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Testing the characteristics of a Lévy process

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  • Reiß, Markus

Abstract

For n equidistant observations of a Lévy process at time distance Δn we consider the problem of testing hypotheses on the volatility, the jump measure and its Blumenthal–Getoor index in a non- or semiparametric manner. Asymptotically as n→∞ we allow for both, the high-frequency regime Δn=1n and the low-frequency regime Δn=1 as well as intermediate cases. The approach via the empirical characteristic function unifies existing theory and sheds new light on diverse results. Particular emphasis is given to asymptotic separation rates which reveal the complexity of these basic, but surprisingly non-standard inference questions.

Suggested Citation

  • Reiß, Markus, 2013. "Testing the characteristics of a Lévy process," Stochastic Processes and their Applications, Elsevier, vol. 123(7), pages 2808-2828.
    • Handle: RePEc:eee:spapps:v:123:y:2013:i:7:p:2808-2828
    DOI: 10.1016/j.spa.2013.03.016
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    References listed on IDEAS

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    1. Denis Belomestny & Markus Reiß, 2006. "Spectral calibration of exponential Lévy models," Finance and Stochastics, Springer, vol. 10(4), pages 449-474, December.
    2. Kappus, Johanna & Reiß, Markus, 2010. "Estimation of the characteristics of a Lévy process observed at arbitrary frequency," SFB 649 Discussion Papers 2010-015, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.
    3. Belomestny, Denis, 2011. "Spectral estimation of the Lévy density in partially observed affine models," Stochastic Processes and their Applications, Elsevier, vol. 121(6), pages 1217-1244, June.
    4. Figueroa-López, José E. & Houdré, Christian, 2009. "Small-time expansions for the transition distributions of Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 119(11), pages 3862-3889, November.
    5. Viktor Todorov & George Tauchen, 2012. "The Realized Laplace Transform of Volatility," Econometrica, Econometric Society, vol. 80(3), pages 1105-1127, May.
    6. Belomestny, Denis, 2009. "Spectral estimation of the fractional order of a Lévy process," SFB 649 Discussion Papers 2009-021, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.
    7. Shota Gugushvili, 2009. "Nonparametric estimation of the characteristic triplet of a discretely observed Lévy process," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 21(3), pages 321-343.
    8. Johanna Kappus & Markus Reiß, 2010. "Estimation of the characteristics of a Lévy process observed at arbitrary frequency," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 64(s1), pages 314-328.
    9. Nickl, Richard & Reiß, Markus, 2012. "A Donsker theorem for Lévy measures," SFB 649 Discussion Papers 2012-003, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.
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    Cited by:

    1. José E. Figueroa-López & Ruoting Gong & Yuchen Han, 2022. "Estimation of Tempered Stable Lévy Models of Infinite Variation," Methodology and Computing in Applied Probability, Springer, vol. 24(2), pages 713-747, June.
    2. Trabs, Mathias, 2015. "Quantile estimation for Lévy measures," Stochastic Processes and their Applications, Elsevier, vol. 125(9), pages 3484-3521.
    3. Jos'e E. Figueroa-L'opez & Ruoting Gong & Yuchen Han, 2021. "Estimation of Tempered Stable L\'{e}vy Models of Infinite Variation," Papers 2101.00565, arXiv.org, revised Feb 2022.

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