Skip to main content
SpringerLink
Log in

Asymptotic Behavior of Weighted Power Variations of Fractional Brownian Motion in Brownian Time

  • Published:
Journal of Theoretical Probability Aims and scope Submit manuscript

Abstract

We study the asymptotic behavior of weighted power variations of fractional Brownian motion in Brownian time \(Z_t:= X_{Y_t},t \geqslant 0\), where X is a fractional Brownian motion and Y is an independent Brownian motion.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Gradinaru, M., Russo, F., Vallois, P.: Generalized covariations, local time and Stratonovich Itô’s formula for fractional Brownian motion with Hurst index \(H \ge 1/4\). Ann. Probab. 31(4), 1772–1820 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  2. Khoshnevisan, D., Lewis, T.M.: Stochastic calculus for Brownian motion on a Brownian fracture. Ann. Appl. Probab. 9(3), 629–667 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  3. Khoshnevisan, D., Lewis, T.M.: Iterated Brownian Motion and its Intrinsic skeletal structure. In: Dalang R.C., Dozzi M., Russo F. (eds) Seminar on Stochastic Analysis, Random Fields and Applications. Progress in Probability, vol 45. Birkhäuser, Basel (1999)

  4. Nourdin, I., Peccati, G.: Weighted power variations of iterated Brownian motion. Electron. J. Probab. 13(43), 1229–1256 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Nourdin, I., Nualart, D., Tudor, C.: Central and non-central limit theorems for weighted power variations of the fractional Brownian motion. Ann. Inst. Henri Poincaré Probab. Stat. 46(4), 1055–1079 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Nourdin, I., Peccati, G.: Normal Approximations Using Malliavin Calculus: From Stein’s Method to the Universality. Cambridge University Press, Cambridge (2012)

    Book  MATH  Google Scholar 

  7. Nourdin, I., Réveillac, A., Swanson, J.: The weak Stratonovich integral with respect to fractional Brownian motion with Hurst parameter 1/6. Electron. J. Probab. 15(70), 2117–2162 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Nourdin, I., Zeineddine, R.: An Itô-type formula for the fractional Brownian motion in Brownian time. Electron. J. Probab. 19(99), 1–15 (2014)

    MATH  Google Scholar 

  9. Zeineddine, R.: Fluctuations of the power variation of fractional Brownian motion in Brownian time. Bernoulli 21(2), 760–780 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  10. Zeineddine, R.: Change-of-variable formula for the bi-dimensional fractional Brownian motion in Brownian time. ALEA Lat. Am. J. Probab. Math. Stat. 12(2), 597–683 (2015)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We are thankful to the referees for their careful reading of the original manuscript and for a number of suggestions. The financial support of the DFG (German Science Foundations) Research Training Group 2131 is gratefully acknowledged.

Author information

Authors and Affiliations

  1. Research Training Group 2131, Fakultät Mathematik, Technische Universität Dortmund, Dortmund, Germany

    Raghid Zeineddine

Authors
  1. Raghid Zeineddine
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Raghid Zeineddine.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zeineddine, R. Asymptotic Behavior of Weighted Power Variations of Fractional Brownian Motion in Brownian Time. J Theor Probab 31, 1539–1589 (2018). https://doi.org/10.1007/s10959-017-0749-1

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10959-017-0749-1

Keywords

Mathematics Subject Classification 2010

Search

Navigation