Perturbed linear rough differential equations

Laure Coutin[1]; Antoine Lejay[2]

  • [1] Institut de Mathématiques de Toulouse, F-31062 Toulouse Cedex 9, France.
  • [2] Université de Lorraine, Institut Élie Cartan, UMR 7502, Vandœuvre-lès-Nancy, F-54600, France CNRS, Institut Élie Cartan, UMR 7502, Vandœuvre-lès-Nancy, F-54600, France Inria, Villers-lès-Nancy, F-54600, France

Annales mathématiques Blaise Pascal (2014)

  • Volume: 21, Issue: 1, page 103-150
  • ISSN: 1259-1734

Abstract

top
We study linear rough differential equations and we solve perturbed linear rough differential equations using the Duhamel principle. These results provide us with a key technical point to study the regularity of the differential of the Itô map in a subsequent article. Also, the notion of linear rough differential equations leads to consider multiplicative functionals with values in Banach algebras more general than tensor algebras and to consider extensions of classical results such as the Magnus and the Chen-Strichartz formula.

How to cite

top

Coutin, Laure, and Lejay, Antoine. "Perturbed linear rough differential equations." Annales mathématiques Blaise Pascal 21.1 (2014): 103-150. <http://eudml.org/doc/275628>.

@article{Coutin2014,
abstract = {We study linear rough differential equations and we solve perturbed linear rough differential equations using the Duhamel principle. These results provide us with a key technical point to study the regularity of the differential of the Itô map in a subsequent article. Also, the notion of linear rough differential equations leads to consider multiplicative functionals with values in Banach algebras more general than tensor algebras and to consider extensions of classical results such as the Magnus and the Chen-Strichartz formula.},
affiliation = {Institut de Mathématiques de Toulouse, F-31062 Toulouse Cedex 9, France.; Université de Lorraine, Institut Élie Cartan, UMR 7502, Vandœuvre-lès-Nancy, F-54600, France CNRS, Institut Élie Cartan, UMR 7502, Vandœuvre-lès-Nancy, F-54600, France Inria, Villers-lès-Nancy, F-54600, France},
author = {Coutin, Laure, Lejay, Antoine},
journal = {Annales mathématiques Blaise Pascal},
keywords = {Rough paths; Rough differential equations; Banach algebra; Magnus formula Chen-Strichartz formula; perturbation formula; Duhamel’s principle; rough paths; rough differential equations; Banach algebras; Magnus formula; Chen-Strichartz formula; Duhamel's principle},
language = {eng},
month = {1},
number = {1},
pages = {103-150},
publisher = {Annales mathématiques Blaise Pascal},
title = {Perturbed linear rough differential equations},
url = {http://eudml.org/doc/275628},
volume = {21},
year = {2014},
}

TY - JOUR
AU - Coutin, Laure
AU - Lejay, Antoine
TI - Perturbed linear rough differential equations
JO - Annales mathématiques Blaise Pascal
DA - 2014/1//
PB - Annales mathématiques Blaise Pascal
VL - 21
IS - 1
SP - 103
EP - 150
AB - We study linear rough differential equations and we solve perturbed linear rough differential equations using the Duhamel principle. These results provide us with a key technical point to study the regularity of the differential of the Itô map in a subsequent article. Also, the notion of linear rough differential equations leads to consider multiplicative functionals with values in Banach algebras more general than tensor algebras and to consider extensions of classical results such as the Magnus and the Chen-Strichartz formula.
LA - eng
KW - Rough paths; Rough differential equations; Banach algebra; Magnus formula Chen-Strichartz formula; perturbation formula; Duhamel’s principle; rough paths; rough differential equations; Banach algebras; Magnus formula; Chen-Strichartz formula; Duhamel's principle
UR - http://eudml.org/doc/275628
ER -

References

top
  1. Shigeki Aida, Notes on Proofs of Continuity Theorem in Rough Path Analysis, (2006) 
  2. Ismaël Bailleul, Flows driven by rough paths, (2012) Zbl06503642
  3. Andrew Baker, Matrix groups. An introduction to Lie group theory, (2002), Springer-Verlag London Ltd., London Zbl1009.22001MR1869885
  4. Fabrice Baudoin, An introduction to the geometry of stochastic flows, (2004), Imperial College Press, London Zbl1085.60002MR2154760
  5. Fabrice Baudoin, Xuejing Zhang, Taylor expansion for the solution of a stochastic differential equation driven by fractional Brownian motions, Electron. J. Probab. 17 (2012) Zbl1252.60052MR2955043
  6. Gérard Ben Arous, Flots et séries de Taylor stochastiques, Probab. Theory Related Fields 81 (1989), 29-77 Zbl0639.60062MR981567
  7. S. Blanes, F. Casas, J. A. Oteo, J. Ros, The Magnus expansion and some of its applications, Phys. Rep. 470 (2009), 151-238 MR2494199
  8. A. Bonfiglioli, E. Lanconelli, F. Uguzzoni, Stratified Lie groups and potential theory for their sub-Laplacians, (2007), Springer, Berlin Zbl1128.43001MR2363343
  9. Andrea Bonfiglioli, Roberta Fulci, Topics in noncommutative algebra. The theorem of Campbell, Baker, Hausdorff and Dynkin, 2034 (2012), Springer, Heidelberg Zbl1231.17001MR2883818
  10. M. Caruana, P. K. Friz, H. Oberhauser, A (rough) pathwise approach to a class of non-linear stochastic partial differential equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 28 (2011), 27-46 Zbl1219.60061MR2765508
  11. Michael Caruana, Peter Friz, Partial differential equations driven by rough paths, J. Differential Equations 247 (2009), 140-173 Zbl1167.35386MR2510132
  12. Fabienne Castell, Asymptotic expansion of stochastic flows, Probab. Theory Related Fields 96 (1993), 225-239 Zbl0794.60054MR1227033
  13. Fabienne Castell, Jessica Gaines, The ordinary differential equation approach to asymptotically efficient schemes for solution of stochastic differential equations, Ann. Inst. H. Poincaré Probab. Statist. 32 (1996), 231-250 Zbl0851.60054MR1386220
  14. Kuo-Tsai Chen, Integration of paths, geometric invariants and a generalized Baker-Hausdorff formula, Ann. of Math. (2) 65 (1957), 163-178 Zbl0077.25301MR85251
  15. Kuo-Tsai Chen, Integration of paths—a faithful representation of paths by non-commutative formal power series, Trans. Amer. Math. Soc. 89 (1958), 395-407 Zbl0097.25803MR106258
  16. Kuo-Tsai Chen, Formal differential equations, Ann. of Math. (2) 73 (1961), 110-133 Zbl0098.05702MR150370
  17. Kuo-Tsai Chen, Expansion of solutions of differential systems, Arch. Rational Mech. Anal. 13 (1963), 348-363 Zbl0117.04802MR157032
  18. L. Coutin, Rough paths via sewing Lemma, ESAIM Probab. Stat. 16 (2012), 479-526 Zbl1277.47081
  19. L. Coutin, A. Lejay, Sensitivity of rough differential equations, (2013) Zbl1322.60084
  20. A.M. Davie, Differential Equations Driven by Rough Signals: an Approach via Discrete Approximation, Appl. Math. Res. Express. AMRX 2 (2007) Zbl1163.34005MR2387018
  21. A. Deya, M. Gubinelli, S. Tindel, Non-linear rough heat equations, Probab. Theory Related Fields 153 (2012), 97-147 Zbl1255.60106MR2925571
  22. Aurélien Deya, Samy Tindel, Rough Volterra equations. I. The algebraic integration setting, Stoch. Dyn. 9 (2009), 437-477 Zbl1181.60105MR2566910
  23. Aurélien Deya, Samy Tindel, Rough Volterra equations 2: Convolutional generalized integrals, Stochastic Process. Appl. 121 (2011), 1864-1899 Zbl1223.60031MR2811027
  24. Ronald G. Douglas, Banach algebra techniques in operator theory, 179 (1998), Springer-Verlag, New York Zbl0920.47001MR1634900
  25. F. J. Dyson, The radiation theories of Tomonaga, Schwinger, and Feynman, Physical Rev. (2) 75 (1949), 486-502 Zbl0032.23702MR28203
  26. Denis Feyel, Arnaud de La Pradelle, Curvilinear integrals along enriched paths, Electron. J. Probab. 11 (2006), no. 34, 860-892 (electronic) Zbl1110.60031MR2261056
  27. Denis Feyel, Arnaud de La Pradelle, Gabriel Mokobodzki, A non-commutative sewing lemma, Electron. Commun. Probab. 13 (2008), 24-34 Zbl1186.26009MR2372834
  28. Peter K. Friz, Nicolas B. Victoir, Multidimensional stochastic processes as rough paths. Theory and applications, 120 (2010), Cambridge University Press, Cambridge Zbl1193.60053MR2604669
  29. Massimiliano Gubinelli, Abstract integration, combinatorics of trees and differential equations, Combinatorics and physics 539 (2011), 135-151, Amer. Math. Soc., Providence, RI Zbl1225.35164MR2790306
  30. Massimiliano Gubinelli, Antoine Lejay, Samy Tindel, Young integrals and SPDEs, Potential Anal. 25 (2006), 307-326 Zbl1103.60062MR2255351
  31. Ernst Hairer, Christian Lubich, Gerhard Wanner, Geometric numerical integration. Structure-preserving algorithms for ordinary differential equations, 31 (2010), Springer, Heidelberg Zbl1228.65237MR2840298
  32. M. Hairer, D. Kelly, Geometric versus non-geometric rough paths, (2012) Zbl1314.60115
  33. Brian C. Hall, Lie groups, Lie algebras, and representations. An elementary introduction, 222 (2003), Springer-Verlag, New York Zbl1026.22001MR1997306
  34. Keisuke Hara, Masanori Hino, Fractional order Taylor’s series and the neo-classical inequality, Bull. Lond. Math. Soc. 42 (2010), 467-477 Zbl1194.26027MR2651942
  35. Antoine Lejay, An introduction to rough paths, Séminaire de Probabilités XXXVII 1832 (2003), 1-59, Springer, Berlin Zbl1041.60051MR2053040
  36. Antoine Lejay, On rough differential equations, Electron. J. Probab. 14 (2009), no. 12, 341-364 Zbl1190.60044MR2480544
  37. Antoine Lejay, Yet another introduction to rough paths, Séminaire de Probabilités XLII 1979 (2009), 1-101, Springer, Berlin Zbl1041.60051MR2599204
  38. Antoine Lejay, Controlled differential equations as Young integrals: a simple approach, J. Differential Equations 249 (2010), 1777-1798 Zbl1216.34058MR2679003
  39. Antoine Lejay, Global solutions to rough differential equations with unbounded vector fields, Séminaire de Probabilités XLIV 2046 (2012), 215-246, Springer, Heidelberg Zbl1254.60059MR2953350
  40. Antoine Lejay, Nicolas Victoir, On ( p , q ) -rough paths, J. Differential Equations 225 (2006), 103-133 Zbl1097.60048MR2228694
  41. Gabriel Lord, Simon J. A. Malham, Anke Wiese, Efficient strong integrators for linear stochastic systems, SIAM J. Numer. Anal. 46 (2008), 2892-2919 Zbl1179.60046MR2439496
  42. Terry Lyons, Zhongmin Qian, System control and rough paths, (2002), Oxford University Press, Oxford Zbl1029.93001MR2036784
  43. Terry J. Lyons, Differential equations driven by rough signals, Rev. Mat. Iberoamericana 14 (1998), 215-310 Zbl0923.34056MR1654527
  44. Terry J. Lyons, Michael Caruana, Thierry Lévy, Differential equations driven by rough paths (Lectures from the 34th Summer School on Probability Theory held in Saint-Flour, July 6–24, 2004), 1908 (2007), Springer, Berlin Zbl1176.60002MR2314753
  45. Terry J. Lyons, Nadia Sidorova, On the radius of convergence of the logarithmic signature, Illinois J. Math. 50 (2006), 763-790 (electronic) Zbl1103.60060MR2247845
  46. Terry J. Lyons, Weijun Xu, A uniform estimate for rough paths, Bull. Sci. Math. 137 (2013), 867-879 Zbl1296.60155MR3116217
  47. Wilhelm Magnus, On the exponential solution of differential equations for a linear operator, Comm. Pure Appl. Math. 7 (1954), 649-673 Zbl0056.34102MR67873
  48. Bogdan Mielnik, Jerzy Plebański, Combinatorial approach to Baker-Campbell-Hausdorff exponents, Ann. Inst. H. Poincaré Sect. A (N.S.) 12 (1970), 215-254 Zbl0206.13602MR273922
  49. Per Christian Moan, Jitse Niesen, Convergence of the Magnus series, Found. Comput. Math. 8 (2008), 291-301 Zbl1154.34307MR2413145
  50. Rimhak Ree, Lie elements and an algebra associated with shuffles, Ann. of Math. (2) 68 (1958), 210-220 Zbl0083.25401MR100011
  51. Christophe Reutenauer, Free Lie algebras, 7 (1993), The Clarendon Press Oxford University Press, New York Zbl0798.17001MR1231799
  52. Pocketbook of mathematical functions (Abridged edition of Handbook of mathematical functions edited by Milton Abramowitz and Irene A. Stegun), (1984), StegunIrene A.I. A., Thun Zbl0643.33002MR768931
  53. Robert S. Strichartz, The Campbell-Baker-Hausdorff-Dynkin formula and solutions of differential equations, J. Funct. Anal. 72 (1987), 320-345 Zbl0623.34058MR886816
  54. L. C. Young, An inequality of the Hölder type, connected with Stieltjes integration, Acta Math. 67 (1936), 251-282 Zbl0016.10404MR1555421

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.