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Understanding limit theorems for semimartingales: a short survey

Author

Listed:
  • Mark Podolskij

    () (ETH Zürich and CREATES)

  • Mathias Vetter

    () (Ruhr-University of Bochum)

Abstract

This paper presents a short survey on limit theorems for certain functionals of semimartingales, which are observed at high frequency. Our aim is to explain the main ideas of the theory to a broader audience. We introduce the concept of stable convergence, which is crucial for our purpose. We show some laws of large numbers (for the continuous and the discontinuous case) that are the most interesting from a practical point of view, and demonstrate the associated stable central limit theorems. Moreover, we state a simple sketch of the proofs and give some examples.

Suggested Citation

  • Mark Podolskij & Mathias Vetter, 2009. "Understanding limit theorems for semimartingales: a short survey," CREATES Research Papers 2009-47, Department of Economics and Business Economics, Aarhus University.
  • Handle: RePEc:aah:create:2009-47
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    File URL: ftp://ftp.econ.au.dk/creates/rp/09/rp09_47.pdf
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    References listed on IDEAS

    as
    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. Ole BARNDORFF-NIELSEN & Svend Erik GRAVERSEN & Jean JACOD & Mark PODOLSKIJ & Neil SHEPHARD, 2004. "A Central Limit Theorem for Realised Power and Bipower Variations of Continuous Semimartingales," OFRC Working Papers Series 2004fe21, Oxford Financial Research Centre.
    3. Barndorff-Nielsen, Ole E. & Shephard, Neil & Winkel, Matthias, 2006. "Limit theorems for multipower variation in the presence of jumps," Stochastic Processes and their Applications, Elsevier, vol. 116(5), pages 796-806, May.
    4. Ole E. Barndorff-Nielsen & Neil Shephard, 2006. "Econometrics of Testing for Jumps in Financial Economics Using Bipower Variation," Journal of Financial Econometrics, Society for Financial Econometrics, vol. 4(1), pages 1-30.
    5. Ole E. Barndorff-Nielsen, 2004. "Power and Bipower Variation with Stochastic Volatility and Jumps," Journal of Financial Econometrics, Society for Financial Econometrics, vol. 2(1), pages 1-37.
    6. Kinnebrock, Silja & Podolskij, Mark, 2008. "A note on the central limit theorem for bipower variation of general functions," Stochastic Processes and their Applications, Elsevier, vol. 118(6), pages 1056-1070, June.
    7. Holger Dette & Mark Podolskij & Mathias Vetter, 2006. "Estimation of Integrated Volatility in Continuous-Time Financial Models with Applications to Goodness-of-Fit Testing," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 33(2), pages 259-278.
    8. Vetter, Mathias, 2010. "Limit theorems for bipower variation of semimartingales," Stochastic Processes and their Applications, Elsevier, vol. 120(1), pages 22-38, January.
    Full references (including those not matched with items on IDEAS)

    Citations

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    Cited by:

    1. Bibinger, Markus, 2012. "An estimator for the quadratic covariation of asynchronously observed Itô processes with noise: Asymptotic distribution theory," Stochastic Processes and their Applications, Elsevier, vol. 122(6), pages 2411-2453.
    2. Randolf Altmeyer & Markus Bibinger, 2014. "Functional stable limit theorems for efficient spectral covolatility estimators," SFB 649 Discussion Papers SFB649DP2014-005, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
    3. Ulrich Hounyo, 2014. "Bootstrapping integrated covariance matrix estimators in noisy jump-diffusion models with non-synchronous trading," CREATES Research Papers 2014-35, Department of Economics and Business Economics, Aarhus University.
    4. Hounyo, Ulrich, 2017. "Bootstrapping integrated covariance matrix estimators in noisy jump–diffusion models with non-synchronous trading," Journal of Econometrics, Elsevier, vol. 197(1), pages 130-152.
    5. Dovonon, Prosper & Gonçalves, Sílvia & Meddahi, Nour, 2013. "Bootstrapping realized multivariate volatility measures," Journal of Econometrics, Elsevier, vol. 172(1), pages 49-65.
    6. Per A. Mykland & Neil Shephard & Kevin Sheppard, 2012. "Efficient and feasible inference for the components of financial variation using blocked multipower variation," Economics Papers 2012-W02, Economics Group, Nuffield College, University of Oxford.
    7. Yuta Koike, 2017. "Time endogeneity and an optimal weight function in pre-averaging covariance estimation," Statistical Inference for Stochastic Processes, Springer, vol. 20(1), pages 15-56, April.
    8. Zu, Yang & Peter Boswijk, H., 2014. "Estimating spot volatility with high-frequency financial data," Journal of Econometrics, Elsevier, vol. 181(2), pages 117-135.
    9. Markus Bibinger & Mathias Vetter, 2015. "Estimating the quadratic covariation of an asynchronously observed semimartingale with jumps," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 67(4), pages 707-743, August.
    10. Mark Podolskij & Katrin Wasmuth, 2013. "Goodness-of-fit testing for fractional diffusions," Statistical Inference for Stochastic Processes, Springer, vol. 16(2), pages 147-159, July.
    11. Imma Valentina Curato, 2012. "Asymptotics for the Fourier estimators of the volatility of volatility and the leverage," Working Papers - Mathematical Economics 2012-11, Universita' degli Studi di Firenze, Dipartimento di Scienze per l'Economia e l'Impresa.
    12. Altmeyer, Randolf & Bibinger, Markus, 2015. "Functional stable limit theorems for quasi-efficient spectral covolatility estimators," Stochastic Processes and their Applications, Elsevier, vol. 125(12), pages 4556-4600.
    13. José Manuel Corcuera & Emil Hedevang & Mikko S. Pakkanen & Mark Podolskij, 2012. "Asymptotic theory for Brownian semi-stationary processes with application to turbulence," CREATES Research Papers 2012-52, Department of Economics and Business Economics, Aarhus University.
    14. Mark Podolskij & Christian Schmidt & Mathias Vetter, 2015. "On U- and V-statistics for discontinuous Itô semimartingale," CREATES Research Papers 2015-52, Department of Economics and Business Economics, Aarhus University.
    15. Rasmus Tangsgaard Varneskov, 2011. "Flat-Top Realized Kernel Estimation of Quadratic Covariation with Non-Synchronous and Noisy Asset Prices," CREATES Research Papers 2011-35, Department of Economics and Business Economics, Aarhus University.
    16. repec:eee:jfinec:v:126:y:2017:i:3:p:563-591 is not listed on IDEAS
    17. Markus Bibinger & Mathias Vetter, 2013. "Estimating the quadratic covariation of an asynchronously observed semimartingale with jumps," SFB 649 Discussion Papers SFB649DP2013-029, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
    18. Caporin, Massimiliano & Kolokolov, Aleksey & Renò, Roberto, 2017. "Systemic co-jumps," Journal of Financial Economics, Elsevier, vol. 126(3), pages 563-591.
      • Caporin, Massimiliano & Kolokolov, Alexey & Renò, Roberto, 2016. "Systemic co-jumps," SAFE Working Paper Series 149, Research Center SAFE - Sustainable Architecture for Finance in Europe, Goethe University Frankfurt.
    19. Andreas Basse-O'Connor & Raphaël Lachièze-Rey & Mark Podolskij, 2015. "Limit theorems for stationary increments Lévy driven moving averages," CREATES Research Papers 2015-56, Department of Economics and Business Economics, Aarhus University.
    20. 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.
    21. Claudio Heinrich & Mark Podolskij, 2014. "On spectral distribution of high dimensional covariation matrices," CREATES Research Papers 2014-54, Department of Economics and Business Economics, Aarhus University.
    22. Anders Bredahl Kock & Timo Teräsvirta, 2010. "Forecasting with nonlinear time series models," CREATES Research Papers 2010-01, Department of Economics and Business Economics, Aarhus University.
    23. Andreas Basse-O'Connor & Mark Podolskij, 2015. "On critical cases in limit theory for stationary increments Lévy driven moving averages," CREATES Research Papers 2015-57, Department of Economics and Business Economics, Aarhus University.

    More about this item

    Keywords

    central limit theorem; high frequency observations; semimartingale; stable convergence;

    JEL classification:

    • C10 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - General
    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General

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