With cedram.org
Annales Mathématiques
Blaise Pascal
Search for an article
Search within the site
Table of contents for this issue | Previous article
Laure Coutin; Antoine Lejay
Perturbed linear rough differential equations
(Équations différentielles linéaires rugueuses perturbées)
Annales mathématiques Blaise Pascal, 21 no. 1 (2014), p. 103-150, doi: 10.5802/ambp.338
Article PDF | Reviews MR 3248224 | Zbl 06329059
Class. Math.: 34A25, 60H10
Keywords: Rough paths, Rough differential equations, Banach algebra, Magnus formula Chen-Strichartz formula, perturbation formula, Duhamel’s principle

Résumé - 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.

Bibliography

[1] Shigeki Aida. Notes on Proofs of Continuity Theorem in Rough Path Analysis. Unpublished note, Osaka University
[2] Ismaël Bailleul. Flows driven by rough paths. Preprint arxiv:1203.0888
[3] Andrew Baker. Matrix groups. An introduction to Lie group theory. Springer-Verlag London Ltd., 2002 Article |  MR 1869885 |  Zbl 1009.22001
[4] Fabrice Baudoin. An introduction to the geometry of stochastic flows. Imperial College Press, 2004 Article |  MR 2154760 |  Zbl 1085.60002
[5] Fabrice Baudoin and Xuejing Zhang. Taylor expansion for the solution of a stochastic differential equation driven by fractional Brownian motions. Electron. J. Probab., 17, 2012. Article |  MR 2955043 |  Zbl 1252.60052
[6] Gérard Ben Arous. Flots et séries de Taylor stochastiques. Probab. Theory Related Fields, 81(1):29-77, 1989. Article |  MR 981567 |  Zbl 0639.60062
[7] S. Blanes, F. Casas, J. A. Oteo and J. Ros. The Magnus expansion and some of its applications. Phys. Rep., 470(5-6):151-238, 2009. Article |  MR 2494199
[8] A. Bonfiglioli, E. Lanconelli and F. Uguzzoni. Stratified Lie groups and potential theory for their sub-Laplacians. Springer, 2007  MR 2363343 |  Zbl 1128.43001
[9] Andrea Bonfiglioli and Roberta Fulci. Topics in noncommutative algebra. The theorem of Campbell, Baker, Hausdorff and Dynkin, volume 2034 of Lecture Notes in Mathematics. Springer, 2012 Article |  MR 2883818 |  Zbl 1231.17001
[10] M. Caruana, P. K. Friz and H. Oberhauser. A (rough) pathwise approach to a class of non-linear stochastic partial differential equations. Ann. Inst. H. Poincaré Anal. Non Linéaire, 28(1):27-46, 2011. Article |  MR 2765508 |  Zbl 1219.60061
[11] Michael Caruana and Peter Friz. Partial differential equations driven by rough paths. J. Differential Equations, 247(1):140-173, 2009. Article |  MR 2510132 |  Zbl 1167.35386
[12] Fabienne Castell. Asymptotic expansion of stochastic flows. Probab. Theory Related Fields, 96(2):225-239, 1993. Article |  MR 1227033 |  Zbl 0794.60054
[13] Fabienne Castell and Jessica Gaines. The ordinary differential equation approach to asymptotically efficient schemes for solution of stochastic differential equations. Ann. Inst. H. Poincaré Probab. Statist., 32(2):231-250, 1996. Numdam |  MR 1386220 |  Zbl 0851.60054
[14] Kuo-Tsai Chen. Integration of paths, geometric invariants and a generalized Baker-Hausdorff formula. Ann. of Math. (2), 65:163-178, 1957. Article |  MR 85251 |  Zbl 0077.25301
[15] Kuo-Tsai Chen. Integration of paths—a faithful representation of paths by non-commutative formal power series. Trans. Amer. Math. Soc., 89:395-407, 1958.  MR 106258 |  Zbl 0097.25803
[16] Kuo-Tsai Chen. Formal differential equations. Ann. of Math. (2), 73:110-133, 1961. Article |  MR 150370 |  Zbl 0098.05702
[17] Kuo-Tsai Chen. Expansion of solutions of differential systems. Arch. Rational Mech. Anal., 13:348-363, 1963. Article |  MR 157032 |  Zbl 0117.04802
[18] L. Coutin. Rough paths via sewing Lemma. ESAIM Probab. Stat., 16:479-526, 2012. Article |  Zbl 1277.47081
[19] L. Coutin and A. Lejay. Sensitivity of rough differential equations. Preprint
[20] A.M. Davie. Differential Equations Driven by Rough Signals: an Approach via Discrete Approximation. Appl. Math. Res. Express. AMRX, 2, 2007.  MR 2387018 |  Zbl 1163.34005
[21] A. Deya, M. Gubinelli and S. Tindel. Non-linear rough heat equations. Probab. Theory Related Fields, 153(1-2):97-147, 2012. Article |  MR 2925571 |  Zbl 1255.60106
[22] Aurélien Deya and Samy Tindel. Rough Volterra equations. I. The algebraic integration setting. Stoch. Dyn., 9(3):437-477, 2009. Article |  MR 2566910 |  Zbl 1181.60105
[23] Aurélien Deya and Samy Tindel. Rough Volterra equations 2: Convolutional generalized integrals. Stochastic Process. Appl., 121(8):1864-1899, 2011. Article |  MR 2811027 |  Zbl 1223.60031
[24] Ronald G. Douglas. Banach algebra techniques in operator theory, volume 179 of Graduate Texts in Mathematics. Springer-Verlag, 1998 Article |  MR 1634900 |  Zbl 0920.47001
[25] F. J. Dyson. The radiation theories of Tomonaga, Schwinger, and Feynman. Physical Rev. (2), 75:486-502, 1949. Article |  MR 28203 |  Zbl 0032.23702
[26] Denis Feyel and Arnaud de La Pradelle. Curvilinear integrals along enriched paths. Electron. J. Probab., 11:no. 34, 860-892 (electronic), 2006. Article |  MR 2261056 |  Zbl 1110.60031
[27] Denis Feyel, Arnaud de La Pradelle and Gabriel Mokobodzki. A non-commutative sewing lemma. Electron. Commun. Probab., 13:24-34, 2008. Article |  MR 2372834 |  Zbl 1186.26009
[28] Peter K. Friz and Nicolas B. Victoir. Multidimensional stochastic processes as rough paths. Theory and applications, volume 120 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, 2010  MR 2604669 |  Zbl 1193.60053
[29] Massimiliano Gubinelli, Abstract integration, combinatorics of trees and differential equations, Combinatorics and physics, Contemp. Math. 539, Amer. Math. Soc., 2011, p. 135–151 Article |  MR 2790306 |  Zbl 1225.35164
[30] Massimiliano Gubinelli, Antoine Lejay and Samy Tindel. Young integrals and SPDEs. Potential Anal., 25(4):307-326, 2006. Article |  MR 2255351 |  Zbl 1103.60062
[31] Ernst Hairer, Christian Lubich and Gerhard Wanner. Geometric numerical integration. Structure-preserving algorithms for ordinary differential equations, volume 31 of Springer Series in Computational Mathematics. Springer, 2010  MR 2840298 |  Zbl 1228.65237
[32] M. Hairer and D. Kelly. Geometric versus non-geometric rough paths. Preprint arxiv:1210.9294
[33] Brian C. Hall. Lie groups, Lie algebras, and representations. An elementary introduction, volume 222 of Graduate Texts in Mathematics. Springer-Verlag, 2003  MR 1997306 |  Zbl 1026.22001
[34] Keisuke Hara and Masanori Hino. Fractional order Taylor’s series and the neo-classical inequality. Bull. Lond. Math. Soc., 42(3):467-477, 2010. Article |  MR 2651942 |  Zbl 1194.26027
[35] Antoine Lejay, An introduction to rough paths, Séminaire de Probabilités XXXVII, Lecture Notes in Math. 1832, Springer, 2003, p. 1–59 Article |  MR 2053040 |  Zbl 1041.60051
[36] Antoine Lejay. On rough differential equations. Electron. J. Probab., 14:no. 12, 341-364, 2009. Article |  MR 2480544 |  Zbl 1190.60044
[37] Antoine Lejay, Yet another introduction to rough paths, Séminaire de Probabilités XLII, Lecture Notes in Math. 1979, Springer, 2009, p. 1–101 Article |  MR 2599204 |  Zbl 1041.60051
[38] Antoine Lejay. Controlled differential equations as Young integrals: a simple approach. J. Differential Equations, 249(8):1777-1798, 2010. Article |  MR 2679003 |  Zbl 1216.34058
[39] Antoine Lejay, Global solutions to rough differential equations with unbounded vector fields, Séminaire de Probabilités XLIV, Lecture Notes in Math. 2046, Springer, 2012, p. 215–246 Article |  MR 2953350 |  Zbl 1254.60059
[40] Antoine Lejay and Nicolas Victoir. On $(p,q)$-rough paths. J. Differential Equations, 225(1):103-133, 2006. Article |  MR 2228694 |  Zbl 1097.60048
[41] Gabriel Lord, Simon J. A. Malham and Anke Wiese. Efficient strong integrators for linear stochastic systems. SIAM J. Numer. Anal., 46(6):2892-2919, 2008. Article |  MR 2439496 |  Zbl 1179.60046
[42] Terry Lyons and Zhongmin Qian. System control and rough paths. Oxford University Press, 2002 Article |  MR 2036784 |  Zbl 1029.93001
[43] Terry J. Lyons. Differential equations driven by rough signals. Rev. Mat. Iberoamericana, 14(2):215-310, 1998. Article |  MR 1654527 |  Zbl 0923.34056
[44] Terry J. Lyons, Michael Caruana and 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), volume 1908 of Lecture Notes in Mathematics. Springer, 2007  MR 2314753 |  Zbl 1176.60002
[45] Terry J. Lyons and Nadia Sidorova. On the radius of convergence of the logarithmic signature. Illinois J. Math., 50(1-4):763-790 (electronic), 2006.  MR 2247845 |  Zbl 1103.60060
[46] Terry J. Lyons and Weijun Xu. A uniform estimate for rough paths. Bull. Sci. Math., 137(7):867-879, 2013. Article |  MR 3116217 |  Zbl pre06243440
[47] Wilhelm Magnus. On the exponential solution of differential equations for a linear operator. Comm. Pure Appl. Math., 7:649-673, 1954. Article |  MR 67873 |  Zbl 0056.34102
[48] Bogdan Mielnik and Jerzy Plebański. Combinatorial approach to Baker-Campbell-Hausdorff exponents. Ann. Inst. H. Poincaré Sect. A (N.S.), 12:215-254, 1970. Numdam |  MR 273922 |  Zbl 0206.13602
[49] Per Christian Moan and Jitse Niesen. Convergence of the Magnus series. Found. Comput. Math., 8(3):291-301, 2008. Article |  MR 2413145 |  Zbl 1154.34307
[50] Rimhak Ree. Lie elements and an algebra associated with shuffles. Ann. of Math. (2), 68:210-220, 1958. Article |  MR 100011 |  Zbl 0083.25401
[51] Christophe Reutenauer. Free Lie algebras, volume 7 of London Mathematical Society Monographs. New Series. The Clarendon Press Oxford University Press, 1993  MR 1231799 |  Zbl 0798.17001
[52] . Pocketbook of mathematical functions (Abridged edition of Handbook of mathematical functions edited by Milton Abramowitz and Irene A. Stegun). Verlag Harri Deutsch, 1984  MR 768931
[53] Robert S. Strichartz. The Campbell-Baker-Hausdorff-Dynkin formula and solutions of differential equations. J. Funct. Anal., 72(2):320-345, 1987. Article |  MR 886816 |  Zbl 0623.34058
[54] L. C. Young. An inequality of the Hölder type, connected with Stieltjes integration. Acta Math., 67(1):251-282, 1936. Article |  MR 1555421 |  Zbl 0016.10404