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Rough paths via sewing Lemma

Published online by Cambridge University Press:  08 November 2012

Laure Coutin
Laure Coutin*
Affiliation:
UniversitéParis 5, Centre Universitaire des Saints Pères, UMR C8145, 45 rue des Saints Pères, 75270 Paris Cedex 06, France. [email protected]
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Abstract

We present the rough path theory introduced by Lyons, using the swewing lemma of Feyel and de Lapradelle.

Keywords

Rough pathsdifferential equations
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 16 , 2012 , pp. 479 - 526
Copyright
© EDP Sciences, SMAI, 2012

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References

F. Baudoin, An introduction to the geometry of stochastic lows. Imperial Press College, London (2004).
Bertoin, J., Sur une intégrale pour les processus à α-variation bornée. Ann. Probab. 17 (1999) 1521–1535. Google Scholar
Chen, K.-T., Integration of paths, geometric invariants and a generalized Baker-Hausdorff formula. Ann. Math. 65 (1957) 163–178. Google Scholar
Chen, K.-T., Integration of paths; a faithful representation of paths by non-commutative formal power series. Trans. Amer. Math Soc. 89 (1958) 395-407. Google Scholar
Chen, K.-T., Integration of paths, Bull. Amer. Math. Soc. 83 (1977) 831–879. CrossRefGoogle Scholar
Ciesielski, Z., Kerkyacharian, G. and Roynette, B., Quelques espaces fonctionnels associés à des processus gaussiens. Studia Math. 107 (1993) 171–204. Google Scholar
Coutin, L. and Qian, Z., Stochastic analysis, rough path analysis and fractional Brownian motions. Probab. Theory Related Fields 122 (2002) 108–140. Google Scholar
Coutin, L. and Victoir, N., Enhanced Gaussian processes and applications. ESAIM Probab. Stat. 13 (2009) 247–260. Google Scholar
Davie, A.M., Differential equations driven by rough paths : an approach via discrete approximation. Appl. Math. Res. Express. AMRX 2 (2007) abm009, 40. Google Scholar
A.M. Davie, Uniqueness of solutions of stochastic differential equations. Int. Math. Res. Not. IMRN 24 (2007) rnm124, 26.
Doss, H., Liens entre équations différentielles stochastiques et ordinaires. Ann. Inst. Henri Poincaré Sect. B (N.S.) 13 (1977) 99–125. Google Scholar
Feyel, D. and de La Pradelle, A., Curvilinear integrals along enriched paths. Electron. J. Probab. 11 (2006) 860–892 (electronic). Google Scholar
P. Friz and N. Victoir, Multidimensional Stochastic Processes as Rough Paths. Theory and Applications. Cambridge University Press (2008).
Friz, P. and Victoir, N., Differential equations driven by Gaussian signals. Ann. Inst. Henri Poincaré Probab. Stat. 46 (2010) 369–413. Google Scholar
Gubinelli, M., Controlling rough paths. J. Funct. Anal. 216 (2004) 86–140. Google Scholar
Hu, Y. and Nualart, D., Rough path analysis via fractional calculus. Trans. Amer. Math. Soc. 361 (2009) 2689–2718. Google Scholar
Inahama, Y. and Kawabi, H., Asymptotic expansions for the Laplace approximations for Itô functionals of Brownian rough paths. J. Funct. Anal. 243 (2007) 270–322. CrossRefGoogle Scholar
Lejay, A., An introduction to rough paths. Séminaire de probabilités, XXXVII 1832 (2003) 1–59. Google Scholar
Lejay, A., Yet another introduction to rough paths. Séminaire de Probabilités, Lect. Notes in Maths XLII (2009) 1–101. Google Scholar
Lejay, A., On rough differential equations. Electron. J. Probab. 14 (2009) 341–364. Google Scholar
Lyons, T., Differential equations driven by rough signals. I. An extension of an inequality of L.C. Young. Math. Res. Lett. 1 (1994) 451–464. Google Scholar
Lyons, T.J., Differential equations driven by rough signals. Rev. Mat. Iberoamericana 14 (1998) 215–310. Google Scholar
T. Lyons and Zhongmin Qian, System control and rough paths. Oxford Mathematical Monographs. Oxford University Press, Oxford, Oxford Science Publications (2002).
T. Lyons, M. Caruana and T. Lévy, Differential equations driven by rough paths Ecole d’été de probabilités de Saint-Flour XXXIV (2004), Lectures Notes in Math 1908. J. Picard Ed., Springer, Berlin (2007).
I. Nourdin, A simple theory for the study of SDEs driven by a fractional Brownian motion, in dimension one, in Séminaire de probabilités XLI, Lecture Notes in Math. 1934. Springer, Berlin (2008) 181–197.
Nualart, D. and Răşcanu, A., Differential equations driven by fractional Brownian motion. Collect. Math. 53 (2002) 55–81. Google Scholar
S.G. Samko, A.A. Kilbas and O.I. Marichev, Fractional integrals and derivatives. Gordon and Breach Science Publishers, Yverdon (1993). Theory and applications, Edited and with a foreword by S.M. Nikolski Ed., Translated from the 1987 Russian original, Revised by the authors.
E.M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series 30. Princeton University Press, Princeton, N.J. (1970).
Sussmann, H., On the gap between deterministic and stochastic ordinary differential equations. Ann. Probab. 6 (1978) 19–41. Google Scholar
Tychonoff, A., Ein Fixpunktsatz. Math. Ann. 111 (1935) 767–776. Google Scholar
Young, L.C., An inequality of the Hölder type, connected with Stieltjes integration. Acta Math. 67 (1936) 251–282. Google Scholar
M. Zähle, On the link between fractional and stochastic calculus, in Stochastic dynamics, Bremen (1997), Springer, New York (1999) 305–325.