Electronic Journal of Probability

On the Convergence of Stochastic Integrals Driven by Processes Converging on account of a Homogenization Property

Antoine Lejay

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Abstract

We study the limit of functionals of stochastic processes for which an homogenization result holds. All these functionals involve stochastic integrals. Among them, we consider more particularly the Levy area and those giving the solutions of some SDEs. The main question is to know whether or not the limit of the stochastic integrals is equal to the stochastic integral of the limit of each of its terms. In fact, the answer may be negative, especially in presence of a highly oscillating first-order differential term. This provides us some counterexamples to the theory of good sequence of semimartingales.

Article information

Source
Electron. J. Probab., Volume 7 (2002), paper no. 18, 18 pp.

Dates
Accepted: 19 September 2002
First available in Project Euclid: 16 May 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1463434891

Digital Object Identifier
doi:10.1214/EJP.v7-117

Mathematical Reviews number (MathSciNet)
MR1943891

Zentralblatt MATH identifier
1007.60018

Subjects
Primary: 60F17: Functional limit theorems; invariance principles
Secondary: 60K40: Other physical applications of random processes

Keywords
stochastic differential equations good sequence of semimartingales conditions UT and UCV Lévy area

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Lejay, Antoine. On the Convergence of Stochastic Integrals Driven by Processes Converging on account of a Homogenization Property. Electron. J. Probab. 7 (2002), paper no. 18, 18 pp. doi:10.1214/EJP.v7-117. https://projecteuclid.org/euclid.ejp/1463434891


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References

  • A. Bensoussan, J.L. Lions, and G. Papanicolaou. Asymptotic Analysis for Periodic Structures. North-Holland, 1978.
  • S.N. Ethier and T.G. Kurtz. Markov Processes, Characterization and Convergence. Wiley, 1986.
  • J. Jacod and A.N. Shiryaev. Limit Theorems for Stochastic Processes. Springer-Verlag, 1987.
  • I. Karatzas and S.E. Shreve. Brownian Motion and Stochastic Calculus. Springer-Verlag, 2 edition, 1991.
  • T.G. Kurtz and P. Protter. Weak convergence of stochastic integrals and differential equations. In D. Talay and L. Tubaro, editors, Probabilistic Models for Nonlinear Partial Differential Equations, Montecatini Terme, 1995, vol. 1629 of Lecture Notes in Mathematics, pp. 1-41. Springer-Verlag, 1996.
  • J. L. Lebowitz and H. Rost. The Einstein relation for the displacement of a test particle in a random environment. Stochastic Process. Appl., 54:183-196, 1994.
  • A. Lejay. Homogenization of divergence-form operators with lower-order terms in random media. Probab. Theory Related Fields, 120(2):255-276, 2001. DOI:10.1007/s004400100135.
  • A. Lejay. A probabilistic approach of the homogenization of divergence-form operators in periodic media. Asymptot. Anal., 28(2):151-162, 2001.
  • A. Lejay. An introduction to rough paths. In Séminaire de Probabilités XXXVII, Lecture Notes in Mathematics. Springer-Verlag, 2003. To appear.
  • A. Lejay and T.J. Lyons. On the importance of the Lévy area for systems controlled by converging stochastic processes. Application to homogenization. In preparation, 2002.
  • T.J. Lyons. Differential equations driven by rough signals. Rev. Mat. Iberoamericana, 14(2):215-310, 1998.
  • T. Lyons and Z. Qian. System Control and Rough Paths. Oxford Mathematical Monographs. Oxford University Press, 2002.
  • S. Olla. Homogenization of diffusion processes in random fields. Cours de l'École doctorale de l'École Polytechnique (Palaiseau, France), 1994. URL: http://www.cmap.polytechnique.fr/~olla/pubolla.html
  • É. Pardoux. Homogenization of linear and semilinear second order PDEs with periodic coefficients: a probabilistic approach. J. Funct. Anal., 167(2):498-520, 1999. DOI:10.1006/jfan.1999.3441.
  • É. Pardoux and A.Y. Veretennikov. Averaging of backward SDEs, with application to semi-linear PDEs. Stochastics, 60(3-4):255-270, 1997.
  • É. Pardoux and A.Y. Veretennikov. On Poisson equation and diffusion approximation I. Ann. Probab., 29(3):1061-1085, 2001.
  • É. Pardoux and A.Y. Veretennikov. On poisson equation and diffusion approximation II. To appear in Ann. Probab., 2002.
  • R.G. Pinsky. Second order elliptic operators with periodic coefficients: criticality theory, perturbations, and positive harmonic functions. J. Funct. Anal., 129(1):80-107, 1995.
  • R.G. Pinsky. Positive Harmonic Functions and Diffusion. Cambridge University Press, 1996.
  • I. Szyszkowski. Weak convergence of stochastic integrals. Theory of Probab. App., 41(4):810-814, 1997.