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A weak law of large numbers for realised covariation in a Hilbert space setting

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  • Benth, Fred Espen
  • Schroers, Dennis
  • Veraart, Almut E.D.

Abstract

This article generalises the concept of realised covariation to Hilbert-space-valued stochastic processes. More precisely, based on high-frequency functional data, we construct an estimator of the trace-class operator-valued integrated volatility process arising in general mild solutions of Hilbert space-valued stochastic evolution equations in the sense of Da Prato and Zabczyk (2014). We prove a weak law of large numbers for this estimator, where the convergence is uniform on compacts in probability with respect to the Hilbert–Schmidt norm. In addition, we determine convergence rates for common stochastic volatility models in Hilbert spaces.

Suggested Citation

  • Benth, Fred Espen & Schroers, Dennis & Veraart, Almut E.D., 2022. "A weak law of large numbers for realised covariation in a Hilbert space setting," Stochastic Processes and their Applications, Elsevier, vol. 145(C), pages 241-268.
    • Handle: RePEc:eee:spapps:v:145:y:2022:i:c:p:241-268
    DOI: 10.1016/j.spa.2021.12.011
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    References listed on IDEAS

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    1. Igor Cialenco, 2018. "Statistical inference for SPDEs: an overview," Statistical Inference for Stochastic Processes, Springer, vol. 21(2), pages 309-329, July.
    2. Fred Espen Benth & Jūratė Šaltytė Benth & Steen Koekebakker, 2008. "Stochastic Modeling of Electricity and Related Markets," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 6811, January.
    3. Bibinger, Markus & Trabs, Mathias, 2020. "Volatility estimation for stochastic PDEs using high-frequency observations," Stochastic Processes and their Applications, Elsevier, vol. 130(5), pages 3005-3052.
    4. Ole E. Barndorff‐Nielsen & Neil Shephard, 2002. "Econometric analysis of realized volatility and its use in estimating stochastic volatility models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 64(2), pages 253-280, May.
    5. Yacine Aït-Sahalia & Jean Jacod, 2014. "High-Frequency Financial Econometrics," Economics Books, Princeton University Press, edition 1, number 10261.
    6. Jacod, Jean, 2008. "Asymptotic properties of realized power variations and related functionals of semimartingales," Stochastic Processes and their Applications, Elsevier, vol. 118(4), pages 517-559, April.
    7. Torben G. Andersen & Tim Bollerslev & Francis X. Diebold & Paul Labys, 2003. "Modeling and Forecasting Realized Volatility," Econometrica, Econometric Society, vol. 71(2), pages 579-625, March.
    8. Yacine Aït-Sahalia & Dacheng Xiu, 2019. "Principal Component Analysis of High-Frequency Data," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 114(525), pages 287-303, January.
    9. Ole E. Barndorff-Nielsen & Neil Shephard, 2004. "Econometric Analysis of Realized Covariation: High Frequency Based Covariance, Regression, and Correlation in Financial Economics," Econometrica, Econometric Society, vol. 72(3), pages 885-925, May.
    10. Yao, Fang & Muller, Hans-Georg & Wang, Jane-Ling, 2005. "Functional Data Analysis for Sparse Longitudinal Data," Journal of the American Statistical Association, American Statistical Association, vol. 100, pages 577-590, June.
    11. Corcuera, José Manuel & Hedevang, Emil & Pakkanen, Mikko S. & Podolskij, Mark, 2013. "Asymptotic theory for Brownian semi-stationary processes with application to turbulence," Stochastic Processes and their Applications, Elsevier, vol. 123(7), pages 2552-2574.
    12. Jim Gatheral & Thibault Jaisson & Mathieu Rosenbaum, 2018. "Volatility is rough," Quantitative Finance, Taylor & Francis Journals, vol. 18(6), pages 933-949, June.
    13. Fred Espen Benth & Paul Kruhner, 2014. "Representation of infinite dimensional forward price models in commodity markets," Papers 1403.4111, arXiv.org.
    14. Benth, Fred Espen & Rüdiger, Barbara & Süss, Andre, 2018. "Ornstein–Uhlenbeck processes in Hilbert space with non-Gaussian stochastic volatility," Stochastic Processes and their Applications, Elsevier, vol. 128(2), pages 461-486.
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    Cited by:

    1. Hildebrandt, Florian & Trabs, Mathias, 2023. "Nonparametric calibration for stochastic reaction–diffusion equations based on discrete observations," Stochastic Processes and their Applications, Elsevier, vol. 162(C), pages 171-217.
    2. Fred Espen Benth & Heidar Eyjolfsson, 2022. "Robustness of Hilbert space-valued stochastic volatility models," Papers 2211.16071, arXiv.org.

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