Parameter Estimation for α-Fractional Bridges

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.





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)MathSciNetMATHCrossRefGoogle 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)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Hu, Y., Nualart, D.: Parameter estimation for fractional Ornstein-Uhlenbeck processes. Statist. Probab. Lett. 80, 1030–1038 (2010)MathSciNetMATHCrossRefGoogle 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)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Nualart, D.: The Malliavin Calculus and Related Topics, 2nd edn. Springer, Berlin (2006)MATHGoogle Scholar
  8. 8.
    Nualart, D., Peccati, G.: Central limit theorems for sequences of multiple stochastic integrals. Ann. Probab. 33(1), 177–193 (2005)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Pipiras, V., Taqqu, M.S.: Integration questions related to fractional Brownian motion. Probab. Theory Rel. Fields 118(2), 251–291 (2000)MathSciNetMATHCrossRefGoogle 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

Personalised recommendations