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Fisher's Information for Discretely Sampled Lévy Processes

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  • Yacine Aït-Sahalia
  • Jean Jacod

Abstract

This paper studies the asymptotic behavior of Fisher's information for a Lévy process discretely sampled at an increasing frequency. As a result, we derive the optimal rates of convergence of efficient estimators of the different parameters of the process and show that the rates are often nonstandard and differ across parameters. We also show that it is possible to distinguish the continuous part of the process from its jumps part, and even different types of jumps from one another. Copyright Copyright 2008 by The Econometric Society.

Suggested Citation

  • Yacine Aït-Sahalia & Jean Jacod, 2008. "Fisher's Information for Discretely Sampled Lévy Processes," Econometrica, Econometric Society, vol. 76(4), pages 727-761, July.
    • Handle: RePEc:ecm:emetrp:v:76:y:2008:i:4:p:727-761
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    File URL: http://hdl.handle.net/10.1111/j.1468-0262.2008.00858.x
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    Cited by:

    1. Mancini, Cecilia, 2011. "The speed of convergence of the Threshold estimator of integrated variance," Stochastic Processes and their Applications, Elsevier, vol. 121(4), pages 845-855, April.
    2. Zhao, Zhibiao & Wu, Wei Biao, 2009. "Nonparametric inference of discretely sampled stable Lévy processes," Journal of Econometrics, Elsevier, vol. 153(1), pages 83-92, November.
    3. Blanke, D. & Bosq, D., 2016. "Detecting and estimating intensity of jumps for discretely observed ARMAD(1,1) processes," Journal of Multivariate Analysis, Elsevier, vol. 146(C), pages 119-137.
    4. Alexandre Brouste & Hiroki Masuda, 2018. "Efficient estimation of stable Lévy process with symmetric jumps," Statistical Inference for Stochastic Processes, Springer, vol. 21(2), pages 289-307, July.
    5. Antoine, Bertille & Renault, Eric, 2012. "Efficient minimum distance estimation with multiple rates of convergence," Journal of Econometrics, Elsevier, vol. 170(2), pages 350-367.
    6. Todorov, Viktor, 2013. "Power variation from second order differences for pure jump semimartingales," Stochastic Processes and their Applications, Elsevier, vol. 123(7), pages 2829-2850.
    7. Lee, Suzanne S. & Hannig, Jan, 2010. "Detecting jumps from Lévy jump diffusion processes," Journal of Financial Economics, Elsevier, vol. 96(2), pages 271-290, May.
    8. Todorov, Viktor & Tauchen, George & Grynkiv, Iaryna, 2011. "Realized Laplace transforms for estimation of jump diffusive volatility models," Journal of Econometrics, Elsevier, vol. 164(2), pages 367-381, October.
    9. Clément, Emmanuelle & Gloter, Arnaud, 2015. "Local Asymptotic Mixed Normality property for discretely observed stochastic differential equations driven by stable Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 125(6), pages 2316-2352.
    10. Reiichiro Kawai, 2013. "Local Asymptotic Normality Property for Ornstein–Uhlenbeck Processes with Jumps Under Discrete Sampling," Journal of Theoretical Probability, Springer, vol. 26(4), pages 932-967, December.
    11. Garcia, René & Renault, Eric & Veredas, David, 2011. "Estimation of stable distributions by indirect inference," Journal of Econometrics, Elsevier, vol. 161(2), pages 325-337, April.
    12. Kawai, Reiichiro & Masuda, Hiroki, 2011. "On the local asymptotic behavior of the likelihood function for Meixner Lévy processes under high-frequency sampling," Statistics & Probability Letters, Elsevier, vol. 81(4), pages 460-469, April.
    13. Mykland, Per A. & Zhang, Lan, 2016. "Between data cleaning and inference: Pre-averaging and robust estimators of the efficient price," Journal of Econometrics, Elsevier, vol. 194(2), pages 242-262.
    14. Chen, Song X. & Delaigle, Aurore & Hall, Peter, 2010. "Nonparametric estimation for a class of Lévy processes," Journal of Econometrics, Elsevier, vol. 157(2), pages 257-271, August.

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