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Order estimate of functionals related to fractional Brownian motion

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  • Yamagishi, Hayate
  • Yoshida, Nakahiro

Abstract

Nualart and Yoshida (2019) presented a general scheme for asymptotic expansion of Skorohod integrals. However, when applying the general theory to a variation of a process related to a fractional Brownian motion, one repeatedly needs estimates of the order of Lp-norms and Sobolev norms of functionals that are complicated as a randomly weighted sum of products of multiple integrals of the fractional Brownian motion. To resolve the difficulties, in this paper, we construct a theory of exponents based on a graphical representation of the structure of the functionals, which enables us to tell an upper bound of the order of functionals by a simple rule. We also show how the exponents change by the actions of the Malliavin derivative and its projection. The exponents are useful in various studies of limit theorems as well as in applications to asymptotic expansion.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:161:y:2023:i:c:p:490-543
    DOI: 10.1016/j.spa.2023.04.014
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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. Yoshida, Nakahiro, 2013. "Martingale expansion in mixed normal limit," Stochastic Processes and their Applications, Elsevier, vol. 123(3), pages 887-933.
    3. 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.
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    Cited by:

    1. 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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