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Parametric Estimation of Lévy Processes

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Lévy Matters IV

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

Abstract

The main purpose of this chapter is to present some theoretical aspects of parametric estimation of Lévy processes based on high-frequency sampling, with a focus on infinite activity pure-jump models. Asymptotics for several classes of explicit estimating functions are discussed. In addition to the asymptotic normality at several rates of convergence, a uniform tail-probability estimate for statistical random fields is given. As specific cases, we discuss method of moments for the stable Lévy processes in much greater detail, with briefly mentioning locally stable Lévy processes too. Also discussed is, due to its theoretical importance, a brief review of how the classical likelihood approach works or does not, beyond the fact that the likelihood function is not explicit.

AMS Subject Classification 2000:

Primary: 60F05, 62F12, 60G51, 60G52

Secondary: 60G18

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Acknowledgements

I extend my thanks to Professor Jean Jacod and the anonymous referee for their detailed suggestions and comments, which not only brought some errors in the first draft to my attention but also led to substantial improvement in the exposition of this chapter. I am grateful to Professor Claudia Klüppelberg for her encouragement. My thanks also go to Sangji Kim, Yuma Uehara, Shoichi Eguchi, and Yusuke Shimizu for proofreading. Needless to say, all remaining errors are of my own. Some materials of this chapter are based on the joint papers with Dr. Reiichiro Kawai, to whom I thank for fruitful discussions during the works.

This work was partly supported by JSPS KAKENHI Grant Numbers 23740082, 26400204.

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    Hiroki Masuda

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Masuda, H. (2015). Parametric Estimation of Lévy Processes. In: Lévy Matters IV. Lecture Notes in Mathematics(), vol 2128. Springer, Cham. https://doi.org/10.1007/978-3-319-12373-8_3

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