Rough paths via sewing Lemma
ESAIM: Probability and Statistics, Tome 16 (2012), pp. 479-526.

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

DOI : 10.1051/ps/2011108
Classification : 47E99, 60G15
Mots-clés : rough paths, differential equations 
@article{PS_2012__16__479_0,
     author = {Coutin, Laure},
     title = {Rough paths \protect\emph{via }sewing {Lemma}},
     journal = {ESAIM: Probability and Statistics},
     pages = {479--526},
     publisher = {EDP-Sciences},
     volume = {16},
     year = {2012},
     doi = {10.1051/ps/2011108},
     zbl = {1277.47081},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/ps/2011108/}
}
TY  - JOUR
AU  - Coutin, Laure
TI  - Rough paths via sewing Lemma
JO  - ESAIM: Probability and Statistics
PY  - 2012
SP  - 479
EP  - 526
VL  - 16
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/ps/2011108/
DO  - 10.1051/ps/2011108
LA  - en
ID  - PS_2012__16__479_0
ER  - 
%0 Journal Article
%A Coutin, Laure
%T Rough paths via sewing Lemma
%J ESAIM: Probability and Statistics
%D 2012
%P 479-526
%V 16
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/ps/2011108/
%R 10.1051/ps/2011108
%G en
%F PS_2012__16__479_0
Coutin, Laure. Rough paths via sewing Lemma. ESAIM: Probability and Statistics, Tome 16 (2012), pp. 479-526. doi : 10.1051/ps/2011108. http://archive.numdam.org/articles/10.1051/ps/2011108/

[1] F. Baudoin, An introduction to the geometry of stochastic lows. Imperial Press College, London (2004). | MR | Zbl

[2] J. Bertoin, Sur une intégrale pour les processus à α-variation bornée. Ann. Probab. 17 (1999) 1521-1535. | MR | Zbl

[3] K.-T. Chen, Integration of paths, geometric invariants and a generalized Baker-Hausdorff formula. Ann. Math. 65 (1957) 163-178. | MR | Zbl

[4] K.-T. Chen, Integration of paths; a faithful representation of paths by non-commutative formal power series. Trans. Amer. Math Soc. 89 (1958) 395-407. | MR | Zbl

[5] K.-T. Chen, Integration of paths, Bull. Amer. Math. Soc. 83 (1977) 831-879. | MR | Zbl

[6] Z. Ciesielski, G. Kerkyacharian and B. Roynette, Quelques espaces fonctionnels associés à des processus gaussiens. Studia Math. 107 (1993) 171-204. | MR | Zbl

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

[8] L. Coutin and N. Victoir, Enhanced Gaussian processes and applications. ESAIM Probab. Stat. 13 (2009) 247-260. | Numdam | MR | Zbl

[9] A.M. Davie, Differential equations driven by rough paths : an approach via discrete approximation. Appl. Math. Res. Express. AMRX 2 (2007) abm009, 40. | MR | Zbl

[10] A.M. Davie, Uniqueness of solutions of stochastic differential equations. Int. Math. Res. Not. IMRN 24 (2007) rnm124, 26. | MR | Zbl

[11] H. Doss, Liens entre équations différentielles stochastiques et ordinaires. Ann. Inst. Henri Poincaré Sect. B (N.S.) 13 (1977) 99-125. | Numdam | MR | Zbl

[12] D. Feyel and A. De La Pradelle, Curvilinear integrals along enriched paths. Electron. J. Probab. 11 (2006) 860-892 (electronic). | MR | Zbl

[13] P. Friz and N. Victoir, Multidimensional Stochastic Processes as Rough Paths. Theory and Applications. Cambridge University Press (2008). | MR | Zbl

[14] P. Friz and N. Victoir, Differential equations driven by Gaussian signals. Ann. Inst. Henri Poincaré Probab. Stat. 46 (2010) 369-413. | Numdam | MR | Zbl

[15] M. Gubinelli, Controlling rough paths. J. Funct. Anal. 216 (2004) 86-140. | MR | Zbl

[16] Y. Hu and D. Nualart, Rough path analysis via fractional calculus. Trans. Amer. Math. Soc. 361 (2009) 2689-2718. | MR | Zbl

[17] Y. Inahama and H. Kawabi, Asymptotic expansions for the Laplace approximations for Itô functionals of Brownian rough paths. J. Funct. Anal. 243 (2007) 270-322. | MR | Zbl

[18] A. Lejay, An introduction to rough paths. Séminaire de probabilités, XXXVII 1832 (2003) 1-59. | MR | Zbl

[19] A. Lejay, Yet another introduction to rough paths. Séminaire de Probabilités, Lect. Notes in Maths XLII (2009) 1-101. | MR | Zbl

[20] A. Lejay, On rough differential equations. Electron. J. Probab. 14 (2009) 341-364. | MR | Zbl

[21] T. Lyons, Differential equations driven by rough signals. I. An extension of an inequality of L.C. Young. Math. Res. Lett. 1 (1994) 451-464. | MR | Zbl

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

[23] T. Lyons and Zhongmin Qian, System control and rough paths. Oxford Mathematical Monographs. Oxford University Press, Oxford, Oxford Science Publications (2002). | MR | Zbl

[24] 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). | MR | Zbl

[25] 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. | MR | Zbl

[26] D. Nualart and A. Răşcanu, Differential equations driven by fractional Brownian motion. Collect. Math. 53 (2002) 55-81. | Zbl

[27] 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. | MR | Zbl

[28] E.M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series 30. Princeton University Press, Princeton, N.J. (1970). | MR | Zbl

[29] H. Sussmann, On the gap between deterministic and stochastic ordinary differential equations. Ann. Probab. 6 (1978) 19-41. | MR | Zbl

[30] A. Tychonoff, Ein Fixpunktsatz. Math. Ann. 111 (1935) 767-776. | MR | Zbl

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

[32] M. Zähle, On the link between fractional and stochastic calculus, in Stochastic dynamics, Bremen (1997), Springer, New York (1999) 305-325. | MR | Zbl

Cité par Sources :