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Modified weighted power variations of the Hermite process and applications to integrated volatility

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  • Ayache, A.
  • Loosveldt, L.
  • Tudor, C.A.

Abstract

We study the asymptotic behaviour of modified weighted power variations of the Hermite process of arbitrary order. By selecting suitable “good” increments and exploiting their decomposition into dominant independent components, we establish a central limit theorem for weighted p-variations using tools from Stein-Malliavin calculus. Our results extend previous works on modified quadratic and wavelet-based variations to general powers and to weighted settings, with explicit bounds in Wasserstein distance. We further apply these limit theorems to construct asymptotically Gaussian estimators of integrated volatility in Hermite-driven models, thereby extending fBm-based methods to non-Gaussian settings. The last part of our work contains numerical simulations which illustrate the practical performance of the proposed estimators.

Suggested Citation

  • Ayache, A. & Loosveldt, L. & Tudor, C.A., 2026. "Modified weighted power variations of the Hermite process and applications to integrated volatility," Stochastic Processes and their Applications, Elsevier, vol. 199(C).
    • Handle: RePEc:eee:spapps:v:199:y:2026:i:c:s0304414926001092
    DOI: 10.1016/j.spa.2026.104977
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    References listed on IDEAS

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    1. Jean Jacod, 2019. "Estimation of volatility in a high-frequency setting: a short review," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 42(2), pages 351-385, December.
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    5. Barndorff-Nielsen, Ole E. & Corcuera, José Manuel & Podolskij, Mark, 2009. "Power variation for Gaussian processes with stationary increments," Stochastic Processes and their Applications, Elsevier, vol. 119(6), pages 1845-1865, June.
    6. Bardet, J.-M. & Tudor, C.A., 2010. "A wavelet analysis of the Rosenblatt process: Chaos expansion and estimation of the self-similarity parameter," Stochastic Processes and their Applications, Elsevier, vol. 120(12), pages 2331-2362, December.
    7. Nourdin, Ivan & Diu Tran, T.T., 2019. "Statistical inference for Vasicek-type model driven by Hermite processes," Stochastic Processes and their Applications, Elsevier, vol. 129(10), pages 3774-3791.
    8. Ole E. Barndorff–Nielsen & Svend Erik Graversen & Jean Jacod & Mark Podolskij & Neil Shephard, 2006. "A Central Limit Theorem for Realised Power and Bipower Variations of Continuous Semimartingales," Springer Books, in: From Stochastic Calculus to Mathematical Finance, pages 33-68, Springer.
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