Geometric versus non-geometric rough paths

Hairer, Martin ; Kelly, David

Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015), p. 207-251 / Harvested from Numdam

Résumé

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 $\gamma $-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 $\mathbf {X}$ définie au-dessus d’une trajectoire $X$, il existe une trajectoire rugueuse géométrique $\bar{\mathbf {X}}$ définie au-dessus d’une trajectoire étendue $\bar{X}$, de manière à ce que $\bar{\mathbf {X}}$ contienne toute l’information de $\mathbf {X}$. Il en suit que toute équation différentielle conduite par $\mathbf {X}$ peut être reformulée comme une équation différentielle modifiée conduite par $\bar{\mathbf {X}}$. On peut interpréter ceci comme une généralisation de la formule de correction Itô–Stratonovich.

Abstract

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 $\gamma $-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 $\mathbf {X}$ lying above a path $X$, there exists a geometric rough path $\bar{\mathbf {X}}$ lying above an extended path $\bar{X}$, such that $\bar{\mathbf {X}}$ contains all the information of $\mathbf {X}$. As a corollary of this result, we show that every RDE driven by a non-geometric rough path $\mathbf {X}$ can be rewritten as an extended RDE driven by a geometric rough path $\bar{\mathbf {X}}$. One could think of this as a generalisation of the Itô–Stratonovich correction formula.

Détail

Publié le : 2015-01-01

MR 3300969 | Zbl 06412903

DOI : https://doi.org/10.1214/13-AIHP564
Classification:  60H10,  34K28,  16T05

Bibliographie

[1] E. Abe. Hopf Algebras. Cambridge Tracts in Mathematics 74. Cambridge Univ. Press, Cambridge, 1980. Translated from the Japanese by Hisae Kinoshita and Hiroko Tanaka. | Zbl 0476.16008

[2] C. Brouder. Trees, renormalization and differential equations. BIT 44 (2004) 425–438. | MR 2106008 | Zbl 1072.16033

[3] K. Burdzy and A. Mpolhkadrecki. Itô formula for an asymptotically $4$-stable process. Ann. Appl. Probab. 6 (1996) 200–217. | MR 1389837 | Zbl 0856.60042

[4] K. Burdzy and J. Swanson. A change of variable formula with Itô correction term. Ann. Probab. 38 (2010) 1817–1869. | MR 2722787 | Zbl 1204.60044

[5] J. C. Butcher. An algebraic theory of integration methods. Math. Comp. 26 (1972) 79–106. | MR 305608 | Zbl 0258.65070

[6] T. Cass, M. Hairer, C. Litterer and S. Tindel. Smoothness of the density for solutions to Gaussian Rough Differential Equations, 2012. | Zbl 06383625

[7] P. Chartier, E. Hairer and G. Vilmart. Algebraic structures of B-series. Found. Comput. Math. 10 (2010) 407–427. | MR 2657947 | Zbl 1201.65124

[8] K. T. Chen. Iterated path integrals. Bull. Amer. Math. Soc. 83 (1977) 831–879. | MR 454968 | Zbl 0389.58001

[9] A. Connes and D. Kreimer. Hopf algebras, renormalization and noncommutative geometry. Comm. Math. Phys. 199 (1998) 203–242. | MR 1660199 | Zbl 0932.16038

[10] L. Coutin and Z. Qian. Stochastic analysis, rough path analysis and fractional Brownian motions. Probab. Theory Related Fields 122 (2002) 108–140. | MR 1883719 | Zbl 1047.60029

[11] S. Dăscălescu, C. Năstăsescu and S. Raianu. Hopf Algebras: An Introduction. Monographs and Textbooks in Pure and Applied Mathematics 235. Marcel Dekker, New York, 2001. | MR 1786197 | Zbl 0962.16026

[12] A. M. Davie. Differential equations driven by rough paths: An approach via discrete approximation. Appl. Math. Res. Express. 2 (2007). | MR 2387018 | Zbl 1163.34005

[13] M. Errami and F. Russo. $n$-covariation, generalized Dirichlet processes and calculus with respect to finite cubic variation processes. Stochastic Process. Appl. 104 (2003) 259–299. | MR 1961622 | Zbl 1075.60531

[14] L. Foissy. An introduction to Hopf algebras of trees. Preprint, 2013.

[15] G. B. Folland and E. M. Stein. Hardy Spaces on Homogeneous Groups. Mathematical Notes 28. Princeton Univ. Press, Princeton, NJ, 1982. | MR 657581 | Zbl 0508.42025

[16] P. Friz and N. Victoir. A note on the notion of geometric rough paths. Probab. Theory Related Fields 136 (2006) 395–416. | MR 2257130 | Zbl 1108.34052

[17] P. Friz and N. Victoir. Differential equations driven by Gaussian signals. Ann. Inst. Henri Poincaré Probab. Stat. 46 (2010) 369–413. | Numdam | MR 2667703 | Zbl 1202.60058

[18] P. Friz and N. Victoir. Multidimensional Stochastic Processes as Rough Paths. Theory and Applications. Cambridge Studies in Advanced Mathematics 120. Cambridge Univ. Press, Cambridge, 2010. | MR 2604669 | Zbl 1193.60053

[19] M. Gradinaru, I. Nourdin, F. Russo and P. Vallois. $m$-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. | Numdam | MR 2144234 | Zbl 1083.60045

[20] R. Grossman and R. G. Larson. Hopf-algebraic structure of families of trees. J. Algebra 126 (1989) 184–210. | MR 1023294 | Zbl 0717.16029

[21] M. Gubinelli. Controlling rough paths. J. Funct. Anal. 216 (2004) 86–140. | MR 2091358 | Zbl 1058.60037

[22] M. Gubinelli. Ramification of rough paths. J. Differential Equations 248 (2010) 693–721. | MR 2578445 | Zbl 05671636

[23] E. Hairer and G. Wanner. On the Butcher group and general multi-value methods. Computing (Arch. Elektron. Rechnen) 13 (1974) 1–15. | MR 403225 | Zbl 0293.65050

[24] A. Kirillov, Jr. An Introduction to Lie Groups and Lie Algebras. Cambridge Studies in Advanced Mathematics 113. Cambridge Univ. Press, Cambridge, 2008. | MR 2440737 | Zbl 1153.17001

[25] A. Lejay and N. Victoir. On $(p,q)$-rough paths. J. Differential Equations 225 (2006) 103–133. | MR 2228694 | Zbl 1097.60048

[26] T. Lyons. Differential equations driven by rough signals. Rev. Mat. Iberoamericana 14 (1998) 215–310. | MR 1654527 | Zbl 0923.34056

[27] T. Lyons, M. Caruana and T. Lévy. 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 2314753 | Zbl 1176.60002

[28] T. Lyons and N. Victoir. An extension theorem to rough paths. Ann. Inst. H. Poincaré Anal. Non Linéaire 24 (2007) 835–847. | Numdam | MR 2348055 | Zbl 1134.60047

[29] D. Manchon. Hopf algebras, from basics to applications to renormalization. ArXiv Mathematics e-prints, 2004.

[30] C. Reutenauer. Free Lie Algebras. London Mathematical Society Monographs. New Series. Oxford Science Publications 7. The Clarendon Press, Oxford Univ. Press, New York, 1993. | MR 1231799 | Zbl 0798.17001

[31] M. E. Sweedler. Hopf Algebras. Mathematics Lecture Note Series. W. A. Benjamin, New York, 1969. | MR 252485 | Zbl 0194.32901

[32] N. Victoir. Levy area for the free Brownian motion: Existence and non-existence. J. Funct. Anal. 208 (2004) 107–121. | MR 2034293 | Zbl 1062.46055