Skip to main content
SpringerLink
Log in

Yet another introduction to rough paths

  • Chapter
  • First Online:
Séminaire de Probabilités XLII

Part of the book series: Lecture Notes in Mathematics ((SEMPROBAB,volume 1979))

Abstract

This specialized course provides another point of view on the theory of rough paths, starting with simple considerations on ordinary integrals, and stressing the importance of the Green-Riemann formula, as in the work of D. Feyel and A. de La Pradelle. This point of view allows us to gently introduce the required algebraic structures and provides alternative ways to understand why the construction of T. Lyons et al. is a natural generalization of the notion of integral of differential forms, in the sense that it shares the same properties as integrals along smooth paths, when we use the “right notion” of a path.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. R. Azencott. Géodésiques et diffusions en temps petit, vol. 84 of Astérisque. Société Mathématique de France, Paris, 1981.

    Google Scholar 

  2. G. Ben Arous. Flots et séries de Taylor stochastiques. Probab. Theory Related Fields, 81:1, 29–77, 1989.

    Article  MathSciNet  MATH  Google Scholar 

  3. A. Baker. Matrix groups: An introduction to Lie group theory. Springer Undergraduate Mathematics Series. Springer-Verlag London Ltd., London, 2002.

    MATH  Google Scholar 

  4. F. Baudoin. An introduction to the geometry of stochastic flows. Imperial College Press, London, 2004.

    Book  MATH  Google Scholar 

  5. D. Burago, Y. Burago and S. Ivanov. A course in metric geometry, vol. 33 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2001.

    MATH  Google Scholar 

  6. J.-M. Bismut. Large deviations and the Malliavin calculus, vol. 45 of Progress in Mathematics. Birkhäuser Boston Inc., Boston, MA, 1984.

    MATH  Google Scholar 

  7. A. Bensoussan, J.-L. Lions and G. Papanicolaou. Asymptotic Analysis for Periodic Structures. North-Holland, 1978.

    Google Scholar 

  8. F. Castell. Asymptotic expansion of stochastic flows. Probab. Theory Related Fields, 96:2, 225–239, 1993.

    Article  MathSciNet  MATH  Google Scholar 

  9. V.V. Chistyakov and O.E. Galkin. On maps of bounded p-variations with p > 1. Positivity, 2:1, 19–45, 1998.

    Article  MathSciNet  MATH  Google Scholar 

  10. K.-T. Chen. Integration of paths, geometric invariants and a generalized Baker-Hausdorff formula. Ann. of Math. (2), 65, 163–178, 1957.

    Article  MathSciNet  MATH  Google Scholar 

  11. K.-T. Chen. Integration of Paths–A Faithful Representation, of Paths by Non-commutative Formal Power Series. Trans. Amer. Math. Soc., 89:2, 395–407, 1958.

    MathSciNet  Google Scholar 

  12. Z. Ciesielski. On the isomorphisms of the spaces H α and m. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 8, 217–222, 1960.

    MathSciNet  MATH  Google Scholar 

  13. L. Coutin and A. Lejay. Semi-martingales and rough paths theory. Electron. J. Probab., 10:23, 761–785, 2005.

    MathSciNet  Google Scholar 

  14. L. Coutin. An introduction to (stochastic) calculus with respect to fractional Brownian motion. In Séminaire de Probabilités XL, vol. 1899 of Lecture Notes in Math., pp. 3–65. Springer, 2007.

    Google Scholar 

  15. L. Coutin and Z. Qian. Stochastic analysis, rough path analysis and fractional Brownian motions. Probab. Theory Related Fields, 122:1, 108–140, 2002. <DOI:10.1007/s004400100158>.

    Article  MathSciNet  MATH  Google Scholar 

  16. A.M. Davie. Differential Equations Driven by Rough Signals: an Approach via Discrete Approximation. Appl. Math. Res. Express. AMRX, 2, Art. ID abm009, 40, 2007.

    MathSciNet  Google Scholar 

  17. J. J. Duistermaat and J. A. C. Kolk. Lie groups. Universitext. Springer-Verlag, Berlin, 2000.

    Book  MATH  Google Scholar 

  18. R.M. Dudley and R. Norvaiša. An introduction to p-variation and Young integrals—with emphasis on sample functions of stochastic processes, 1998. Lecture given at the Centre for Mathematical Physics and Stochastics, Department of Mathematical Sciences, University of Aarhus. Available on the web site <www.maphysto.dk>.

    MATH  Google Scholar 

  19. H. Doss. Liens entre équations différentielles stochastiques et ordinaires. Ann. Inst. H. Poincaré Sect. B (N.S.), 13:2, 99–125, 1977.

    MathSciNet  MATH  Google Scholar 

  20. C. T. J. Dodson and T. Poston. Tensor geometry: The geometric viewpoint and its uses, vol. 130 of Graduate Texts in Mathematics. Springer-Verlag, Berlin, 2nd edition, 1991.

    MATH  Google Scholar 

  21. D. Feyel and A. de La Pradelle. Curvilinear integrals along enriched paths. Electron. J. Probab., 11:35, 860–892, 2006.

    MathSciNet  Google Scholar 

  22. D. Feyel, A. de La Pradelle and G. Mokobodzki. A non-commutative sewing lemma. Electron. Commun. Probab., 13, 24–34, 2008. <ArXiv: 0706. 0202>.

    MathSciNet  MATH  Google Scholar 

  23. M. Fliess. Fonctionnelles causales non linéaires et indéterminées non communtatives. Bull. Soc. Math. France, 109:1, 3–40, 1981.

    MathSciNet  MATH  Google Scholar 

  24. M. Fliess and D. Normand-Cyrot. Algèbres de Lie nilpotentes, formule de Baker-Campbell-Hausdorff et intégrales itérées de K. T. Chen. In Séminaire de Probabilités, XVI, vol. 920, pp. 257–267. Springer, Berlin, 1982.

    Google Scholar 

  25. G. B. Folland. Harmonic analysis in phase space, vol. 122 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 1989.

    Google Scholar 

  26. P. Friz. Continuity of the Itâ-Map for Hölder rough paths with applications to the support theorem in Hölder norm. In Probability and Partial Differential Equations in Modern Applied Mathematics, vol. 140 of IMA Volumes in Mathematics and its Applications, pp. 117–135. Springer, 2005.

    Google Scholar 

  27. G. B. Folland and E. M. Stein. Hardy spaces on homogeneous groups, vol. 28 of Mathematical Notes. Princeton University Press, 1982.

    Google Scholar 

  28. P. Friz and N. Victoir. Multidimensional Stochastic Processes as Rough Paths. Theory and Applications. Cambridge University Press, 2009.

    Google Scholar 

  29. P. Friz and N. Victoir. A Note on the Notion of Geometric Rough Paths. Probab. Theory Related Fields, 136:3, 395–416, 2006. <doi:10.1007/s00440-005-0487-7>, <ArXiv: math.PR/0403115>.

    Article  MathSciNet  MATH  Google Scholar 

  30. P. Friz and N. Victoir. Differential Equations Driven by Gaussian Signals I. <ArXiv: 0707.0313>, Cambridge University (preprint), 2007.

    Google Scholar 

  31. P. Friz and N. Victoir. Differential Equations Driven by Gaussian Signals II. <ArXiv: 0711.0668>, Cambridge University (preprint), 2007.

    Google Scholar 

  32. P. Friz and N. Victoir. Euler Estimates of Rough Differential Equations. J. Differential Equations, 244:2, 388–412, 2008. <ArXiv: math. CA/0602345>.

    Article  MathSciNet  MATH  Google Scholar 

  33. B. Gaveau. Principle de moindre action, propagation de la chaleur et estimées sous elliptiques sur certains groupes nilpotents. Acta Math., 139:1–2, 95–153, 1977.

    Article  MathSciNet  MATH  Google Scholar 

  34. R. Goodman. Filtrations and asymptotic automorphisms on nilpotent Lie groups. J. Differential Geometry, 12:2, 183–196, 1977.

    MathSciNet  Google Scholar 

  35. M. Gromov. Carnot-Carathéodory spaces seen from within. In Sub-Riemannian geometry, vol. 144 of Progr. Math., pp. 79–323. Birkhäuser, 1996.

    Google Scholar 

  36. M. Gubinelli. Controlling rough paths. J. Funct. Anal., 216:1, 86–140, 2004.

    Article  MathSciNet  MATH  Google Scholar 

  37. B. C. Hall. Lie groups, Lie algebras, and representations, vol. 222 of Graduate Texts in Mathematics. Springer-Verlag, New York, 2003.

    Book  MATH  Google Scholar 

  38. B.M. Hambly and T.J. Lyons. Uniqueness for the signature of a path of bounded variation and continuous analogues for the free group. Oxford University (preprint), 2006.

    Google Scholar 

  39. A. Isidori. Nonlinear control systems. Springer-Verlag, Berlin, 3rd edition, 1995.

    MATH  Google Scholar 

  40. N. Ikeda and S. Watanabe. Stochastic differential equations and diffusion processes, vol. 24 of North-Holland Mathematical Library. North-Holland Publishing Co., Second edition, 1989.

    Google Scholar 

  41. M. Kawski. Nonlinear control and combinatorics of words. In Geometry of feedback and optimal control, vol. 207 of Monogr. Textbooks Pure Appl. Math., pp. 305–346. Dekker, New York, 1998.

    Google Scholar 

  42. T. Lyons, M. Caruana and T. Lévy. Differential Equations Driven by Rough Paths. In École d'été de probabilités de Saint-Flour XXXIV—2004, edited by J. Picard, vol. 1908 of Lecture Notes in Math., Berlin, 2007. Springer.

    Google Scholar 

  43. A. Lejay. On the convergence of stochastic integrals driven by processes converging on account of a homogenization property. Electron. J. Probab., 7:18, 1–18, 2002.

    MathSciNet  Google Scholar 

  44. A. Lejay. An introduction to rough paths. In Séminaire de probabilités, XXXVII, vol. 1832 of Lecture Notes in Mathematics, pp. 1–59. Springer-Verlag, 2003.

    Google Scholar 

  45. A. Lejay. Stochastic Differential Equations driven by processes generated by divergence form operators I: a Wong-Zakai theorem. ESAIM Probab. Stat., 10, 356–379, 2006. <doi: 10.1051/ps:2006015>.

    Article  MathSciNet  MATH  Google Scholar 

  46. A. Lejay. Stochastic Differential Equations driven by processes generated by divergence form operators II: convergence results. ESAIM Probab. Stat., 12, 387–411, 2008. <doi: 10.1051/ps:2007040>.

    Article  MathSciNet  MATH  Google Scholar 

  47. P. Lévy. Processus stochastiques et mouvement brownien. Gauthier-Villars & Cie, Paris, 2e édition, 1965.

    MATH  Google Scholar 

  48. A. Lejay and T. Lyons. On the importance of the Lévy area for systems controlled by converging stochastic processes. Application to homogenization. In New Trends in Potential Theory, Conference Proceedings, Bucharest, September 2002 and 2003, edited by D. Bakry, L. Beznea, Gh. Bucur and M. Röckner, pp. 63–84. The Theta Foundation, 2006.

    Google Scholar 

  49. X.D. Li and T. J. Lyons. Smoothness of Itâ maps and diffusion processes on path spaces. I. Ann. Sci. École Norm. Sup., 39:4, 649–677, 2006.

    MathSciNet  MATH  Google Scholar 

  50. M. Ledoux, T. Lyons and Z. Qian. Lévy area of Wiener processes in Banach spaces. Ann. Probab., 30:2, 546–578, 2002.

    Article  MathSciNet  MATH  Google Scholar 

  51. T. Lyons and Z. Qian. Flow of diffeomorphisms induced by a geometric multiplicative functional. Proba. Theory Related Fields, 112:1, 91–119, 1998.

    Article  MathSciNet  MATH  Google Scholar 

  52. T. Lyons and Z. Qian. System Control and Rough Paths. Oxford Mathematical Monographs. Oxford University Press, 2002.

    Book  MATH  Google Scholar 

  53. A. Lejay and N. Victoir. On (p, q)-rough paths. J. Differential Equations, 225:1, 103–133, 2006. <doi: 10.1016/j.jde.2006.01.018>.

    Article  MathSciNet  MATH  Google Scholar 

  54. T. Lyons and N. Victoir. An Extension Theorem to Rough Paths. Ann. Inst. H. Poincaré Anal. Non Linéaire, 24:5, 835–847, 2007. <doi: 10.1016/j.anihpc.2006.07.004>.

    Article  MathSciNet  MATH  Google Scholar 

  55. T.J. Lyons. Differential equations driven by rough signals. Rev. Mat. Iberoamericana, 14:2, 215–310, 1998.

    Article  MathSciNet  MATH  Google Scholar 

  56. T.J. Lyons and T. Zhang. Decomposition of Dirichlet Processes and its Application. Ann. Probab., 22:1, 494–524, 1994.

    Article  MathSciNet  MATH  Google Scholar 

  57. W. Magnus. On the exponential solution of differential equations for a linear operator. Comm. Pure Appl. Math., 7, 649–673, 1954.

    Article  MathSciNet  MATH  Google Scholar 

  58. R. Mahony and J.H. Manton. The Geometry of the Newton Method on Non-Compact Lie Groups J. Global Optim., 23, 309–327, 2002.

    Article  MathSciNet  MATH  Google Scholar 

  59. E. J. McShane. Stochastic differential equations and models of random processes. In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. III: Probability theory, pp. 263–294. Univ. California Press, 1972.

    Google Scholar 

  60. R. Montgomery. A tour of subriemannian geometries, their geodesics and applications, vol. 91 of Mathematical Surveys and Monographs. American Mathematical Society, 2002.

    Google Scholar 

  61. J. Musielak and Z. Semadeni. Some classes of Banach spaces depending on a parameter. Studia Math., 20, 271–284, 1961.

    MathSciNet  MATH  Google Scholar 

  62. A. Millet and M. Sanz-solé. Approximation of rough paths of fractional Brownian motion. In Seminar on Stochastic Analysis, Random Fields and Applications V, pp. 275–303, Progr. Probab., Birkhäuser, 2008.

    Chapter  Google Scholar 

  63. C. Reutenauer. Free Lie algebras, vol. 7 of London Mathematical Society Monographs. New Series. Oxford University Press, 1993.

    Google Scholar 

  64. R.A. Ryan. Introduction to Tensor Products of Banach Spaces. Springer-Verlag, 2002.

    Google Scholar 

  65. E.-M. Sipiläinen. A pathwise view of solutions of stochastic differential equations. PhD thesis, University of Edinburgh, 1993.

    Google Scholar 

  66. R. S. Strichartz. The Campbell-Baker-Hausdorff-Dynkin formula and solutions of differential equations. J. Funct. Anal., 72:2, 320–345, 1987.

    Article  MathSciNet  MATH  Google Scholar 

  67. D. H. Sattinger and O. L. Weaver. Lie groups and algebras with applications to physics, geometry, and mechanics, vol. 61 of Applied Mathematical Sciences. Springer-Verlag, New York, 1993. Corrected reprint of the 1986 original.

    Google Scholar 

  68. K. Tapp. Matrix groups for undergraduates, vol. 29 of Student Mathematical Library. American Mathematical Society, 2005.

    Google Scholar 

  69. V. S. Varadarajan. Lie groups, Lie algebras, and their representations, vol. 102 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1984. Reprint of the 1974 edition.

    MATH  Google Scholar 

  70. N. Victoir. Lévy area for the free Brownian motion: existence and non-existence. J. Funct. Anal., 208:1, 107–121, 2004.

    Article  MathSciNet  MATH  Google Scholar 

  71. F.W. Warner. Foundations of differentiable manifolds and Lie groups, vol. 94 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1983. Corrected reprint of the 1971 edition.

    MATH  Google Scholar 

  72. Y. Yamato. Stochastic differential equations and nilpotent Lie algebras. Z. Wahrsch. Verw. Gebiete, 47:2, 213–229, 1979.

    Article  MathSciNet  MATH  Google Scholar 

  73. L.C. Young. An inequality of the Hölder type, connected with Stieltjes integration. Acta Math., 67, 251–282, 1936.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Èquipe-Projet TOSCA, Institut Èlie Cartan (Nancy-Universitè, CNRS, INRIA) Campus scientifique, BP 239, 54506, Vandoeuvre-lès-Nancy, Cedex, France

    Antoine Lejay

Authors
  1. Antoine Lejay
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Antoine Lejay .

Editor information

Editors and Affiliations

  1. Laboratoire de Mathématiques, Université de Versailles-St-Quentin, avenue des États-Unis 45, Versailles, 78035, France

    Catherine Donati-Martin

  2. Inst. Recherche, Mathématiques Avancées, Université Strasbourg I, rue René Descartes 7, Strasbourg CX, 67084, France

    Michel Émery

  3. Laboratoire de Mathematiques, Université de Versailles, av. des Etats-Unis 45, Versailles CX, 78035, France

    Alain Rouault

  4. CNRS UMR 6623, Lab. de Mathématiques, Université Besancon, route de Gray 16, Besancon CX, 25030, France

    Christophe Stricker

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Lejay, A. (2009). Yet another introduction to rough paths. In: Donati-Martin, C., Émery, M., Rouault, A., Stricker, C. (eds) Séminaire de Probabilités XLII. Lecture Notes in Mathematics(), vol 1979. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-01763-6_1

Download citation

Publish with us

Policies and ethics

Search

Navigation