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

Antoine Ayache, laurent Loosveldt, Ciprian Tudor

TL;DR

This work derives central limit theorems for modified weighted power variations of Hermite processes of arbitrary order, by isolating good increments that decompose into dominant independent components and applying Stein–Malliavin calculus to obtain explicit Wasserstein bounds. It extends prior results on modified quadratic and wavelet variations to general powers and weighted settings, and proves convergence to Wiener integrals against independent Brownian motion. The theory is then applied to asymptotically Gaussian estimators of integrated volatility in Hermite-driven (non-Gaussian) models, with numerical experiments illustrating practical performance. The results provide a rigorous foundation for statistical inference in non-Gaussian self-similar models with long-range dependence and offer practical tools for volatility estimation in such settings.

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.

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

TL;DR

This work derives central limit theorems for modified weighted power variations of Hermite processes of arbitrary order, by isolating good increments that decompose into dominant independent components and applying Stein–Malliavin calculus to obtain explicit Wasserstein bounds. It extends prior results on modified quadratic and wavelet variations to general powers and weighted settings, and proves convergence to Wiener integrals against independent Brownian motion. The theory is then applied to asymptotically Gaussian estimators of integrated volatility in Hermite-driven (non-Gaussian) models, with numerical experiments illustrating practical performance. The results provide a rigorous foundation for statistical inference in non-Gaussian self-similar models with long-range dependence and offer practical tools for volatility estimation in such settings.

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 -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.
Paper Structure (10 sections, 17 theorems, 221 equations, 5 tables)

This paper contains 10 sections, 17 theorems, 221 equations, 5 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Hermite process and its special increments
  4. The special increments of the Hermite process
  5. Strong independence
  6. The modified power variation of the Hermite process
  7. Weighted power variation of the Hermite process
  8. Volatility estimation
  9. Numerical experiments
  10. Appendix: Multiple stochastic integrals and the Malliavin derivative

Key Result

Lemma 1

For $N\geq 1$ and $l=1,2,..., 2 ^{[N ^{\gamma}]}$, let $\check{\Delta}Z_{l, N}$ be given by (dc). Then for every $N\geq 1$ and $l=1,2,..., 2 ^{[N ^{\gamma}]}$.

Theorems & Definitions (24)

  • Remark 1
  • Lemma 1
  • Lemma 2
  • Proposition 1
  • Lemma 3
  • Lemma 4
  • Proposition 2
  • Lemma 5
  • Theorem 1
  • Proposition 3
  • ...and 14 more