Annals of Probability

Stochastic calculus for fractional Brownian motion with Hurst exponent H>¼: A rough path method by analytic extension

Jérémie Unterberger

Full-text: Open access


The d-dimensional fractional Brownian motion (FBM for short) Bt=((Bt(1), …, Bt(d)), t∈ℝ) with Hurst exponent α, α∈(0, 1), is a d-dimensional centered, self-similar Gaussian process with covariance ${\mathbb{E}}[B_{s}^{(i)}B_{t}^{(j)}]=\frac{1}{2}\delta_{i,j}(|s|^{2\alpha}+|t|^{2\alpha}-|t-s|^{2\alpha})$. The long-standing problem of defining a stochastic integration with respect to FBM (and the related problem of solving stochastic differential equations driven by FBM) has been addressed successfully by several different methods, although in each case with a restriction on the range of either d or α. The case α=½ corresponds to the usual stochastic integration with respect to Brownian motion, while most computations become singular when α gets under various threshhold values, due to the growing irregularity of the trajectories as α→0.

We provide here a new method valid for any d and for α>¼ by constructing an approximation Γ(ɛ)t, ɛ→0, of FBM which allows to define iterated integrals, and then applying the geometric rough path theory. The approximation relies on the definition of an analytic process Γz on the cut plane z∈ℂ∖ℝ of which FBM appears to be a boundary value, and allows to understand very precisely the well-known (see [5]) but as yet a little mysterious divergence of Lévy’s area for α→¼.

Article information

Ann. Probab., Volume 37, Number 2 (2009), 565-614.

First available in Project Euclid: 30 April 2009

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems 60G15: Gaussian processes 60G18: Self-similar processes 60H05: Stochastic integrals

Fractional Brownian motion stochastic integrals


Unterberger, Jérémie. Stochastic calculus for fractional Brownian motion with Hurst exponent H >¼: A rough path method by analytic extension. Ann. Probab. 37 (2009), no. 2, 565--614. doi:10.1214/08-AOP413.

Export citation


  • [1] Abramowitz, M., Stegun, A., Danos, M. and Rafelski, J. (1984). Handbook of Mathematical Functions. Harri Deutsch, Frankfurt.
  • [2] Adler, R. J. (1981). The Geometry of Random Fields. Wiley, Chichester.
  • [3] Baudoin, F. and Coutin, L. (2005). Étude en temps petit des solutions d’EDS conduites par des mouvements browniens fractionnaires. C. R. Math. Acad. Sci. Paris 341 39–42.
  • [4] Cheridito, P. and Nualart, D. (2005). Stochastic integral of divergence type with respect to fractional Brownian motion with Hurst parameter H∈(0, ½). Ann. Inst. H. Poincaré Probab. Statist. 41 1049–1081.
  • [5] Coutin, L. and Qian, Z. (2002). Stochastic analysis, rough path analysis and fractional Brownian motions. Probab. Theory Related Fields 122 108–140.
  • [6] Decreusefond, L. and Üstünel, A. S. (1999). Stochastic analysis of the fractional Brownian motion. Potential Anal. 10 177–214.
  • [7] Dzhaparidze, K. and van Zanten, H. (2004). A series expansion of fractional Brownian motion. Probab. Theory Related Fields 130 39–55.
  • [8] Feyel, D. and de la Pradelle, A. (2001). The FBM Itô’s formula through analytic continuation. Electron. J. Probab. 6 22.
  • [9] Gradinaru, M., Nourdin, I., Russo, F. and Vallois, P. (2005). m-order integrals and generalized Itô’s formula: The case of a fractional Brownian motion with any Hurst index. Ann. Inst. H. Poincaré Probab. Statist. 41 781–806.
  • [10] Kahane, J.-P. (1985). Some Random Series of Functions, 2nd ed. Cambridge Studies in Advanced Mathematics 5. Cambridge Univ. Press, Cambridge.
  • [11] Kühn, T. and Linde, W. (2002). Optimal series representation of fractional Brownian sheets. Bernoulli 8 669–696.
  • [12] Lejay, A. (2003). An introduction to rough paths. In Séminaire de Probabilités XXXVII. Lecture Notes in Mathematics 1832 1–59. Springer, Berlin.
  • [13] Ledoux, M., Lyons, T. and Qian, Z. (2002). Lévy area of Wiener processes in Banach spaces. Ann. Probab. 30 546–578.
  • [14] Lyons, T. J. (1998). Differential equations driven by rough signals. Rev. Mat. Iberoamericana 14 215–310.
  • [15] Lyons, T. and Qian, Z. (2002). System Control and Rough Paths. Oxford Univ. Press, Oxford.
  • [16] Nualart, D. (1995). The Malliavin Calculus and Related Topics. Springer, New York.
  • [17] Pipiras, V. and Taqqu, M. S. (2000). Integration questions related to fractional Brownian motion. Probab. Theory Related Fields 118 251–291.
  • [18] Russo, F. and Vallois, P. (1993). Forward, backward and symmetric stochastic integration. Probab. Theory Related Fields 97 403–421.
  • [19] Russo, F. and Vallois, P. (2000). Stochastic calculus with respect to continuous finite quadratic variation processes. Stochastics Stochastics Rep. 70 1–40.
  • [20] van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes. Springer, New York.