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Estimation of Lévy Processes via Stochastic Programming and Kalman Filtering

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  • Mark Anthony Caruana

    (University of Malta)

Abstract

The estimation of Lévy process has received a lot of attention in recent years. Evidence of this is the extensive amount of literature concerning this problem which can be classified in two categories: the nonparametric approach, and the parametric approach. In this paper, we shall concentrate on the latter, and in particular the parameters will be estimated within a stochastic programming framework. To be more specific, the first derivative of the characteristic function and its empirical version shall be used in objective function. Furthermore, the parameter estimates are recursively estimated by making use of a modified extended Kalman filter (MEKF). Some properties of the parameter estimates are studied. Finally, a number of simulations will be carried out and the results are presented and discussed.

Suggested Citation

  • Mark Anthony Caruana, 2017. "Estimation of Lévy Processes via Stochastic Programming and Kalman Filtering," Methodology and Computing in Applied Probability, Springer, vol. 19(4), pages 1211-1225, December.
    • Handle: RePEc:spr:metcap:v:19:y:2017:i:4:d:10.1007_s11009-017-9552-9
    DOI: 10.1007/s11009-017-9552-9
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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. Vaart,A. W. van der, 2000. "Asymptotic Statistics," Cambridge Books, Cambridge University Press, number 9780521784504.
    3. Florian Ueltzhöfer & Claudia Klüppelberg, 2011. "An oracle inequality for penalised projection estimation of Lévy densities from high-frequency observations," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 23(4), pages 967-989.
    4. Denis Belomestny & Markus Reiß, 2006. "Spectral calibration of exponential Lévy models," Finance and Stochastics, Springer, vol. 10(4), pages 449-474, December.
    5. 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.
    6. Comte, F. & Genon-Catalot, V., 2009. "Nonparametric estimation for pure jump Lévy processes based on high frequency data," Stochastic Processes and their Applications, Elsevier, vol. 119(12), pages 4088-4123, December.
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