Electronic Communications in Probability

Weak approximation of fractional SDEs: the Donsker setting

Xavier Bardina, Carles Rovira, and Samy Tindel

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Abstract

In this note, we take up the study of weak convergence for stochastic differential equations driven by a (Liouville) fractional Brownian motion $B$ with Hurst parameter $H\in(1/3,1/2)$, initiated in a paper of Bardina et al. . In the current paper, we approximate the $d$-dimensional fBm by the convolution of a rescaled random walk with Liouville's kernel. We then show that the corresponding differential equation converges in law to a fractional SDE driven by $B$.

Article information

Source
Electron. Commun. Probab., Volume 15 (2010), paper no. 30, 314-329.

Dates
Accepted: 23 July 2010
First available in Project Euclid: 6 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465243973

Digital Object Identifier
doi:10.1214/ECP.v15-1561

Mathematical Reviews number (MathSciNet)
MR2670198

Zentralblatt MATH identifier
1225.60091

Subjects
Primary: 60H10: Stochastic ordinary differential equations [See also 34F05]
Secondary: 60H05: Stochastic integrals

Keywords
Weak approximation Kac-Stroock type approximation fractional Brownian motion rough paths

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

Citation

Bardina, Xavier; Rovira, Carles; Tindel, Samy. Weak approximation of fractional SDEs: the Donsker setting. Electron. Commun. Probab. 15 (2010), paper no. 30, 314--329. doi:10.1214/ECP.v15-1561. https://projecteuclid.org/euclid.ecp/1465243973


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References

  • Alòs, Elisa; Mazet, Olivier; Nualart, David. Stochastic calculus with respect to fractional Brownian motion with Hurst parameter lesser than $frac 12$. Stochastic Process. Appl. 86 (2000), no. 1, 121–139.
  • Bardina, Xavier; Jolis, Maria; Quer-Sardanyons, Lluísl. Weak convergence for the stochasticheat equation driven by Gaussian white noise. Preprint.(2009)
  • Bardina, X.; Nourdin, I.; Rovira, C.; Tindel, S. Weak approximation of a fractional SDE. Stochastic Process. Appl. 120 (2010), no. 1, 39–65.
  • Breuillard, Emmanuel; Friz, Peter; Huesmann, Martin. From random walks to rough paths. Proc. Amer. Math. Soc. 137 (2009), no. 10, 3487–3496.
  • Delgado, Rosario; Jolis, Maria. Weak approximation for a class of Gaussian processes. J. Appl. Probab. 37 (2000), no. 2, 400–407.
  • Ethier, Stewart N.; Kurtz, Thomas G. Markov processes. Characterization and convergence. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & Sons, Inc., New York, 1986. x+534 pp. ISBN: 0-471-08186-8
  • Feyel, Denis; de La Pradelle, Arnaud. On fractional Brownian processes. Potential Anal. 10 (1999), no. 3, 273–288.
  • Friz, Peter K.; Victoir, Nicolas B. Differential equations driven by Gaussian signals (I). Ann. Inst. H. Poincaré Probab. Statist., to appear.
  • 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. xiv+656 pp. ISBN: 978-0-521-87607-0
  • Gubinelli, M. Controlling rough paths. J. Funct. Anal. 216 (2004), no. 1, 86–140.
  • Kac, Mark. A stochastic model related to the telegrapher's equation. Reprinting of an article published in 1956. Papers arising from a Conference on Stochastic Differential Equations (Univ. Alberta, Edmonton, Alta., 1972). Rocky Mountain J. Math. 4 (1974), 497–509.
  • Kurtz, Thomas G.; Protter, Philip E. Weak convergence of stochastic integrals and differential equations. Probabilistic models for nonlinear partial differential equations (Montecatini Terme, 1995), 1–41, Lecture Notes in Math., 1627, Springer, Berlin, 1996.
  • Lyons, Terry; Qian, Zhongmin. System control and rough paths. Oxford Mathematical Monographs. Oxford Science Publications. Oxford University Press, Oxford, 2002. x+216 pp. ISBN: 0-19-850648-1
  • Sottinen, Tommi. Fractional Brownian motion, random walks and binary market models. Finance Stoch. 5 (2001), no. 3, 343–355.
  • Stroock, Daniel W. Lectures on topics in stochastic differential equations. With notes by Satyajit Karmakar. Tata Institute of Fundamental Research Lectures on Mathematics and Physics, 68. Tata Institute of Fundamental Research, Bombay; by Springer-Verlag, Berlin-New York, 1982. iii+93 pp. ISBN: 3-540-11549-8
  • Tessitore, Gianmario; Zabczyk, Jerzy. Wong-Zakai approximations of stochastic evolution equations. J. Evol. Equ. 6 (2006), no. 4, 621–655.
  • Wong, Eugene; Zakai, Moshe. On the relation between ordinary and stochastic differential equations. Internat. J. Engrg. Sci. 3 1965 213–229.