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Lévy Matters IV pp 1–76Cite as

Estimation and Calibration of Lévy Models via Fourier Methods

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Part of the book series: Lecture Notes in Mathematics ((LEVY,volume 2128))

Abstract

In this chapter we discuss different aspects of statistical estimation for Lévy-based processes based on low-frequency observations. In particular, we consider the estimation of the Lévy triplet and the Blumenthal-Getoor index in Lévy and time-changed Lévy models. Moreover, a calibration problem in exponential Lévy models based on option data is studied. The common feature of all these statistical problems is that they can be conveniently formulated in the Fourier domain. We introduce a general spectral estimation/calibration approach that can be applied to these and many other statistical problems related to Lévy processes. On the theoretical side, we provide a comprehensive convergence analysis of the proposed algorithms and address each time the question of optimality.

AMS Subject Classification 2000:

Primary: 60G10, 60G70, 60J10

Secondary: 91B28, 91B84

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Notes

  1. 1.

    Provided through the SFB 649 “Economic Risk”, Humboldt-Universität zu Berlin.

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Acknowledgements

We are indebted to Jakob Söhl, Mathias Trabs and the reviewers for careful proofreading, all remaining mistakes are our own.

Partial financial support by the Deutsche Forschungsgemeinschaft through SFB 649 “Economic Risk”, Research Unit 1749 “Structural Inference in Statistics” and through SFB 823 “Statistical modelling of nonlinear dynamic processes”, by Laboratory for Structural Methods of Data Analysis in Predictive Modeling, MIPT, RF government grant, ag. 11.G34.31.0073 and by the International Laboratory of Qualitative Finance, NRU HSE, RF government grant, ag. 14.A12.31.0007 is gratefully acknowledged.

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Authors and Affiliations

  1. Faculty of Mathematics, Duisburg-Essen University, Thea-Leymann-Str. 9, D-45127, Essen, Germany

    Denis Belomestny

  2. National University Higher School of Economics, Moscow, Russia

    Denis Belomestny

  3. Humboldt-Universität zu Berlin, Institut für Mathematik, Unter den Linden 6, D-10099, Berlin, Germany

    Markus Reiß

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  1. Denis Belomestny
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Correspondence to Denis Belomestny .

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Belomestny, D., Reiß, M. (2015). Estimation and Calibration of Lévy Models via Fourier Methods. In: Lévy Matters IV. Lecture Notes in Mathematics(), vol 2128. Springer, Cham. https://doi.org/10.1007/978-3-319-12373-8_1

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