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On estimation of quadratic variation for multivariate pure jump semimartingales

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  • Heiny, Johannes
  • Podolskij, Mark

Abstract

In this paper we present the asymptotic analysis of the realised quadratic variation for multivariate symmetric β-stable Lévy processes, β∈(0,2), and certain pure jump semimartingales. The main focus is on derivation of functional limit theorems for the realised quadratic variation and its spectrum. We will show that the limiting process is a matrix-valued β-stable Lévy process when the original process is symmetric β-stable, while the limit is conditionally β-stable in case of integrals with respect to locally β-stable motions. These asymptotic results are mostly related to the work (Diop et al., 2013), which investigates the univariate version of the problem. Furthermore, we will show the implications for estimation of eigenvalues and eigenvectors of the quadratic variation matrix, which is a useful result for the principle component analysis. Finally, we propose a consistent subsampling procedure in the Lévy setting to obtain confidence regions.

Suggested Citation

  • Heiny, Johannes & Podolskij, Mark, 2021. "On estimation of quadratic variation for multivariate pure jump semimartingales," Stochastic Processes and their Applications, Elsevier, vol. 138(C), pages 234-254.
    • Handle: RePEc:eee:spapps:v:138:y:2021:i:c:p:234-254
    DOI: 10.1016/j.spa.2021.04.016
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    References listed on IDEAS

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    1. 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.
    2. Yuri Kabanov & Robert Liptser, 2006. "From Stochastic Calculus to Mathematical Finance. The Shiryaev Festschrift," Post-Print hal-00488295, HAL.
    3. Ole E. Barndorff-Nielsen, 2004. "Power and Bipower Variation with Stochastic Volatility and Jumps," Journal of Financial Econometrics, Oxford University Press, vol. 2(1), pages 1-37.
    4. Todorov, Viktor, 2019. "Nonparametric inference for the spectral measure of a bivariate pure-jump semimartingale," Stochastic Processes and their Applications, Elsevier, vol. 129(2), pages 419-451.
    5. Diop, Assane & Jacod, Jean & Todorov, Viktor, 2013. "Central Limit Theorems for approximate quadratic variations of pure jump Itô semimartingales," Stochastic Processes and their Applications, Elsevier, vol. 123(3), pages 839-886.
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    Cited by:

    1. Zhang, Yi & Zhou, Long & Chen, Yajiao & Liu, Fang, 2022. "The contagion effect of jump risk across Asian stock markets during the Covid-19 pandemic," The North American Journal of Economics and Finance, Elsevier, vol. 61(C).

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