Electronic Journal of Probability

Taylor expansion for the solution of a stochastic differential equation driven by fractional Brownian motions

Fabrice Baudoin and Xuejing Zhang

Full-text: Open access


We study the Taylor expansion for the solution of a differential equation driven by a multi-dimensional Hölder path with exponent  $H> 1/2$. We derive a convergence criterion that enables us to write the solution as an infinite sum of iterated integrals on a non empty interval. We apply our deterministic results to stochastic differential equations driven by fractional Brownian motions with Hurst parameter $H > 1/2$. We also study the convergence in L2 of the stochastic Taylor expansion by using L2 estimates of iterated integrals and Borel-Cantelli type arguments.

Article information

Electron. J. Probab., Volume 17 (2012), paper no. 51, 21 pp.

Accepted: 6 July 2012
First available in Project Euclid: 4 June 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

taylor expansion fractional Brownian motion

This work is licensed under aCreative Commons Attribution 3.0 License.


Baudoin, Fabrice; Zhang, Xuejing. Taylor expansion for the solution of a stochastic differential equation driven by fractional Brownian motions. Electron. J. Probab. 17 (2012), paper no. 51, 21 pp. doi:10.1214/EJP.v17-2136. https://projecteuclid.org/euclid.ejp/1465062373

Export citation


  • Azencott, Robert. Formule de Taylor stochastique et développement asymptotique d'intégrales de Feynman. (French) [Stochastic Taylor formula and asymptotic expansion of Feynman integrals] Seminar on Probability, XVI, Supplement, pp. 237–285, Lecture Notes in Math., 921, Springer, Berlin-New York, 1982.
  • 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.
  • 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.
  • Baudoin, Fabrice; Coutin, Laure. Self-similarity and fractional Brownian motions on Lie groups. Electron. J. Probab. 13 (2008), no. 38, 1120–1139.
  • Castell, Fabienne. Asymptotic expansion of stochastic flows. Probab. Theory Related Fields 96 (1993), no. 2, 225–239.
  • Decreusefond, Laurent; Nualart, David. Flow properties of differential equations driven by fractional Brownian motion. Stochastic differential equations: theory and applications, 249–262, Interdiscip. Math. Sci., 2, World Sci. Publ., Hackensack, NJ, 2007.
  • Duistermaat, J. J.; Kolk, J. A. C. Lie groups. Universitext. Springer-Verlag, Berlin, 2000. viii+344 pp. ISBN: 3-540-15293-8
  • Y. Hu. Multiple integrals and expansion of solutions of differential equations driven by rough paths and by fractional Brownian motions. phJournal of Theoretical Probability, To appear.
  • Magnus, Wilhelm. On the exponential solution of differential equations for a linear operator. Comm. Pure Appl. Math. 7, (1954). 649–673.
  • Neuenkirch, A.; Nourdin, I.; Rössler, A.; Tindel, S. Trees and asymptotic expansions for fractional stochastic differential equations. Ann. Inst. Henri Poincaré Probab. Stat. 45 (2009), no. 1, 157–174.
  • Nualart, David; Răşcanu, Aurel. Differential equations driven by fractional Brownian motion. Collect. Math. 53 (2002), no. 1, 55–81.
  • Hu, Yaozhong; Nualart, David. Differential equations driven by Hölder continuous functions of order greater than 1/2. Stochastic analysis and applications, 399–413, Abel Symp., 2, Springer, Berlin, 2007.
  • Nualart, David; Saussereau, Bruno. Malliavin calculus for stochastic differential equations driven by a fractional Brownian motion. Stochastic Process. Appl. 119 (2009), no. 2, 391–409.
  • Ruzmaikina, A. A. Stieltjes integrals of Hölder continuous functions with applications to fractional Brownian motion. J. Statist. Phys. 100 (2000), no. 5-6, 1049–1069.
  • Strichartz, Robert S. The Campbell-Baker-Hausdorff-Dynkin formula and solutions of differential equations. J. Funct. Anal. 72 (1987), no. 2, 320–345.
  • 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. I. Probab. Theory Related Fields 111 (1998), no. 3, 333–374.