic Analysis of the Fractional Brownian Motion, Potential Analysis 10, 177–214 (1997)
[4] Dudley, R.M., Norvaisa, R.: An introduction to p-variation and Young integrals. Lecture notes.
[5] Folland, G.B., Stein, E.M.: Hardy spaces on homogeneous groups. Mathematical Notes, 28. Princeton University Press, 1982 · Zbl 0508.42025
[6] De La Pradelle, A., Feyel, D.: Curvilinear Integrals along Rough Paths. Preprint · Zbl 1110.60031
[7] Friz, P.: Continuity of the Ito-map for Hoelder rough paths with applications to the support theorem in Hölder norm, Probability and PDEs in Modern Applied Mathematics, The IMA Volumes in Mathematics and its Applications, Vol. 140, 2005
[8] Friz, P., Lyons, T., Stroock, D.: Lévy’s area under conditioning, Annales de l’Institut Henri Poincare (B), Probability and Statistics, 2005. To appear · Zbl 1099.60054
[9] Friz, P., Victoir, N.: Approximations of the Brownian Rough Path with Applications to Stochastic Analysis, Annales de l’Institut Henri Poincare (B), Probability and Statistics, Volume 41, Issue 4, 2005, pp. 703–724 · Zbl 1080.60021
[10] Goodman, R.: Filtrations and Asymptotic Automorphisms on Nilpotent Lie Groups. J.Diff.Geometry 12, 183–196 (1977) · Zbl 0389.22015
[11] Haynes, G.W., Hermes H.: Nonlinear Controllability via Lie Theory. SIAM J. Control Optim. 8 (4), 450–460 (1970) · Zbl 0229.93012
[12] Kurzweil, J., Jarnik, J.: Limit Process in Ordinary Differential Equations”. J. Appl. Math. Phys. 38, 241–256 (1987) · Zbl 0616.34004 · doi:10.1007/BF00945409
[13] Kurzweil, J., Jarnik, J.: Iterated Lie Brackets in Limit Processes in Ordinary Differential Equations. Results in Mathematics 14, 125–137 (1988) · Zbl 0663.34043
[14] Kurzweil, J., Jarnik, J.: A Convergence Effect in Ordinary Differential Equations. Asym. Meth. Math. Physics (Russian), 301, ’Naukova Pumka’, Kiev, 1989 · Zbl 0319.34066
[15] Lejay, A.: Introduction to Rough Paths, Séminaire de Probabilité, Springer. To appear
[16] Ledoux, M., Qian, Z., Zhang, T.: Large deviations and support theorem for diffusion processes via rough paths. Stochastic Process. Appl. 102 (2), 265–283 (2002) · Zbl 1075.60510 · doi:10.1016/S0304-4149(02)00176-X
[17] Liu, W.: An approximation algorithm for nonholonomic systems. SIAM J. Control Optim. 35 (4), 1328–1365 (1997) · Zbl 0887.34063 · doi:10.1137/S0363012993260501
[18] Liu, W.S.: Averaging Theorems for Highly Oscillatory Differential Equations and Iterated Lie Brackets.” SIAM J. Control Optim. 35 (6), (1997) · Zbl 0888.34037
[19] Liu, W., Sussman, H.: Shortest paths for sub-Riemannian metrics on rank-two distributions. Mem. Amer. Math. Soc. 118 (564), (1995) · Zbl 0843.53038
[20] Lyons, T.: Differential equations driven by rough signals. Rev. Mat. Iberoamericana 14 (2), 215–310 (1998) · Zbl 0923.34056
[21] Lyons, T., Qian, Z.: System Control and Rough Paths, Oxford University Press, 2002 · Zbl 1029.93001
[22] Lyons, T., Victoir, N.: An Extension Theorem to Rough Path. Preprint · Zbl 1134.60047
[23] Malliavin, P.: Stochastic Analysis, Springer, 1997 · Zbl 0878.60001
[24] Montgomery, R.: A tour of subriemannian geometries, their geodesics and applications. Math.Surveys and Monographs 91. AMS, 2002 · Zbl 1044.53022
[25] Musielak, J., Semadeni, Z.: Some classes of Banach spaces depending on a parameter. Studia Math. 20, 271–284 (1961) · Zbl 0099.09401
[26] Sussmann, H.J., Liu, W.: Limits of highly oscillatory controls and the approximation of general paths by admissible trajectories.” IEEE Publications, New York, 1991, pp. 437–442
[27] Wiener, N.: The quadratic variation of a function and its Fourier coefficients. J. Math. and Phys. 3, 72–94 (1924) · JFM 50.0203.01
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.