Convergence in Dirichlet Law of Certain Stochastic Integrals

Christophe Chorro (Cermsem, Université Paris 1, France)

Abstract


Recently, Nicolas Bouleau has proposed an extension of the Donsker's invariance principle in the framework of Dirichlet forms. He proves that an erroneous random walk of i.i.d random variables converges in Dirichlet law toward the Ornstein-Uhlenbeck error structure on the Wiener space. The aim of this paper is to extend this result to some families of stochastic integrals.

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Pages: 1005-1025

Publication Date: July 21, 2005

DOI: 10.1214/EJP.v10-272

References

  • Alòs, Elisa; Mazet, Olivier; Nualart, David. Stochastic calculus with respect to Gaussian processes. Ann. Probab. 29 (2001), no. 2, 766--801. MR1849177
  • Bardina, Xavier; Jolis, Maria. Weak convergence to the multiple Stratonovich integral. Stochastic Process. Appl. 90 (2000), no. 2, 277--300. MR1794540
  • Billingsley, Patrick. Convergence of probability measures. Second edition. Wiley Series in Probability and Statistics: Probability and Statistics. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1999. x+277 pp. ISBN: 0-471-19745-9 MR1700749
  • Bouleau, Nicolas. Théorème de Donsker et formes de Dirichlet. (French) [Donsker's theorem and Dirichlet forms] Bull. Sci. Math. 129 (2005), no. 5, 369--380. MR2146156
  • Bouleau, Nicolas; Chorro, Christophe. Error structures and parameter estimation. C. R. Math. Acad. Sci. Paris 338 (2004), no. 4, 305--310. MR2076501
  • Bouleau, Nicolas. Error calculus for finance and physics: the language of Dirichlet forms. de Gruyter Expositions in Mathematics, 37. Walter de Gruyter & Co., Berlin, 2003. x+234 pp. ISBN: 3-11-018036-7 MR2079474
  • Bouleau, Nicolas; Hirsch, Francis. Dirichlet forms and analysis on Wiener space. de Gruyter Studies in Mathematics, 14. Walter de Gruyter & Co., Berlin, 1991. x+325 pp. ISBN: 3-11-012919-1 MR1133391
  • textscC. Chorro : phOn an extension of the central limit theorem to Dirichlet forms, Cahiers de la mse, 2004.80, Univ. Paris 1, 2004.
  • Delgado, Rosario; Jolis, Maria. Weak approximation for a class of Gaussian processes. J. Appl. Probab. 37 (2000), no. 2, 400--407. MR1780999
  • textscD. Dacunha-Castelle, M. Duflo : phProbabilités et statistiques 2, Masson, 1993.
  • Feyel, D.; de La Pradelle, A. Espaces de Sobolev Gaussiens. (French) [Gaussian Sobolev spaces] Ann. Inst. Fourier (Grenoble) 39 (1989), no. 4, 875--908. MR1036336
  • Fukushima, Masatoshi. Dirichlet forms and Markov processes. North-Holland Mathematical Library, 23. North-Holland Publishing Co., Amsterdam-New York; Kodansha, Ltd., Tokyo, 1980. x+196 pp. ISBN: 0-444-85421-5 MR0569058
  • Jacod, Jean; Shiryaev, Albert N. Limit theorems for stochastic processes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 288. Springer-Verlag, Berlin, 1987. xviii+601 pp. ISBN: 3-540-17882-1 MR0959133
  • Jakubowski, A.; Mémin, J.; Pagès, G. Convergence en loi des suites d'intégrales stochastiques sur l'espace ${\bf D}^ 1$ de Skorokhod. (French) [Convergence in law of sequences of stochastic integrals on the Skorokhod space ${\bf D}^ 1$] Probab. Theory Related Fields 81 (1989), no. 1, 111--137. MR0981569
  • textscO. Jonhson, R. Samworth : phCentral limit theorem and convergence to stable laws in Mallows distance, Preprint, 2004.
  • Kurtz, Thomas G.; Protter, Philip. Weak limit theorems for stochastic integrals and stochastic differential equations. Ann. Probab. 19 (1991), no. 3, 1035--1070. MR1112406
  • textscZ. Ma, M. Rddotmboxockner : phIntroduction to the theory of (non-symmetric) Dirichlet forms, Springer-Verlag, 1992.
  • Nualart, David. The Malliavin calculus and related topics. Probability and its Applications (New York). Springer-Verlag, New York, 1995. xii+266 pp. ISBN: 0-387-94432-X MR1344217
  • Nualart, David; Zakai, Moshe. Multiple Wiener-Itô integrals possessing a continuous extension. Probab. Theory Related Fields 85 (1990), no. 1, 131--145. MR1044305
  • van der Vaart, Aad W.; Wellner, Jon A. Weak convergence and empirical processes. With applications to statistics. Springer Series in Statistics. Springer-Verlag, New York, 1996. xvi+508 pp. ISBN: 0-387-94640-3 MR1385671
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