Dans cet article, nous considérons des équations différentielles conduites par des trajectoires rugueuses non-géométriques en utilisant le concept de trajectoire rugueuse ramifiée introduit dans (J. Differential Equations 248 (2010) 693–721). Nous montrons d’abord que celles-ci peuvent être définies de manière équivalente comme une fonction -Hölderienne à valeurs dans un certain groupe de Lie, comme c’est le cas pour les trajectoires rugueuses dites « géométriques » . Nous montrons ensuite que toute trajectoire rugueuse ramifiée peut être encodée par une trajectoire rugueuse géométrique. Plus précisément, pour toute trajectoire rugueuse ramifiée définie au-dessus d’une trajectoire , il existe une trajectoire rugueuse géométrique définie au-dessus d’une trajectoire étendue , de manière à ce que contienne toute l’information de . Il en suit que toute équation différentielle conduite par peut être reformulée comme une équation différentielle modifiée conduite par . On peut interpréter ceci comme une généralisation de la formule de correction Itô–Stratonovich.
In this article we consider rough differential equations (RDEs) driven by non-geometric rough paths, using the concept of branched rough paths introduced in (J. Differential Equations 248 (2010) 693–721). We first show that branched rough paths can equivalently be defined as -Hölder continuous paths in some Lie group, akin to geometric rough paths. We then show that every branched rough path can be encoded in a geometric rough path. More precisely, for every branched rough path lying above a path , there exists a geometric rough path lying above an extended path , such that contains all the information of . As a corollary of this result, we show that every RDE driven by a non-geometric rough path can be rewritten as an extended RDE driven by a geometric rough path . One could think of this as a generalisation of the Itô–Stratonovich correction formula.
Mots clés : rough paths, Hopf algebra, integration
@article{AIHPB_2015__51_1_207_0, author = {Hairer, Martin and Kelly, David}, title = {Geometric versus non-geometric rough paths}, journal = {Annales de l'I.H.P. Probabilit\'es et statistiques}, pages = {207--251}, publisher = {Gauthier-Villars}, volume = {51}, number = {1}, year = {2015}, doi = {10.1214/13-AIHP564}, mrnumber = {3300969}, zbl = {06412903}, language = {en}, url = {http://www.numdam.org/articles/10.1214/13-AIHP564/} }
TY - JOUR AU - Hairer, Martin AU - Kelly, David TI - Geometric versus non-geometric rough paths JO - Annales de l'I.H.P. Probabilités et statistiques PY - 2015 SP - 207 EP - 251 VL - 51 IS - 1 PB - Gauthier-Villars UR - http://www.numdam.org/articles/10.1214/13-AIHP564/ DO - 10.1214/13-AIHP564 LA - en ID - AIHPB_2015__51_1_207_0 ER -
%0 Journal Article %A Hairer, Martin %A Kelly, David %T Geometric versus non-geometric rough paths %J Annales de l'I.H.P. Probabilités et statistiques %D 2015 %P 207-251 %V 51 %N 1 %I Gauthier-Villars %U http://www.numdam.org/articles/10.1214/13-AIHP564/ %R 10.1214/13-AIHP564 %G en %F AIHPB_2015__51_1_207_0
Hairer, Martin; Kelly, David. Geometric versus non-geometric rough paths. Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 1, pp. 207-251. doi : 10.1214/13-AIHP564. http://www.numdam.org/articles/10.1214/13-AIHP564/
[1] Hopf Algebras. Cambridge Tracts in Mathematics 74. Cambridge Univ. Press, Cambridge, 1980. Translated from the Japanese by Hisae Kinoshita and Hiroko Tanaka. | Zbl
.[2] Trees, renormalization and differential equations. BIT 44 (2004) 425–438. | DOI | MR | Zbl
.[3] Itô formula for an asymptotically -stable process. Ann. Appl. Probab. 6 (1996) 200–217. | DOI | MR | Zbl
and .[4] A change of variable formula with Itô correction term. Ann. Probab. 38 (2010) 1817–1869. | DOI | MR | Zbl
and .[5] An algebraic theory of integration methods. Math. Comp. 26 (1972) 79–106. | DOI | MR | Zbl
.[6] Smoothness of the density for solutions to Gaussian Rough Differential Equations, 2012. | Zbl
, , and .[7] Algebraic structures of B-series. Found. Comput. Math. 10 (2010) 407–427. | DOI | MR | Zbl
, and .[8] Iterated path integrals. Bull. Amer. Math. Soc. 83 (1977) 831–879. | DOI | MR | Zbl
.[9] Hopf algebras, renormalization and noncommutative geometry. Comm. Math. Phys. 199 (1998) 203–242. | DOI | MR | Zbl
and .[10] Stochastic analysis, rough path analysis and fractional Brownian motions. Probab. Theory Related Fields 122 (2002) 108–140. | DOI | MR | Zbl
and .[11] Hopf Algebras: An Introduction. Monographs and Textbooks in Pure and Applied Mathematics 235. Marcel Dekker, New York, 2001. | MR | Zbl
, and .[12] Differential equations driven by rough paths: An approach via discrete approximation. Appl. Math. Res. Express. 2 (2007). | MR | Zbl
.[13] -covariation, generalized Dirichlet processes and calculus with respect to finite cubic variation processes. Stochastic Process. Appl. 104 (2003) 259–299. | DOI | MR | Zbl
and .[14] An introduction to Hopf algebras of trees. Preprint, 2013.
.[15] Hardy Spaces on Homogeneous Groups. Mathematical Notes 28. Princeton Univ. Press, Princeton, NJ, 1982. | MR | Zbl
and .[16] A note on the notion of geometric rough paths. Probab. Theory Related Fields 136 (2006) 395–416. | DOI | MR | Zbl
and .[17] Differential equations driven by Gaussian signals. Ann. Inst. Henri Poincaré Probab. Stat. 46 (2010) 369–413. | DOI | Numdam | MR | Zbl
and .[18] Multidimensional Stochastic Processes as Rough Paths. Theory and Applications. Cambridge Studies in Advanced Mathematics 120. Cambridge Univ. Press, Cambridge, 2010. | DOI | MR | Zbl
and .[19] -order integrals and generalized Itô’s formula: The case of a fractional Brownian motion with any Hurst index. Ann. Inst. Henri Poincaré Probab. Stat. 41 (2005) 781–806. | DOI | Numdam | MR | Zbl
, , and .[20] Hopf-algebraic structure of families of trees. J. Algebra 126 (1989) 184–210. | DOI | MR | Zbl
and .[21] Controlling rough paths. J. Funct. Anal. 216 (2004) 86–140. | DOI | MR | Zbl
.[22] Ramification of rough paths. J. Differential Equations 248 (2010) 693–721. | DOI | MR | Zbl
.[23] On the Butcher group and general multi-value methods. Computing (Arch. Elektron. Rechnen) 13 (1974) 1–15. | MR | Zbl
and .[24] An Introduction to Lie Groups and Lie Algebras. Cambridge Studies in Advanced Mathematics 113. Cambridge Univ. Press, Cambridge, 2008. | MR | Zbl
[25] On -rough paths. J. Differential Equations 225 (2006) 103–133. | DOI | MR | Zbl
and .[26] Differential equations driven by rough signals. Rev. Mat. Iberoamericana 14 (1998) 215–310. | DOI | MR | Zbl
.[27] Differential Equations Driven by Rough Paths. Lecture Notes in Mathematics 1908. Springer, Berlin, 2007. Lectures from the 34th Summer School on Probability Theory held in Saint-Flour, July 6–24, 2004. With an introduction concerning the Summer School by Jean Picard. | MR | Zbl
, and .[28] An extension theorem to rough paths. Ann. Inst. H. Poincaré Anal. Non Linéaire 24 (2007) 835–847. | DOI | Numdam | MR | Zbl
and .[29] Hopf algebras, from basics to applications to renormalization. ArXiv Mathematics e-prints, 2004.
.[30] Free Lie Algebras. London Mathematical Society Monographs. New Series. Oxford Science Publications 7. The Clarendon Press, Oxford Univ. Press, New York, 1993. | MR | Zbl
.[31] Hopf Algebras. Mathematics Lecture Note Series. W. A. Benjamin, New York, 1969. | MR | Zbl
.[32] Levy area for the free Brownian motion: Existence and non-existence. J. Funct. Anal. 208 (2004) 107–121. | DOI | MR | Zbl
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