Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Geometric versus non-geometric rough paths

Martin Hairer and David Kelly

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

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.

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.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist. Volume 51, Number 1 (2015), 207-251.

Dates
First available in Project Euclid: 14 January 2015

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1421244404

Digital Object Identifier
doi:10.1214/13-AIHP564

Mathematical Reviews number (MathSciNet)
MR3300969

Zentralblatt MATH identifier
1314.60115

Subjects
Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 34K28: Numerical approximation of solutions 16T05: Hopf algebras and their applications [See also 16S40, 57T05]

Keywords
Rough paths Hopf algebra Integration

Citation

Hairer, Martin; Kelly, David. Geometric versus non-geometric rough paths. Ann. Inst. H. Poincaré Probab. Statist. 51 (2015), no. 1, 207--251. doi:10.1214/13-AIHP564. https://projecteuclid.org/euclid.aihp/1421244404


Export citation

References

  • [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.
  • [2] C. Brouder. Trees, renormalization and differential equations. BIT 44 (2004) 425–438.
  • [3] K. Burdzy and A. Mpolhkadrecki. Itô formula for an asymptotically $4$-stable process. Ann. Appl. Probab. 6 (1996) 200–217.
  • [4] K. Burdzy and J. Swanson. A change of variable formula with Itô correction term. Ann. Probab. 38 (2010) 1817–1869.
  • [5] J. C. Butcher. An algebraic theory of integration methods. Math. Comp. 26 (1972) 79–106.
  • [6] T. Cass, M. Hairer, C. Litterer and S. Tindel. Smoothness of the density for solutions to Gaussian Rough Differential Equations, 2012.
  • [7] P. Chartier, E. Hairer and G. Vilmart. Algebraic structures of B-series. Found. Comput. Math. 10 (2010) 407–427.
  • [8] K. T. Chen. Iterated path integrals. Bull. Amer. Math. Soc. 83 (1977) 831–879.
  • [9] A. Connes and D. Kreimer. Hopf algebras, renormalization and noncommutative geometry. Comm. Math. Phys. 199 (1998) 203–242.
  • [10] L. Coutin and Z. Qian. Stochastic analysis, rough path analysis and fractional Brownian motions. Probab. Theory Related Fields 122 (2002) 108–140.
  • [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.
  • [12] A. M. Davie. Differential equations driven by rough paths: An approach via discrete approximation. Appl. Math. Res. Express. 2 (2007).
  • [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.
  • [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.
  • [16] P. Friz and N. Victoir. A note on the notion of geometric rough paths. Probab. Theory Related Fields 136 (2006) 395–416.
  • [17] P. Friz and N. Victoir. Differential equations driven by Gaussian signals. Ann. Inst. Henri Poincaré Probab. Stat. 46 (2010) 369–413.
  • [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.
  • [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.
  • [20] R. Grossman and R. G. Larson. Hopf-algebraic structure of families of trees. J. Algebra 126 (1989) 184–210.
  • [21] M. Gubinelli. Controlling rough paths. J. Funct. Anal. 216 (2004) 86–140.
  • [22] M. Gubinelli. Ramification of rough paths. J. Differential Equations 248 (2010) 693–721.
  • [23] E. Hairer and G. Wanner. On the Butcher group and general multi-value methods. Computing (Arch. Elektron. Rechnen) 13 (1974) 1–15.
  • [24] A. Kirillov, Jr. An Introduction to Lie Groups and Lie Algebras. Cambridge Studies in Advanced Mathematics 113. Cambridge Univ. Press, Cambridge, 2008.
  • [25] A. Lejay and N. Victoir. On $(p,q)$-rough paths. J. Differential Equations 225 (2006) 103–133.
  • [26] T. Lyons. Differential equations driven by rough signals. Rev. Mat. Iberoamericana 14 (1998) 215–310.
  • [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.
  • [28] T. Lyons and N. Victoir. An extension theorem to rough paths. Ann. Inst. H. Poincaré Anal. Non Linéaire 24 (2007) 835–847.
  • [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.
  • [31] M. E. Sweedler. Hopf Algebras. Mathematics Lecture Note Series. W. A. Benjamin, New York, 1969.
  • [32] N. Victoir. Levy area for the free Brownian motion: Existence and non-existence. J. Funct. Anal. 208 (2004) 107–121.