Electronic Journal of Probability

Time Reversal for Drifted Fractional Brownian Motion with Hurst Index $H>1/2$

Sebastien Darses and Bruno Saussereau

Full-text: Open access

Abstract

Let $X$ be a drifted fractional Brownian motion with Hurst index $H > 1/2$. We prove that there exists a fractional backward representation of $X$, i.e. the time reversed process is a drifted fractional Brownian motion, which continuously extends the one obtained in the theory of time reversal of Brownian diffusions when $H=1/2$. We then apply our result to stochastic differential equations driven by a fractional Brownian motion.

Article information

Source
Electron. J. Probab., Volume 12 (2007), paper no. 43, 1181-1211.

Dates
Accepted: 7 September 2007
First available in Project Euclid: 1 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464818514

Digital Object Identifier
doi:10.1214/EJP.v12-439

Mathematical Reviews number (MathSciNet)
MR2346508

Zentralblatt MATH identifier
1130.60044

Subjects
Primary: 60G18: Self-similar processes
Secondary: 60H07: Stochastic calculus of variations and the Malliavin calculus 60H10: Stochastic ordinary differential equations [See also 34F05] 60J60: Diffusion processes [See also 58J65]

Keywords
Fractional Brownian motion Time reversal Malliavin Calculus

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

Citation

Darses, Sebastien; Saussereau, Bruno. Time Reversal for Drifted Fractional Brownian Motion with Hurst Index $H>1/2$. Electron. J. Probab. 12 (2007), paper no. 43, 1181--1211. doi:10.1214/EJP.v12-439. https://projecteuclid.org/euclid.ejp/1464818514


Export citation

References

  • Alòs, Elisa; Nualart, David. Stochastic integration with respect to the fractional Brownian motion. Stoch. Stoch. Rep. 75 (2003), no. 3, 129–152.
  • Carmona, Philippe; Coutin, Laure; Montseny, Gérard. Stochastic integration with respect to fractional Brownian motion. Ann. Inst. H. Poincaré Probab. Statist. 39 (2003), no. 1, 27–68.
  • Cresson, Jacky; Darses, Sébastien. Plongement stochastique des systèmes lagrangiens. (French) [Stochastic embedding of Lagrangian systems] C. R. Math. Acad. Sci. Paris 342 (2006), no. 5, 333–336.
  • Darses S., Nourdin I. : Stochastic derivatives for fractional diffusions. Ann. Prob. To appear.
  • Decreusefond, L. Stochastic integration with respect to Volterra processes. Ann. Inst. H. Poincaré Probab. Statist. 41 (2005), no. 2, 123–149.
  • Decreusefond, L.; Üstünel, A. S. Stochastic analysis of the fractional Brownian motion. Potential Anal. 10 (1999), no. 2, 177–214.
  • Föllmer, H. Time reversal on Wiener space. Stochastic processes–-mathematics and physics (Bielefeld, 1984), 119–129, Lecture Notes in Math., 1158, Springer, Berlin, 1986.
  • Garsia, A. M.; Rodemich, E.; Rumsey, H., Jr. A real variable lemma and the continuity of paths of some Gaussian processes. Indiana Univ. Math. J. 20 1970/1971 565–578.
  • Haussmann, U. G.; Pardoux, É. Time reversal of diffusions. Ann. Probab. 14 (1986), no. 4, 1188–1205.
  • Millet, A.; Nualart, D.; Sanz, M. Integration by parts and time reversal for diffusion processes. Ann. Probab. 17 (1989), no. 1, 208–238.
  • Moret, Sílvia; Nualart, David. Onsager-Machlup functional for the fractional Brownian motion. Probab. Theory Related Fields 124 (2002), no. 2, 227–260.
  • Nelson, Edward. Dynamical theories of Brownian motion. Princeton University Press, Princeton, N.J. 1967 iii+142 pp.
  • Nualart, D.: The Malliavin Calculus and Related Topics. Springer Verlag (1996).
  • Nualart, David. Stochastic integration with respect to fractional Brownian motion and applications. Stochastic models (Mexico City, 2002), 3–39, Contemp. Math., 336, Amer. Math. Soc., Providence, RI, 2003.
  • Nualart, David; Ouknine, Youssef. Regularization of differential equations by fractional noise. Stochastic Process. Appl. 102 (2002), no. 1, 103–116.
  • Nualart, David; Rascanu, Aurel. Differential equations driven by fractional Brownian motion. Collect. Math. 53 (2002), no. 1, 55–81.
  • Pardoux, É. Grossissement d'une filtration et retournement du temps d'une diffusion. (French) [Enlargement of a filtration and time reversal of a diffusion] Séminaire de Probabilités, XX, 1984/85, 48–55, Lecture Notes in Math., 1204, Springer, Berlin, 1986.