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On some maximal inequalities for fractional Brownian motions

Author

Listed:
  • Novikov, Alexander
  • Valkeila, Esko

Abstract

We prove some maximal inequalities for fractional Brownian motions. These extend the Burkholder-Davis-Gundy inequalities for fractional Brownian motions. The methods are based on the integral representations of fractional Brownian motions with respect to a certain Gaussian martingale in terms of beta kernels.

Suggested Citation

  • Novikov, Alexander & Valkeila, Esko, 1999. "On some maximal inequalities for fractional Brownian motions," Statistics & Probability Letters, Elsevier, vol. 44(1), pages 47-54, August.
    • Handle: RePEc:eee:stapro:v:44:y:1999:i:1:p:47-54
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    Citations

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    Cited by:

    1. Jim Gatheral & Thibault Jaisson & Mathieu Rosenbaum, 2014. "Volatility is rough," Papers 1410.3394, arXiv.org.
    2. Michael J. Klass & Ming Yang, 2012. "Maximal Inequalities for Additive Processes," Journal of Theoretical Probability, Springer, vol. 25(4), pages 981-1012, December.
    3. Yang, Ming, 2002. "Occupation times and beyond," Stochastic Processes and their Applications, Elsevier, vol. 97(1), pages 77-93, January.
    4. Orimar Sauri, 2024. "Asymptotic Error Distribution of the Euler Scheme for Fractional Stochastic Delay Differential Equations with Additive Noise," Papers 2402.08513, arXiv.org.
    5. Yulia Mishura & Esko Valkeila, 2000. "An Isometric Approach to Generalized Stochastic Integrals," Journal of Theoretical Probability, Springer, vol. 13(3), pages 673-693, July.
    6. Wood, Andrew T. A., 2001. "Acknowledgement of priority," Stochastic Processes and their Applications, Elsevier, vol. 93(2), pages 349-349, June.
    7. Yan, Litan, 2004. "Maximal inequalities for the iterated fractional integrals," Statistics & Probability Letters, Elsevier, vol. 69(1), pages 69-79, August.
    8. Slominski, Leszek & Ziemkiewicz, Bartosz, 2005. "Inequalities for the norms of integrals with respect to a fractional Brownian motion," Statistics & Probability Letters, Elsevier, vol. 73(1), pages 79-90, June.
    9. Mémin, Jean & Mishura, Yulia & Valkeila, Esko, 2001. "Inequalities for the moments of Wiener integrals with respect to a fractional Brownian motion," Statistics & Probability Letters, Elsevier, vol. 51(2), pages 197-206, January.
    10. Dzhaparidze, Kacha & van Zanten, Harry & Zareba, Pawel, 2005. "Representations of fractional Brownian motion using vibrating strings," Stochastic Processes and their Applications, Elsevier, vol. 115(12), pages 1928-1953, December.
    11. Lee, Chihoon, 2012. "Bounds on exponential moments of hitting times for reflected processes on the positive orthant," Statistics & Probability Letters, Elsevier, vol. 82(6), pages 1120-1128.
    12. Dorival Le~ao & Alberto Ohashi & Francesco Russo, 2017. "Discrete-type approximations for non-Markovian optimal stopping problems: Part I," Papers 1707.05234, arXiv.org, revised Jun 2019.

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