Bernoulli

  • Bernoulli
  • Volume 12, Number 4 (2006), 713-735.

Power variation of some integral fractional processes

José Manuel Corcuera, David Nualart, and Jeannette H.C. Woerner

Full-text: Open access

Abstract

We consider the asymptotic behaviour of the realized power variation of processes of the form 0 t u s dB s H , where B H is a fractional Brownian motion with Hurst parameter H (0,1) , and u is a process with finite q -variation, q <1/(1-H) . We establish the stable convergence of the corresponding fluctuations. These results provide new statistical tools to study and detect the long-memory effect and the Hurst parameter.

Article information

Source
Bernoulli, Volume 12, Number 4 (2006), 713-735.

Dates
First available in Project Euclid: 16 August 2006

Permanent link to this document
https://projecteuclid.org/euclid.bj/1155735933

Digital Object Identifier
doi:10.3150/bj/1155735933

Mathematical Reviews number (MathSciNet)
MR2248234

Zentralblatt MATH identifier
1130.60058

Keywords
central and non-central limit theorems fractional Brownian motion long memory p-variation realized power variation

Citation

Manuel Corcuera, José; Nualart, David; Woerner, Jeannette H.C. Power variation of some integral fractional processes. Bernoulli 12 (2006), no. 4, 713--735. doi:10.3150/bj/1155735933. https://projecteuclid.org/euclid.bj/1155735933


Export citation

References

  • [1] Aldous, D.J. and Eagleson, G.K. (1978) On mixing and stability of limit theorems. Ann. Probab., 6(2), 325-331.
  • [2] Barndorff-Nielsen, O.E. and Shephard, N. (2002) Econometric analysis of realized volatility and its use in estimating stochastic volatility models. J. Roy. Statist. Soc., Ser. B, 64, 253-280.
  • [3] Barndorff-Nielsen, O.E. and Shephard, N. (2003) Realized power variation and stochastic volatility models. Bernoulli, 9, 243-265.
  • [4] Barndorff-Nielsen, O.E. and Shephard, N. (2004a) Power and bipower variation with stochastic volatility and jumps (with discussion). J. Financial Econometrics, 2, 1-48.
  • [5] Barndorff-Nielsen, O.E. and Shephard, N. (2004b) Econometric analysis of realised covariation: high frequency covariance, regression and correlation in financial economics. Econometrica, 72, 885-925.
  • [6] Barndorff-Nielsen, O.E., Graversen S. E., Jacod, J., Podolskij, M. and Shephard, N. (2006) A central limit theorem for realised power and bipower variations of continuous semimartingales. In Y. Kabanov, R. Liptser and J. Stoyanov (eds), From Stochastic Analysis to Mathematical Finance: The Shiryaev Festschrift. Berlin: Springer-Verlag.
  • [7] Billingsley, P. (1968). Convergence of Probability Measures. New York: Wiley.
  • [8] Breuer, P. and Major, P. (1983) Central limit theorems for non-linear functionals of Gaussian fields. J. Multivariate Anal., 13, 425-441.
  • [9] Dobrushin, R.L. and Major, P. (1979) Non-central limit theorems for non-linear functionals of Gaussian fields, Z. Wahrscheinlichkeitstheorie Verw. Geb., 50, 27-52.
  • [10] Fernique, X. (1975) Regularité des trajectoires des fonctions aléatoires gaussiennes. In P.L. Hennequin, École d´Été de Probabilités de Saint-Flour IV - 1974, Lecture Notes in Math. 480. Berlin: Springer-Verlag.
  • [11] Giraitis, L. and Surgailis, D. (1985) CTL and other limit theorems for functionals of Gaussian processes. Z. Wahrscheinlichkeitstheorie Verw. Geb., 70, 191-212.
  • [12] Hu, Y. and Nualart, D. (2005) Renormalized self-intersection local time for fractional Brownian motion. Ann. Probab., 33, 948-983.
  • [13] Hudson, W.N. and Mason J.D. (1976) Variational sums for additive processes. Proc. Amer. Math. Soc., 55, 395-399.
  • [14] Leo´n, J.R. and Ludena, C. (2004) Stable convergence of certain functionals of diffusions driven by fBm. Stochastic Anal. Appl., 22, 289-314.
  • [15] Nualart, D. and Peccati, G. (2005) Central limit theorems for sequences of multiple stochastic integrals. Ann. Probab., 33, 177-193.
  • [16] Peccati, G. and Tudor, C.A. (2005) Gaussian limits for vector-valued multiple stochastic integrals. In M. É mery, M. Ledoux and M. Yor (eds), Séminaire de Probabilités XXXVIII, Lecture Notes in Math. 1857, pp. 247-262. Berlin: Springer-Verlag.
  • [17] Rosenblatt, M. (1961) Independence and dependence. In J. Neyman (ed.), Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Vol. 2, pp. 411-443. Berkeley: University of California Press.
  • [18] Shiryaev, A.N. (1999) Essentials of Stochastic Finance. Singapore: World Scientific.
  • [19] Sun, T.C. and Ho, H.C. (1986) On central and non-central limit theorems for non-linear functions of
  • [20] stationary Gaussian process. In E. Eberlein and M.S. Taqqu (eds.). Dependence in Probability and Statistics: A Survey of recent results, Progr. Probab. 11. Boston: Birkhäuser.
  • [21] Taqqu, M.S. (1975) Weak convergence to fractional Brownian motion and to the Rosenblatt process. Z. Wahrscheinlichkeitstheorie Verw. Geb., 31, 287-302.
  • [22] Taqqu, M.S. (1977) Law of the iterated logarithm for sums of non-linear functions of Gaussian variables that exhibit a long range dependence. Z. Wahrscheinlichkeitstheorie Verw. Geb., 40, 203-238.
  • [23] Taqqu, M.S. (1979) Convergence of integrated processes of arbitrary Hermite rank. Z. Wahrscheinlichkeitstheorie Verw. Geb., 50, 53-83.
  • [24] Woerner, J.H.C. (2003) Variational sums and power variation: a unifying approach to model selection and estimation in semimartingale models. Statist. Decisions, 21, 47-68.
  • [25] Woerner, J.H.C. (2005) Estimation of Integrated Volatility in Stochastic Volatility Models. Appl. Stochastic Models Business Industry, 21, 27-44.
  • [26] Young, L.C. (1936) An inequality of the Hölder type connected with Stieltjes integration, Acta Math., 67, 251-282.