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Asymptotic expansion of the quadratic variation of a mixed fractional Brownian motion

Author

Listed:
  • Ciprian A. Tudor

    (Université de Lille 1)

  • Nakahiro Yoshida

    (University of Tokyo
    CREST, Japan Science and Technology Agency
    The Institute of Statistical Mathematics)

Abstract

We obtain the high-order asymptotic expansion for the distribution of the quadratic variation of the mixed fractional Brownian motion, which is defined as the sum of a Brownian motion and an independent fractional Brownian motion. Our approach is based on the analysis of the cumulants of this sequence. We show that both the Brownian and fractional Brownian parts contribute to the asymptotic expansion.

Suggested Citation

  • Ciprian A. Tudor & Nakahiro Yoshida, 2020. "Asymptotic expansion of the quadratic variation of a mixed fractional Brownian motion," Statistical Inference for Stochastic Processes, Springer, vol. 23(2), pages 435-463, July.
    • Handle: RePEc:spr:sistpr:v:23:y:2020:i:2:d:10.1007_s11203-020-09220-6
    DOI: 10.1007/s11203-020-09220-6
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    References listed on IDEAS

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    1. Podolskij, Mark & Veliyev, Bezirgen & Yoshida, Nakahiro, 2017. "Edgeworth expansion for the pre-averaging estimator," Stochastic Processes and their Applications, Elsevier, vol. 127(11), pages 3558-3595.
    2. Breuer, Péter & Major, Péter, 1983. "Central limit theorems for non-linear functionals of Gaussian fields," Journal of Multivariate Analysis, Elsevier, vol. 13(3), pages 425-441, September.
    3. Mark Podolskij & Bezirgen Veliyev & Nakahiro Yoshida, 2018. "Edgeworth expansion for Euler approximation of continuous diffusion processes," CREATES Research Papers 2018-28, Department of Economics and Business Economics, Aarhus University.
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    Cited by:

    1. Yamagishi, Hayate & Yoshida, Nakahiro, 2023. "Order estimate of functionals related to fractional Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 161(C), pages 490-543.
    2. Yoshida, Nakahiro, 2023. "Asymptotic expansion and estimates of Wiener functionals," Stochastic Processes and their Applications, Elsevier, vol. 157(C), pages 176-248.
    3. Tudor, Ciprian A. & Yoshida, Nakahiro, 2023. "High order asymptotic expansion for Wiener functionals," Stochastic Processes and their Applications, Elsevier, vol. 164(C), pages 443-492.
    4. Yamagishi, Hayate & Yoshida, Nakahiro, 2024. "Asymptotic expansion of the quadratic variation of fractional stochastic differential equation," Stochastic Processes and their Applications, Elsevier, vol. 175(C).

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