UniversitéParis 5, Centre Universitaire des Saints Pères, UMR C8145, 45 rue des Saints Pères, 75270 Paris Cedex 06, France. laure.Coutin@math-info.univ-paris5.fr
Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.
We present the rough path theory introduced by Lyons, using the swewing lemma of Feyel and de Lapradelle.
F. Baudoin, An introduction to the geometry
of stochastic lows. Imperial Press College, London (2004).
[2]
Bertoin, J., Sur une intégrale pour les
processus à α-variation bornée. Ann.
Probab.17 (1999)
1521–1535. Google Scholar
[3]
Chen, K.-T., Integration of paths, geometric
invariants and a generalized Baker-Hausdorff formula.
Ann. Math.65 (1957)
163–178. Google Scholar
[4]
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
Ciesielski, Z., Kerkyacharian, G. and
Roynette, B., Quelques espaces fonctionnels
associés à des processus gaussiens. Studia
Math.107 (1993)
171–204. Google Scholar
[7]
Coutin, L. and Qian, Z., Stochastic analysis, rough path
analysis and fractional Brownian motions. Probab. Theory
Related Fields122 (2002)
108–140. Google Scholar
[8]
Coutin, L. and Victoir, N., Enhanced Gaussian processes and
applications. ESAIM Probab. Stat.13 (2009)
247–260. Google Scholar
[9]
Davie, A.M., Differential equations driven by rough
paths : an approach via discrete approximation. Appl. Math. Res. Express.
AMRX2 (2007) abm009, 40. Google Scholar
[10]
A.M. Davie, Uniqueness of solutions of stochastic
differential equations. Int. Math. Res. Not. IMRN24 (2007) rnm124, 26.
[11]
Doss, H., Liens entre équations
différentielles stochastiques et ordinaires. Ann. Inst.
Henri Poincaré Sect. B (N.S.)13 (1977) 99–125.
Google Scholar
[12]
Feyel, D. and de La Pradelle, A., Curvilinear
integrals along enriched paths. Electron. J. Probab.11 (2006) 860–892 (electronic). Google Scholar
[13]
P. Friz and N. Victoir, Multidimensional
Stochastic Processes as Rough Paths. Theory and Applications. Cambridge
University Press (2008).
[14]
Friz, P. and Victoir, N., Differential equations driven
by Gaussian signals. Ann. Inst. Henri Poincaré Probab.
Stat.46 (2010)
369–413. Google Scholar
[15]
Gubinelli, M., Controlling rough
paths. J. Funct. Anal.216 (2004)
86–140. Google Scholar
[16]
Hu, Y. and Nualart, D., Rough path analysis
via fractional calculus. Trans. Amer.
Math. Soc.361 (2009)
2689–2718. Google Scholar
[17]
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
[18]
Lejay, A., An introduction to rough paths.
Séminaire de probabilités, XXXVII1832 (2003) 1–59. Google Scholar
[19]
Lejay, A., Yet another introduction to rough paths.
Séminaire de Probabilités, Lect. Notes in MathsXLII (2009) 1–101. Google Scholar
[20]
Lejay, A., On rough differential
equations. Electron. J. Probab.14 (2009)
341–364. Google Scholar
[21]
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
[22]
Lyons, T.J., Differential equations driven by
rough signals. Rev. Mat.
Iberoamericana14 (1998)
215–310. Google Scholar
[23]
T. Lyons and Zhongmin Qian, System control
and rough paths. Oxford Mathematical Monographs. Oxford University Press,
Oxford, Oxford Science Publications (2002).
[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
Math1908. J. Picard Ed., Springer, Berlin (2007).
[25]
I. Nourdin, A simple theory for the study of SDEs
driven by a fractional Brownian motion, in dimension one, in Séminaire de
probabilitésXLI, Lecture Notes in Math.1934. Springer, Berlin (2008) 181–197.
[26]
Nualart, D. and Răşcanu, A., Differential equations driven
by fractional Brownian motion. Collect.
Math.53 (2002) 55–81.
Google Scholar
[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.
[28]
E.M. Stein, Singular integrals and
differentiability properties of functions, Princeton Mathematical
Series30. Princeton University Press, Princeton, N.J. (1970).
[29]
Sussmann, H., On the gap between
deterministic and stochastic ordinary differential equations.
Ann. Probab.6 (1978) 19–41.
Google Scholar
[30]
Tychonoff, A., Ein
Fixpunktsatz. Math. Ann.111 (1935)
767–776. Google Scholar
[31]
Young, L.C., An inequality of the Hölder type,
connected with Stieltjes integration. Acta
Math.67 (1936)
251–282. Google Scholar
[32]
M. Zähle, On the link between fractional and
stochastic calculus, in Stochastic dynamics, Bremen (1997), Springer, New
York (1999) 305–325.