Perturbed linear rough differential equations

Coutin, Laure; Lejay, Antoine

Annales mathématiques Blaise Pascal, Tome 21 (2014) no. 1, pp. 103-150.

Nous étudions les équations différentielles linéaires rugueuses et résolvons des équations linéaires rugueuses perturbées à l’aide du principe de Duhamel. Ces résultats donnent un argument technique pour étudier la différentiabilité de l’application d’Itô. La notion d’équation différentielle rugueuses nous condition à considérer des fonctionnelles multiplicatives à valeurs dans des algèbres de Banach plus générales que celle des algèbres tensorielles, ainsi que des extensions de résultats classiques tels que les formules de Magnus et Chen-Strichartz.

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.

Classification : 34A25, 60H10
Keywords: Rough paths, Rough differential equations, Banach algebra, Magnus formula Chen-Strichartz formula, perturbation formula, Duhamel’s principle
Mot clés : Trajectoires rugueuses, Équations différentielles rugueuses, algèbre de Banach, formule de Magnus, formule de Chen-Strichartz, formule de perturbation, principe de Duhamel

[1] Aida, Shigeki Notes on Proofs of Continuity Theorem in Rough Path Analysis, 2006 (Unpublished note, Osaka University)

[2] Bailleul, Ismaël Flows driven by rough paths, 2012 (Preprint arxiv:1203.0888)

[3] Baker, Andrew Matrix groups. An introduction to Lie group theory, Springer Undergraduate Mathematics Series, Springer-Verlag London Ltd., London, 2002, pp. xii+330 | DOI | MR | Zbl

[4] Baudoin, Fabrice An introduction to the geometry of stochastic flows, Imperial College Press, London, 2004, pp. x+140 | DOI | MR | Zbl

[5] Baudoin, Fabrice; Zhang, Xuejing Taylor expansion for the solution of a stochastic differential equation driven by fractional Brownian motions, Electron. J. Probab., Volume 17 (2012), pp. no. 51, 21 | DOI | MR | Zbl

[6] Ben Arous, Gérard Flots et séries de Taylor stochastiques, Probab. Theory Related Fields, Volume 81 (1989) no. 1, pp. 29-77 | DOI | MR | Zbl

[7] Blanes, S.; Casas, F.; Oteo, J. A.; Ros, J. The Magnus expansion and some of its applications, Phys. Rep., Volume 470 (2009) no. 5-6, pp. 151-238 | DOI | MR

[8] Bonfiglioli, A.; Lanconelli, E.; Uguzzoni, F. Stratified Lie groups and potential theory for their sub-Laplacians, Springer Monographs in Mathematics, Springer, Berlin, 2007, pp. xxvi+800 | MR | Zbl

[9] Bonfiglioli, Andrea; Fulci, Roberta Topics in noncommutative algebra. The theorem of Campbell, Baker, Hausdorff and Dynkin, Lecture Notes in Mathematics, 2034, Springer, Heidelberg, 2012, pp. xxii+539 | DOI | MR | Zbl

[10] Caruana, M.; Friz, P. K.; Oberhauser, H. A (rough) pathwise approach to a class of non-linear stochastic partial differential equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 28 (2011) no. 1, pp. 27-46 | DOI | Numdam | MR | Zbl

[11] Caruana, Michael; Friz, Peter Partial differential equations driven by rough paths, J. Differential Equations, Volume 247 (2009) no. 1, pp. 140-173 | DOI | MR | Zbl

[12] Castell, Fabienne Asymptotic expansion of stochastic flows, Probab. Theory Related Fields, Volume 96 (1993) no. 2, pp. 225-239 | DOI | MR | Zbl

[13] Castell, Fabienne; Gaines, Jessica The ordinary differential equation approach to asymptotically efficient schemes for solution of stochastic differential equations, Ann. Inst. H. Poincaré Probab. Statist., Volume 32 (1996) no. 2, pp. 231-250 | EuDML | Numdam | MR | Zbl

[14] Chen, Kuo-Tsai Integration of paths, geometric invariants and a generalized Baker-Hausdorff formula, Ann. of Math. (2), Volume 65 (1957), pp. 163-178 | DOI | MR | Zbl

[15] Chen, Kuo-Tsai Integration of paths—a faithful representation of paths by non-commutative formal power series, Trans. Amer. Math. Soc., Volume 89 (1958), pp. 395-407 | MR | Zbl

[16] Chen, Kuo-Tsai Formal differential equations, Ann. of Math. (2), Volume 73 (1961), pp. 110-133 | DOI | MR | Zbl

[17] Chen, Kuo-Tsai Expansion of solutions of differential systems, Arch. Rational Mech. Anal., Volume 13 (1963), pp. 348-363 | DOI | MR | Zbl

[18] Coutin, L. Rough paths via sewing Lemma, ESAIM Probab. Stat., Volume 16 (2012), pp. 479-526 | DOI | EuDML | Numdam | Zbl

[19] Coutin, L.; Lejay, A. Sensitivity of rough differential equations, 2013 (Preprint)

[20] Davie, A.M. Differential Equations Driven by Rough Signals: an Approach via Discrete Approximation, Appl. Math. Res. Express. AMRX, Volume 2 (2007), pp. Art. ID abm009 | MR | Zbl

[21] Deya, A.; Gubinelli, M.; Tindel, S. Non-linear rough heat equations, Probab. Theory Related Fields, Volume 153 (2012) no. 1-2, pp. 97-147 | DOI | MR | Zbl

[22] Deya, Aurélien; Tindel, Samy Rough Volterra equations. I. The algebraic integration setting, Stoch. Dyn., Volume 9 (2009) no. 3, pp. 437-477 | DOI | MR | Zbl

[23] Deya, Aurélien; Tindel, Samy Rough Volterra equations 2: Convolutional generalized integrals, Stochastic Process. Appl., Volume 121 (2011) no. 8, pp. 1864-1899 | DOI | MR | Zbl

[24] Douglas, Ronald G. Banach algebra techniques in operator theory, Graduate Texts in Mathematics, 179, Springer-Verlag, New York, 1998, pp. xvi+194 | DOI | MR | Zbl

[25] Dyson, F. J. The radiation theories of Tomonaga, Schwinger, and Feynman, Physical Rev. (2), Volume 75 (1949), pp. 486-502 | DOI | MR | Zbl

[26] Feyel, Denis; de La Pradelle, Arnaud Curvilinear integrals along enriched paths, Electron. J. Probab., Volume 11 (2006), p. no. 34, 860-892 (electronic) | DOI | EuDML | MR | Zbl

[27] Feyel, Denis; de La Pradelle, Arnaud; Mokobodzki, Gabriel A non-commutative sewing lemma, Electron. Commun. Probab., Volume 13 (2008), pp. 24-34 | DOI | EuDML | MR | Zbl

[28] Friz, Peter K.; Victoir, Nicolas B. Multidimensional stochastic processes as rough paths. Theory and applications, Cambridge Studies in Advanced Mathematics, 120, Cambridge University Press, Cambridge, 2010, pp. xiv+656 | MR | Zbl

[29] Gubinelli, Massimiliano Abstract integration, combinatorics of trees and differential equations, Combinatorics and physics (Contemp. Math.), Volume 539, Amer. Math. Soc., Providence, RI, 2011, pp. 135-151 | DOI | MR | Zbl

[30] Gubinelli, Massimiliano; Lejay, Antoine; Tindel, Samy Young integrals and SPDEs, Potential Anal., Volume 25 (2006) no. 4, pp. 307-326 | DOI | MR | Zbl

[31] Hairer, Ernst; Lubich, Christian; Wanner, Gerhard Geometric numerical integration. Structure-preserving algorithms for ordinary differential equations, Springer Series in Computational Mathematics, 31, Springer, Heidelberg, 2010, pp. xviii+644 Reprint of the second (2006) edition | MR | Zbl

[32] Hairer, M.; Kelly, D. Geometric versus non-geometric rough paths, 2012 (Preprint arxiv:1210.9294) | Zbl

[33] Hall, Brian C. Lie groups, Lie algebras, and representations. An elementary introduction, Graduate Texts in Mathematics, 222, Springer-Verlag, New York, 2003, pp. xiv+351 | MR | Zbl

[34] Hara, Keisuke; Hino, Masanori Fractional order Taylor’s series and the neo-classical inequality, Bull. Lond. Math. Soc., Volume 42 (2010) no. 3, pp. 467-477 | DOI | MR | Zbl

[35] Lejay, Antoine An introduction to rough paths, Séminaire de Probabilités XXXVII (Lecture Notes in Math.), Volume 1832, Springer, Berlin, 2003, pp. 1-59 | DOI | MR | Zbl

[36] Lejay, Antoine On rough differential equations, Electron. J. Probab., Volume 14 (2009), pp. no. 12, 341-364 | DOI | EuDML | MR | Zbl

[37] Lejay, Antoine Yet another introduction to rough paths, Séminaire de Probabilités XLII (Lecture Notes in Math.), Volume 1979, Springer, Berlin, 2009, pp. 1-101 | DOI | MR | Zbl

[38] Lejay, Antoine Controlled differential equations as Young integrals: a simple approach, J. Differential Equations, Volume 249 (2010) no. 8, pp. 1777-1798 | DOI | MR | Zbl

[39] Lejay, Antoine Global solutions to rough differential equations with unbounded vector fields, Séminaire de Probabilités XLIV (Lecture Notes in Math.), Volume 2046, Springer, Heidelberg, 2012, pp. 215-246 | DOI | MR | Zbl

[40] Lejay, Antoine; Victoir, Nicolas On $(p,q)$-rough paths, J. Differential Equations, Volume 225 (2006) no. 1, pp. 103-133 | DOI | MR | Zbl

[41] Lord, Gabriel; Malham, Simon J. A.; Wiese, Anke Efficient strong integrators for linear stochastic systems, SIAM J. Numer. Anal., Volume 46 (2008) no. 6, pp. 2892-2919 | DOI | MR | Zbl

[42] Lyons, Terry; Qian, Zhongmin System control and rough paths, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2002, pp. x+216 | DOI | MR | Zbl

[43] Lyons, Terry J. Differential equations driven by rough signals, Rev. Mat. Iberoamericana, Volume 14 (1998) no. 2, pp. 215-310 | DOI | EuDML | MR | Zbl

[44] Lyons, Terry J.; Caruana, Michael; Lévy, Thierry Differential equations driven by rough paths (Lectures from the 34th Summer School on Probability Theory held in Saint-Flour, July 6–24, 2004), Lecture Notes in Mathematics, 1908, Springer, Berlin, 2007, pp. xviii+109 | MR | Zbl

[45] Lyons, Terry J.; Sidorova, Nadia On the radius of convergence of the logarithmic signature, Illinois J. Math., Volume 50 (2006) no. 1-4, p. 763-790 (electronic) http://projecteuclid.org/getRecord?id=euclid.ijm/1258059491 | MR | Zbl

[46] Lyons, Terry J.; Xu, Weijun A uniform estimate for rough paths, Bull. Sci. Math., Volume 137 (2013) no. 7, pp. 867-879 | DOI | MR | Zbl

[47] Magnus, Wilhelm On the exponential solution of differential equations for a linear operator, Comm. Pure Appl. Math., Volume 7 (1954), pp. 649-673 | DOI | MR | Zbl

[48] Mielnik, Bogdan; Plebański, Jerzy Combinatorial approach to Baker-Campbell-Hausdorff exponents, Ann. Inst. H. Poincaré Sect. A (N.S.), Volume 12 (1970), pp. 215-254 | EuDML | Numdam | MR | Zbl

[49] Moan, Per Christian; Niesen, Jitse Convergence of the Magnus series, Found. Comput. Math., Volume 8 (2008) no. 3, pp. 291-301 | DOI | MR | Zbl

[50] Ree, Rimhak Lie elements and an algebra associated with shuffles, Ann. of Math. (2), Volume 68 (1958), pp. 210-220 | DOI | MR | Zbl

[51] Reutenauer, Christophe Free Lie algebras, London Mathematical Society Monographs. New Series, 7, The Clarendon Press Oxford University Press, New York, 1993, pp. xviii+269 | MR | Zbl

[52] Stegun, Irene A. Pocketbook of mathematical functions (Abridged edition of Handbook of mathematical functions edited by Milton Abramowitz and Irene A. Stegun), Verlag Harri Deutsch, Thun, 1984, pp. 468 | MR | Zbl

[53] Strichartz, Robert S. The Campbell-Baker-Hausdorff-Dynkin formula and solutions of differential equations, J. Funct. Anal., Volume 72 (1987) no. 2, pp. 320-345 | DOI | MR | Zbl

[54] Young, L. C. An inequality of the Hölder type, connected with Stieltjes integration, Acta Math., Volume 67 (1936) no. 1, pp. 251-282 | DOI | JFM | MR | Zbl