Parameter Estimation for α-Fractional Bridges

  • Khalifa Es-Sebaiy
  • Ivan NourdinEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 34)


Let α, T > 0. We study the asymptotic properties of a least squares estimator for the parameter α of a fractional bridge defined as \(\mathrm{d}X_{t} = -\alpha \, \frac{X_{t}} {T-t}\,\mathrm{d}t + \mathrm{d}B_{t}\), 0 ≤ t < T, where B is a fractional Brownian motion of Hurst parameter \(H > \frac{1} {2}\). Depending on the value of α, we prove that we may have strong consistency or not as t → T. When we have consistency, we obtain the rate of this convergence as well. Also, we compare our results to the (known) case where B is replaced by a standard Brownian motion W.


Brownian Motion Fractional Brownian Motion Standard Brownian Motion Strong Consistency Hurst Parameter 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



We thank an anonymous referee for his/her careful reading of the manuscript and for his/her valuable suggestions and remarks. Ivan Nourdin was supported in part by the (French) ANR grant ‘Exploration des Chemins Rugueux’


  1. 1.
    Alòs, E., Nualart, D.: Stochastic integration with respect to the fractional Brownian motion. Stoch. Stoch. Reports 75(3), 129–152 (2003)Google Scholar
  2. 2.
    Barczy, M., Pap, G.: Explicit formulas for Laplace transforms of certain functionals of some time inhomogeneous diffusions. J. Math. Anal. Appl. 380(2), 405–424 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Barczy, M., Pap, G.: α-Wiener bridges: singularity of induced measures and sample path properties. Stoch. Anal. Appl. 28(3), 447–466 (2010)Google Scholar
  4. 4.
    Barczy, M., Pap, G.: Asymptotic behavior of maximum likelihood estimator for time inhomogeneous diffusion processes. J. Statist. Plan. Infer. 140(6), 1576–1593 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Hu, Y., Nualart, D.: Parameter estimation for fractional Ornstein-Uhlenbeck processes. Statist. Probab. Lett. 80, 1030–1038 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Mansuy, R.: On a one-parameter generalization of the Brownian bridge and associated quadratic functionals. J. Theoret. Probab. 17(4), 1021–1029 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Nualart, D.: The Malliavin Calculus and Related Topics, 2nd edn. Springer, Berlin (2006)zbMATHGoogle Scholar
  8. 8.
    Nualart, D., Peccati, G.: Central limit theorems for sequences of multiple stochastic integrals. Ann. Probab. 33(1), 177–193 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Pipiras, V., Taqqu, M.S.: Integration questions related to fractional Brownian motion. Probab. Theory Rel. Fields 118(2), 251–291 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Russo, F., Vallois, P.: Elements of stochastic calculus via regularization. Séminaire de Probabilités XL. In: Lecture Notes in Math, vol. 1899, pp. 147–185. Springer, Berlin (2007)Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Institut de Mathématiques de BourgogneUniversité de BourgogneDijonFrance

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