Differential equations driven by gaussian signals
Annales de l'I.H.P. Probabilités et statistiques, Volume 46 (2010) no. 2, pp. 369-413.

We consider multi-dimensional gaussian processes and give a new condition on the covariance, simple and sharp, for the existence of Lévy area(s). gaussian rough paths are constructed with a variety of weak and strong approximation results. Together with a new RKHS embedding, we obtain a powerful - yet conceptually simple - framework in which to analyze differential equations driven by gaussian signals in the rough paths sense.

Nous donnons une condition simple et optimale sur la covariance d'un processus gaussien pour que celui-ci puisse être associé naturellement à un rough path. Une fois ce processus construit, nous démontrons un principe de grandes déviations, un théorème du support, et plusieurs résultats d'approximations. Avec la théorie des rough paths de T. Lyons, nous obtenons ainsi un cadre puissant, bien que conceptuellement simple, dans lequel nous pouvons analyser les équations différentielles conduites par des signaux gaussiens dans le sens des rough paths.

DOI: 10.1214/09-AIHP202
Classification: 60G15, 60H99
Keywords: rough paths, gaussian processes
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     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {369--413},
     publisher = {Gauthier-Villars},
     volume = {46},
     number = {2},
     year = {2010},
     doi = {10.1214/09-AIHP202},
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Friz, Peter; Victoir, Nicolas. Differential equations driven by gaussian signals. Annales de l'I.H.P. Probabilités et statistiques, Volume 46 (2010) no. 2, pp. 369-413. doi : 10.1214/09-AIHP202. http://archive.numdam.org/articles/10.1214/09-AIHP202/

[1] S. Aida, S. Kusuoka and D. Stroock. On the support of Wiener functionals. In Asymptotic Problems in Probability Theory: Wiener Functionals and Asymptotics (Sanda/Kyoto, 1990) 3-34. Pitman Res. Notes Math. Ser. 284. Longman Sci. Tech., Harlow, 1993. | MR | Zbl

[2] P. Billingsley. Convergence of Probability Measures. Wiley, New York, 1968. | MR | Zbl

[3] C. Borell. Tail probabilities in Gauss space. In Vector Space Measures and Applications (Proc. Conf., Univ. Dublin, Dublin, 1977), II 73-82. Lecture Notes in Phys. 77. Springer, Berlin, 1978. | MR | Zbl

[4] C. Borell. On polynomial chaos and integrability. Probab. Math. Statist. 3 (1984) 191-203. | MR | Zbl

[5] C. Borell. On the Taylor series of a Wiener polynomial. In Seminar Notes on Multiple Stochastic Integration, Polynomial Chaos and Their Integration. Case Western Reserve University, Cleveland, 1984.

[6] T. Cass and P. Friz. Densities for rough differential equations under Hoermander's condition. Ann. of Math. Accepted, 2008. Available at http://pjm.math.berkeley.edu/scripts/coming.php?jpath=annals. | Zbl

[7] T. Cass, P. Friz and N. Victoir. Non-degeneracy of Wiener functionals arising from rough differential equations. Trans. Amer. Math. Soc. 361 (2009) 3359-3371. | MR | Zbl

[8] K.-T. Chen. Integration of paths, geometric invariants and a generalized Baker-Hausdorff formula. Ann. of Math. (2) 65 (1957) 163-178. | MR | Zbl

[9] L. Coutin, P. Friz and N. Victoir. Good rough path sequences and applications to anticipating stochastic calculus. Ann. Probab. 35 (2007) 1172-1193. | MR | Zbl

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

[11] L. Coutin and N. Victoir. Enhanced Gaussian processes and applications. Preprint, 2005. | Numdam | MR | Zbl

[12] J.-D. Deuschel and D. W. Stroock. Large Deviations. Pure and Applied Mathematics 137. Academic Press, Boston, MA, 1989. | MR | Zbl

[13] R. M. Dudley and R. Norvaiša. Differentiability of Six Operators on Nonsmooth Functions and p-variation. Lecture Notes in Mathematics 1703. Springer, Berlin, 1999. (With the collaboration of Jinghua Qian). | MR | Zbl

[14] D. Feyel and A. De La Pradelle. Curvilinear integrals along enriched paths. Electron. J. Probab. 11 (2006) 860-892. | MR | Zbl

[15] P. Friz and H. Oberhauser. Isoperimetry and rough path regularity. arXiv-preprint, 2007.

[16] P. Friz and N. Victoir. Approximations of the Brownian rough path with applications to stochastic analysis. Ann. Inst. H. Poincaré Probab. Statist. 41 (2005) 703-724. | Numdam | MR | Zbl

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

[18] P. Friz and N. Victoir. A variation embedding theorem and applications. J. Funct. Anal. 239 (2006) 631-637. | MR | Zbl

[19] P. Friz and N. Victoir. Large deviation principle for enhanced Gaussian processes. Ann. Inst. H. Poincaré Probab. Statist. 43 (2007) 775-785. | Zbl

[20] P. K. Friz. Continuity of the Itô-map for Hölder rough paths with applications to the support theorem in Hölder norm. In Probability and Partial Differential Equations in Modern Applied Mathematics 117-135. IMA Vol. Math. Appl. 140. Springer, New York, 2005. | MR | Zbl

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

[22] N. C. Jain and D. Monrad. Gaussian measures in Bp. Ann. Probab. 11 (1983) 46-57. | MR | Zbl

[23] M. Ledoux. Isoperimetry and Gaussian analysis. In Lectures on Probability Theory and Statistics (Saint-Flour, 1994) 165-294. Lecture Notes in Math. 1648. Springer, Berlin, 1996. | MR | Zbl

[24] M. Ledoux and M. Talagrand. Probability in Banach spaces. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] 23. Springer, Berlin, 1991. (Isoperimetry and processes.) | MR | Zbl

[25] A. Lejay. An introduction to rough paths. In Séminaire de Probabilités XXXVII 1-59. Lecture Notes in Math. 1832. Springer, Berlin, 2003. | MR | Zbl

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

[27] T. Lyons and Z. Qian. System Control and Rough Paths. Oxford Mathematical Monographs. Oxford Univ. Press, 2002. | MR | Zbl

[28] A. Millet and M. Sanz-Solé. Large deviations for rough paths of the fractional Brownian motion. Ann. Inst. H. Poincaré Probab. Statist. 42 (2006) 245-271. | Numdam | MR | Zbl

[29] A. Millet and M. Sanz-Sole. Approximation of rough paths of fractional Brownian motion. In Seminar on Stochastic Analysis, Random Fields and Application 275-303. Progress in Probability 59. Birkhäuser, Basel, 2008. | MR | Zbl

[30] J. Musielak and W. Orlicz. On generalized variations. I. Studia Math. 18 (1959) 11-41. | MR | Zbl

[31] J. Musielak and Z. Semadeni. Some classes of Banach spaces depending on a parameter. Studia Math. 20 (1961) 271-284. | MR | Zbl

[32] D. Nualart. The Malliavin Calculus and Related Topics. Probability and Its Applications. Springer, New York, 1995. | MR | Zbl

[33] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion, 3rd edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 293. Springer, Berlin, 1999. | MR | Zbl

[34] M. Schreiber. Fermeture en probabilité de certains sous-espaces d'un espace L2. Application aux chaos de Wiener. Z. Wahrsch. Verw. Gebiete 14 (1969/70) 36-48. | MR | Zbl

[35] D. W. Stroock and S. R. S. Varadhan. Multidimensional Diffusion Processes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 233. Springer, Berlin, 1979. | MR | Zbl

[36] N. Towghi. Multidimensional extension of L. C. Young's inequality. JIPAM. J. Inequal. Pure Appl. Math. 3 (2002), article 22 (electronic). | MR | Zbl

[37] L. C. Young. An inequality of Hölder type connected with Stieltjes integration. Acta Math. 67 (1936) 251-282. | MR | Zbl

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