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New and refined bounds for expected maxima of fractional Brownian motion

Author

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  • Borovkov, Konstantin
  • Mishura, Yuliya
  • Novikov, Alexander
  • Zhitlukhin, Mikhail

Abstract

For the fractional Brownian motion BH with the Hurst parameter value H in (0,1∕2), we derive new upper and lower bounds for the difference between the expectations of the maximum of BH over [0,1] and the maximum of BH over the discrete set of values in−1, i=1,…,n. We use these results to improve our earlier upper bounds for the expectation of the maximum of BH over [0,1] and derive new upper bounds for Pickands’ constant.

Suggested Citation

  • Borovkov, Konstantin & Mishura, Yuliya & Novikov, Alexander & Zhitlukhin, Mikhail, 2018. "New and refined bounds for expected maxima of fractional Brownian motion," Statistics & Probability Letters, Elsevier, vol. 137(C), pages 142-147.
    • Handle: RePEc:eee:stapro:v:137:y:2018:i:c:p:142-147
    DOI: 10.1016/j.spl.2018.01.025
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    References listed on IDEAS

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    1. Christian Bayer & Peter Friz & Jim Gatheral, 2016. "Pricing under rough volatility," Quantitative Finance, Taylor & Francis Journals, vol. 16(6), pages 887-904, June.
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    Cited by:

    1. Krzysztof Bisewski & Krzysztof Dȩbicki & Tomasz Rolski, 2022. "Derivative of the expected supremum of fractional Brownian motion at $$H=1$$ H = 1," Queueing Systems: Theory and Applications, Springer, vol. 102(1), pages 53-68, October.
    2. Krzysztof Bisewski & Grigori Jasnovidov, 2023. "On the speed of convergence of Piterbarg constants," Queueing Systems: Theory and Applications, Springer, vol. 105(1), pages 129-137, October.
    3. Pérez-Cendejas, Ulises & Pérez-Suárez, Gerardo, 2024. "Stochastic ordering for hitting times of fractional Brownian motions," Statistics & Probability Letters, Elsevier, vol. 208(C).

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