• Bernoulli
  • Volume 11, Number 6 (2005), 987-1008.

Identification of multifractional Brownian motion

Jean-François Coeurjolly

Full-text: Open access


We develop a method for estimating the Hurst function of a multifractional Brownian motion, which is an extension of the fractional Brownian motion in the sense that the path regularity can now vary with time. This method is based on a local estimation of the second-order moment of a unique discretized filtered path. The effectiveness of our procedure is investigated in a short simulation study.

Article information

Bernoulli, Volume 11, Number 6 (2005), 987-1008.

First available in Project Euclid: 16 January 2006

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

filtering fractional Brownian motion functional estimation multifractional Brownian motion


Coeurjolly, Jean-François. Identification of multifractional Brownian motion. Bernoulli 11 (2005), no. 6, 987--1008. doi:10.3150/bj/1137421637.

Export citation


  • [1] Benassi, A., Cohen, S. and Istas, J. (1998) Identifying the multifractional function of a Gaussian process. Statist. Probab. Lett., 39, 337-345.
  • [2] Breuer, P. and Major, P. (1983) Central limit theorems for non-linear functionals of Gaussian fields. J. Multivariate Anal., 13, 425-441.
  • [3] Coeurjolly, J.-F. (2000) Inférence statistique pour les mouvements browniens fractionnaires et multifractionnaires. Doctoral thesis, Université de Grenoble.
  • [4] Coeurjolly, J.-F. (2001) Estimating the parameters of a fractional Brownian motion by discrete variations of its sample paths. Statist. Inference Stochastic Process., 4, 199-227.
  • [5] Coeurjolly, J.-F. and Istas, J. (2001) Cramér-Rao bounds for fractional Brownian motions. Statist. Probab. Lett., 53, 435-448.
  • [6] Cohen, S. (1999) From self-similarity to local self-similarity: the estimation problem. In M. Dekking, J. Lévy Véhel, E. Lutton, and C. Tricot (eds), Fractals: Theory and Applications in Engineering, pp. 3-16. London: Springer-Verlag.
  • [7] Collins, J.J. and De Luca, C.J. (1994) Upright, correlated random walks: A statistical-biomechanics approach to the human postural control system. Chaos, 5(1), 57-63.
  • [8] Doob, J.L. (1953) Stochastic Processes. New York: Wiley.
  • [9] Frisch, U. (1997) Turbulence; The Legacy of A.N. Kolmogorov. Cambridge: Cambridge University Press.
  • [10] Istas, J. and Lang, G. (1997) Quadratic variations and estimation of the Hölder index of a Gaussian process. Ann. Inst. H. Poincaré Probab. Statist., 33, 407-436.
  • [11] Jaffard, S. (1990) Propriétés ds matrices 'bien localisées´ près de leur diagonale et quelques applications. Ann. Inst. H. Poincaré Anal. Non Lineaire, 7, 461-476.
  • [12] Kent. J.T. and Wood, A.T.A. (1997) Estimating the fractal dimension of a locally self-similar Gaussian process using increments. J. Roy. Statist. Soc. Ser. B, 59, 679-700.
  • [13] Mandelbrot, B. and Van Ness, J. (1968) Fractional Brownian motions, fractional noises and applications. SIAM Rev., 10, 422-437.
  • [14] Peltier, R.-F. and Lévy Véhel, J. (1995) Multifractional Brownian motion: definition and preliminary results. INRIA Research Report No. 2645.
  • [15] Taqqu, M.S. (1975) Weak convergence to fractional Brownian motion and to the Rosenblatt process. Z. Wahrscheinlichkeitstheorie Verw. Geb., 40, 203-238.
  • [16] Von Bahr, B. and Esseen, C.G. (1965) Inequalities for the rth moment of a sum of random variables, 1 ≤ r ≤ 2. Ann. Math. Statist., 36, 299-303.