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

Hermite process; multiple Wiener-Itˆo integrals; Stein-Malliavin calculus; asymptotic normality; central limit theorem; integrated volatility estimation; strong consistency

Abstract :

[en] 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.

Disciplines :

Mathematics

Author, co-author :

Ayache, Antoine

Loosveldt, Laurent ; Université de Liège - ULiège > Département de mathématique > Probabilités - Analyse stochastique

Tudor, Ciprian

Language :

English

Title :

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

Publication date :

September 2026

Journal title :

Stochastic Processes and Their Applications

ISSN :

0304-4149

eISSN :

1879-209X

Publisher :

Elsevier, Amsterdam, Netherlands

Volume :

199

Peer reviewed :

Peer Reviewed verified by ORBi

Funders :

F.R.S.-FNRS - Fonds de la Recherche Scientifique

Funding number :

J.0136.26

Available on ORBi :

since 06 January 2026

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Bibliography

Pipiras, V., Taqqu, M., Long-range dependence and self-similarity. Cambridge Series in Statistical and Probabilistic Mathematics, 2017, Cambridge University Press.
Tudor, C.A., Non-Gaussian selfsimilar stochastic processes. SpringerBriefs in Probability and Statistics, 2023, Springer, Cham.
Chronopoulou, A., Tudor, C.A., Viens, F., Self-similarity parameter estimation and reproduction property for non-Gaussian Hermite processes. Commun. Stoch. Anal. 5:1 (2011), 161–185.
Tudor, C.A., Viens, F., Variations and estimators through Malliavin calculus. Ann. Probab. 37 (2008), 2093–2134.
Bardet, J.-M., Tudor, C.A., A wavelet analysis of the rosenblatt process: chaos expansion and estimation of the self-similarity parameter. Stoch. Process. Appl. 120:12 (2010), 2331–2362.
Clausel, M., Roueß, F., Taqqu, M.S., Tudor, C.A., Wavelet estimation of the long memory parameter for Hermite polynomial of Gaussian processes. ESAIM. Probab. Stat. 18 (2014), 42–76.
Es-Sebaiy, K., Tudor, C.A., Noncentral limit theorems for the cubic variation of a class of self-similar stochastic processes. Teor. Veroyatn. Primen. 55:3 (2010), 507–529 Translation in Theory Probab. Appl. 55 (2011), no. 3, 411–431.
Chronopoulou, A., Viens, F., Tudor, C.A., Variations and hurst index estimation for a rosenblatt process using longer filters. Electron. J. Stat. 3 (2009), 1393–1435.
Nourdin, I., Tran, T.T.D., Statistical inference for Vasicek-type model driven by Hermite processes. Stochastic Process. Appl. 129:10 (2019), 3774–3791.
Diu Tran, T.T., Non-central limit theorems for quadratic functionals of Hermite-driven long memory moving average processes. Stoch. Dyn., 18(4), 2018, 18 1850028.
Ayache, A., Tudor, C.A., Asymptotic normality for a modified quadratic variation of the Hermite process. Bernoulli 30:2 (2024), 1154–1176.
Ayache, A., Lower bound for local oscillations of Hermite processes. Stochastic Process. Appl. 130:8 (2020), 4593–4607.
Loosveldt, L., Tudor, C.A., Modified wavelet variation for the Hermite processes. Electron. J. Stat. 19:1 (2025), 2411–2455.
Barndorff-Nielsen, O., Corcuera, J.M., Podolskij, M., Power variation for Gaussian processes with stationary increments. Stochastic Process. Appl. 119 (2009), 1845–1865.
Barndorff-Nielsen, O., Graversen, S., Jacod, J., Podolskij, M., Shephard, N., A central limit theorem for realised power and bipower variations of continuous semimartingales. From Stochastic Calculus to Mathematical Finance: The Shiryaev Festschrift, 2006, Springer, Berlin, 33–68.
Podolskij, M., Vetter, M., Estimation of volatility functionals in the simultaneous presence of microstructure noise and jumps. Bernoulli 15:3 (2009), 634–658.
Jacod, J., Estimation of volatility in a high-frequency setting: a short review. Decis. Econ. Finance 42:2 (2019), 351–385.
Corcuera, J.M., Nualart, D., Woerner, J., Power variation of some integral fractional processes. Bernoulli 12:4 (2006), 713–735.
Ustunel, A.S., Zakai, M., On independence and conditioning on wiener space. Ann. Probab. 4 (1989), 1441–1453.
Bourguin, S., Diez, C.-P., Tudor, C.A., Limiting behavior of large correlated wishart matrices with chaotic entries. Bernoulli 27:2 (2021), 1077–1102.
Nourdin, I., Peccati, G., Normal Approximations with Malliavin Calculus From Stein's Method to Universality. 2012, Cambridge University Press.
Barndorff-Nielsen, O., Shephard, N., Econometric analysis of realised volatility and its use in estimating stochastic volatility models. J. Roy. Statist. Soc. Ser. B 64 (2002), 253–280.
Lyons, T.J., Caruana, M., Lévy, T., Differential Equations Driven by Rough Paths. Lecture Notes in Mathematics, 1908, 2007, Springer.
Ayache, A., Hamonier, J., Loosveldt, L., Numerical simulation of generalized Hermite processes. Ann. Appl. Probab., 2026 In press.
Ayache, A., Hamonier, J., Loosveldt, L., Wavelet-type expansion of generalized Hermite processes with rate of convergence. Constr. Approx. 61:3 (2025), 535–601.
Nualart, D., Malliavin Calculus and Related Topics. second edition, 2006, Springer.
Meyer, Y., Sellan, F., Taqqu, M.S., Wavelets, generalized white noise and fractional integration: the synthesis of fractional Brownian motion. J. Fourier Anal. Appl. 5:5 (1999), 465–494.
Lemarié-Rieusset, P.G., Meyer, Y., Ondelettes et bases hilbertiennes. Rev. Matem/’Atica Iberoam. 2 (1986), 1–18.
Davies, R.B., Harte, D.S., Tests for hurst effect. Biometrika 74 (1987), 95–101.
Wood, A.T.A., Chan, G., Simulation of stationary Gaussian processes in [0, 1]d. J. Comput. Graph. Stat. 3 (1994), 409–432.