MR2565851). 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 $B$.">
Home | Contents | Submissions, editors, etc. | Login | Search | EJP
 Electronic Communications in Probability > Vol. 15(2010) > Paper 30 open journal systems 


Weak approximation of fractional SDEs: the Donsker setting

Xavier Bardina, Universitat Autònoma de Barcelona
Carles Rovira, Universitat de Barcelona
Samy Tindel, Institut Elie Cartan Nancy


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 $H∈ (1/3,1/2)$, initiated in a paper of Bardina et al. (2010, MR2565851). 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 $B$.


Full text: PDF

Pages: 314-329

Published on: July 23, 2010
Bibliography
  1. 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. MR1741199 (2000m:60059)
  2. Bardina, Xavier; Jolis, Maria; Quer-Sardanyons, Lluísl. Weak convergence for the stochasticheat equation driven by Gaussian white noise. Preprint.(2009)
  3. Bardina, X.; Nourdin, I.; Rovira, C.; Tindel, S. Weak approximation of a fractional SDE. Stochastic Process. Appl. 120 (2010), no. 1, 39--65. MR2565851
  4. Breuillard, Emmanuel; Friz, Peter; Huesmann, Martin. From random walks to rough paths. Proc. Amer. Math. Soc. 137 (2009), no. 10, 3487--3496. MR2515418 (2010f:60015)
  5. Delgado, Rosario; Jolis, Maria. Weak approximation for a class of Gaussian processes. J. Appl. Probab. 37 (2000), no. 2, 400--407. MR1780999 (2001j:60064)
  6. 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 MR0838085 (88a:60130)
  7. Feyel, Denis; de La Pradelle, Arnaud. On fractional Brownian processes. Potential Anal. 10 (1999), no. 3, 273--288. MR1696137 (2000j:60051)
  8. Friz, Peter K.; Victoir, Nicolas B. Differential equations driven by Gaussian signals (I). Ann. Inst. H. Poincaré Probab. Statist., to appear.
  9. 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 MR2604669
  10. Gubinelli, M. Controlling rough paths. J. Funct. Anal. 216 (2004), no. 1, 86--140. MR2091358 (2005k:60169)
  11. 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. MR0510166 (58 #23185)
  12. 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. MR1431298 (98h:60073)
  13. 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 MR2036784 (2005f:93001)
  14. Sottinen, Tommi. Fractional Brownian motion, random walks and binary market models. Finance Stoch. 5 (2001), no. 3, 343--355. MR1849425 (2002j:91068)
  15. 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 MR0685758 (84m:60071)
  16. Tessitore, Gianmario; Zabczyk, Jerzy. Wong-Zakai approximations of stochastic evolution equations. J. Evol. Equ. 6 (2006), no. 4, 621--655. MR2267702 (2008d:60088)
  17. Wong, Eugene; Zakai, Moshe. On the relation between ordinary and stochastic differential equations. Internat. J. Engrg. Sci. 3 1965 213--229. MR0183023 (32 #505)
















Research
Support Tool
Capture Cite
View Metadata
Printer Friendly
Context
Author Address
Action
Email Author
Email Others


Home | Contents | Submissions, editors, etc. | Login | Search | EJP

Electronic Communications in Probability. ISSN: 1083-589X