Electronic Journal of Probability

Self-similarity and fractional Brownian motion on Lie groups

Fabrice Baudoin and Laure Coutin

Full-text: Open access

Abstract

The goal of this paper is to define and study a notion of fractional Brownian motion on a Lie group. We define it as at the solution of a stochastic differential equation driven by a linear fractional Brownian motion. We show that this process has stationary increments and satisfies a local self-similar property. Furthermore the Lie groups for which this self-similar property is global are characterized.

Article information

Source
Electron. J. Probab., Volume 13 (2008), paper no. 38, 1120-1139.

Dates
Accepted: 22 July 2008
First available in Project Euclid: 1 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464819111

Digital Object Identifier
doi:10.1214/EJP.v13-530

Mathematical Reviews number (MathSciNet)
MR2424989

Zentralblatt MATH identifier
1189.60083

Subjects
Primary: 60G15: Gaussian processes
Secondary: 60G18: Self-similar processes

Keywords
Fractional Brownian motion Lie group

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

Citation

Baudoin, Fabrice; Coutin, Laure. Self-similarity and fractional Brownian motion on Lie groups. Electron. J. Probab. 13 (2008), paper no. 38, 1120--1139. doi:10.1214/EJP.v13-530. https://projecteuclid.org/euclid.ejp/1464819111


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References

  • Baudoin, Fabrice. An introduction to the geometry of stochastic flows.Imperial College Press, London, 2004. x+140 pp. ISBN: 1-86094-481-7
  • Baudoin, Fabrice; Coutin, Laure. Operators associated with a stochastic differential equation driven by fractional Brownian motions. Stochastic Process. Appl. 117 (2007), no. 5, 550–574.
  • Ben Arous, Gérard. Flots et séries de Taylor stochastiques.(French) [Flows and stochastic Taylor series] Probab. Theory Related Fields 81 (1989), no. 1, 29–77.
  • Borell, Christer. On polynomial chaos and integrability. Probab. Math. Statist. 3 (1984), no. 2, 191–203.
  • Castell, Fabienne. Asymptotic expansion of stochastic flows. Probab. Theory Related Fields 96 (1993), no. 2, 225–239.
  • Cheridito, Patrick; Nualart, David. Stochastic integral of divergence type with respect to fractional Brownian motion with Hurst parameter $H\in(0,{1\over2})$. Ann. Inst. H. Poincaré Probab. Statist. 41 (2005), no. 6, 1049–1081.
  • 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.
  • L. Coutin, P Friz and N. Victoir, Good Rough Path Sequences ans Applications to Anticipating and Fractional Stochastic Calculus; preprint 2005
  • L. Coutin, Z. Qian: Stochastic rough path analysis and fractional Brownian motion, Probab. Theory Relat. Fields 122, 108-140, (2002).
  • Decreusefond, L.; Üstünel, A. S. Stochastic analysis of the fractional Brownian motion. Potential Anal. 10 (1999), no. 2, 177–214.
  • Friz P., Victoir N. (2006): Euler estimates for rough differential equations, preprint.
  • M. Gradinaru, I. Nourdin, F. Russo, P. Vallois: m-order integrals and generalized Itô's formula: the case of fractional Brownian motion with any Hurst parameter, To appear in Ann. Inst. H. Poincare.
  • Hunt G.A.: Markov processes and potentials:, Illinois J. Math, 1, 44-93, 316-369 (1957); 2, 151-213, (1958).
  • Itô, Kiyosi. Stochastic differential equations in a differentiable manifold. Nagoya Math. J. 1, (1950). 35–47. (12,425g)
  • Kunita, Hiroshi. Stochastic flows with self-similar properties. Stochastic analysis and applications (Powys, 1995), 286–300, World Sci. Publ., River Edge, NJ, 1996.
  • Kunita, Hiroshi. Asymptotic self-similarity and short time asymptotics of stochastic flows. J. Math. Sci. Univ. Tokyo 4 (1997), no. 3, 595–619.
  • Lawton, Wayne. Infinite convolution products and refinable distributions on Lie groups. Trans. Amer. Math. Soc. 352 (2000), no. 6, 2913–2936.
  • Lejay, Antoine. An introduction to rough paths. Séminaire de Probabilités XXXVII, 1–59, Lecture Notes in Math., 1832, Springer, Berlin, 2003.
  • X.D. Li, T. Lyons: Smoothness of Ito maps and simulated annealing on path spaces (I), $1<2$, preprint, 2003.
  • Lyons, Terry J. Differential equations driven by rough signals. Rev. Mat. Iberoamericana 14 (1998), no. 2, 215–310.
  • 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
  • Malliavin, Paul. Stochastic analysis.Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 313. Springer-Verlag, Berlin, 1997. xii+343 pp. ISBN: 3-540-57024-1
  • Nourdin, Ivan. Schémas d'approximation associés à une équation différentielle dirigée par une fonction höldérienne; cas du mouvement brownien fractionnaire.(French) [Approximation schemes associated with a differential equation governed by a Holder function; the case of fractional Brownian motion] C. R. Math. Acad. Sci. Paris 340 (2005), no. 8, 611–614.
  • Nourdin, Ivan; Tudor, Ciprian A. Some linear fractional stochastic equations. Stochastics 78 (2006), no. 2, 51–65.
  • Nualart, David; Rascanu, Aurel. Differential equations driven by fractional Brownian motion. Collect. Math. 53 (2002), no. 1, 55–81.
  • Rogers, L. C. G.; Williams, David. Diffusions, Markov processes, and martingales. Vol. 2.Itô calculus.Reprint of the second (1994) edition. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 2000. xiv+480 pp. ISBN: 0-521-77593-0
  • Strichartz, Robert S. The Campbell-Baker-Hausdorff-Dynkin formula and solutions of differential equations. J. Funct. Anal. 72 (1987), no. 2, 320–345.
  • Yamato, Yuiti. Stochastic differential equations and nilpotent Lie algebras. Z. Wahrsch. Verw. Gebiete 47 (1979), no. 2, 213–229.
  • Yosida, Kôsaku. On Brownian motion in a homogeneous Riemannian space. Pacific J. Math. 2, (1952). 263–270. (14,387f)
  • Young, L. C. An inequality of the Hölder type, connected with Stieltjes integration. Acta Math. 67 (1936), no. 1, 251–282.
  • Zähle, M. Integration with respect to fractal functions and stochastic calculus. II. Math. Nachr. 225 (2001), 145–183.