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Inference for Observations of Integrated Diffusion Processes

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  • Susanne Ditlevsen
  • Michael Sørensen

Abstract

Estimation of parameters in diffusion models is investigated when the observations are integrals over intervals of the process with respect to some weight function. This type of observations can, for example, be obtained when the process is observed after passage through an electronic filter. Another example is provided by the ice-core data on oxygen isotopes used to investigate paleo-temperatures. Finally, such data play a role in connection with the stochastic volatility models of finance. The integrated process is not a Markov process. Therefore, prediction-based estimating functions are applied to estimate parameters in the underlying diffusion model. The estimators are shown to be consistent and asymptotically normal. The theory developed in the paper also applies to integrals of processes other than diffusions. The method is applied to inference based on integrated data from Ornstein-Uhlenbeck processes and from the Cox-Ingersoll-Ross model, for both of which an explicit optimal estimating function is found. Copyright 2004 Board of the Foundation of the Scandinavian Journal of Statistics..

Suggested Citation

  • Susanne Ditlevsen & Michael Sørensen, 2004. "Inference for Observations of Integrated Diffusion Processes," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 31(3), pages 417-429.
  • Handle: RePEc:bla:scjsta:v:31:y:2004:i:3:p:417-429
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    File URL: http://www.blackwell-synergy.com/doi/abs/10.1111/j.1467-9469.2004.02_023.x
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    Citations

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    Cited by:

    1. Julie Lyng Forman & Michael Sørensen, 2008. "The Pearson Diffusions: A Class of Statistically Tractable Diffusion Processes," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 35(3), pages 438-465.
    2. Nicolau, João, 2008. "Modeling financial time series through second-order stochastic differential equations," Statistics & Probability Letters, Elsevier, vol. 78(16), pages 2700-2704, November.
    3. Samson, Adeline & Thieullen, Michèle, 2012. "A contrast estimator for completely or partially observed hypoelliptic diffusion," Stochastic Processes and their Applications, Elsevier, vol. 122(7), pages 2521-2552.
    4. Blanke, Delphine & Vial, Céline, 2008. "Assessing the number of mean square derivatives of a Gaussian process," Stochastic Processes and their Applications, Elsevier, vol. 118(10), pages 1852-1869, October.
    5. Friedrich Hubalek & Petra Posedel, 2008. "Asymptotic analysis for a simple explicit estimator in Barndorff-Nielsen and Shephard stochastic volatility models," Papers 0807.3479, arXiv.org.
    6. Comte, F. & Genon-Catalot, V. & Rozenholc, Y., 2009. "Nonparametric adaptive estimation for integrated diffusions," Stochastic Processes and their Applications, Elsevier, vol. 119(3), pages 811-834, March.
    7. Jean Jacod & Mark Podolskij, 2012. "A Test for the Rank of the Volatility Process: The Random Perturbation Approach," Global COE Hi-Stat Discussion Paper Series gd12-268, Institute of Economic Research, Hitotsubashi University.
    8. Song, Yuping & Lin, Zhengyan, 2013. "Empirical likelihood inference for the second-order jump-diffusion model," Statistics & Probability Letters, Elsevier, vol. 83(1), pages 184-195.
    9. Jean Jacod & Mark Podolskij, 2012. "A test for the rank of the volatility process: the random perturbation approach," CREATES Research Papers 2012-57, Department of Economics and Business Economics, Aarhus University.
    10. Yunyan Wang & Lixin Zhang & Mingtian Tang, 2012. "Re-weighted functional estimation of second-order diffusion processes," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 75(8), pages 1129-1151, November.

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