Abstract and Applied Analysis

Least Squares Estimation for α -Fractional Bridge with Discrete Observations

Guangjun Shen and Xiuwei Yin

Full-text: Open access

Abstract

We consider a fractional bridge defined as d X t = - α (X t / ( T - t )) d t + d B t H ,   0 t < T , where B H is a fractional Brownian motion of Hurst parameter H > 1 / 2 and parameter α > 0 is unknown. We are interested in the problem of estimating the unknown parameter α > 0 . Assume that the process is observed at discrete time t i = i Δ n , i = 0 , , n , and T n = n Δ n denotes the length of the “observation window.” We construct a least squares estimator α ^ n of α which is consistent; namely, α ^ n converges to α in probability as n .

Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2013), Article ID 748376, 8 pages.

Dates
First available in Project Euclid: 26 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1395858396

Digital Object Identifier
doi:10.1155/2014/748376

Mathematical Reviews number (MathSciNet)
MR3166653

Zentralblatt MATH identifier
07023012

Citation

Shen, Guangjun; Yin, Xiuwei. Least Squares Estimation for $\alpha $ -Fractional Bridge with Discrete Observations. Abstr. Appl. Anal. 2014, Special Issue (2013), Article ID 748376, 8 pages. doi:10.1155/2014/748376. https://projecteuclid.org/euclid.aaa/1395858396


Export citation

References

  • F. Biagini, Y. Hu, B. ${\text{\O}}$ksendal, and T. Zhang, Stochastic Calculus for fBm and Applications. Probability and Its Application, Springer, Berlin, Germany, 2008.
  • Y. Hu, “Integral transformations and anticipative calculus for fractional Brownian motions,” Memoirs of the American Mathematical Society, vol. 175, no. 825, 2005.
  • Y. S. Mishura, Stochastic Calculus for Fractional Brownian Motion and Related Processes, vol. 1929 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 2008.
  • D. Nualart, The Malliavin Calculus and Related Topics, Springer, Berlin, Germany, 2nd edition, 2006.
  • K. Es-Sebaiy and I. Nourdin, “Parameter estimation for $\alpha $ fractional bridges,” in Malliavin Calculus and Stochastic Analysis: A Festschrift in Honor of David Nualart, F. Viens, J. Feng, Y. Hu, and E. Nualart, Eds., vol. 34 of Springer Proceedings in Mathematics and Statistics, pp. 385–412, 2013.
  • M. Barczy and G. Pap, “Asymptotic behavior of maximum likelihood estimator for time inhomogeneous diffusion processes,” Journal of Statistical Planning and Inference, vol. 140, no. 6, pp. 1576–1593, 2010.
  • M. Barczy and G. Pap, “$\alpha $-Wiener bridges: singularity of induced measures and sample path properties,” Stochastic Analysis and Applications, vol. 28, no. 3, pp. 447–466, 2010.
  • C. A. Tudor and F. G. Viens, “Statistical aspects of the fractional stochastic calculus,” The Annals of Statistics, vol. 35, no. 3, pp. 1183–1212, 2007.
  • Y. Hu and D. Nualart, “Parameter estimation for fractional Ornstein-Uhlenbeck processes,” Statistics & Probability Letters, vol. 80, no. 11-12, pp. 1030–1038, 2010.
  • R. Belfadli, K. Es-Sebaiy, and Y. Ouknine, “Parameter estimation for fractional Ornstein-Uhlenbeck processes: non-ergodic case,” Frontiers in Science and Engineering, vol. 1, pp. 1–16, 2011.
  • Y. Hu and J. Song, “Parameter estimation for fractional Orn-stein-Uhlenbeck processes with discrete observations,” in Malliavin Calculus and Stochastic Analysis: A Festschrift in Honor of David Nualart, F. Viens, J. Feng, Y. Hu, and E. Nualart, Eds., vol. 34 of Springer Proceedings in Mathematics and Statistics, pp. 427–442, 2013.
  • K. Es-Sebaiy, “Berry-Esséen bounds for the least squares estimator for discretely observed fractional Ornstein-Uhlenbeck processes,” Statistics & Probability Letters, vol. 83, no. 10, pp. 2372–2385, 2013.
  • E. Alòs, O. Mazet, and D. Nualart, “Stochastic calculus with respect to Gaussian processes,” The Annals of Probability, vol. 29, no. 2, pp. 766–801, 2001.
  • V. Pipiras and M. S. Taqqu, “Integration questions related to fractional Brownian motion,” Probability Theory and Related Fields, vol. 118, no. 2, pp. 251–291, 2000.
  • E. Alòs and D. Nualart, “Stochastic integration with respect to the fractional Brownian motion,” Stochastics and Stochastics Reports, vol. 75, no. 3, pp. 129–152, 2003. \endinput