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A Potential Theoretic Approach to Tanaka Formula for Asymmetric Lévy Processes

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Séminaire de Probabilités XLIX

Part of the book series: Lecture Notes in Mathematics ((SEMPROBAB,volume 2215))

Abstract

In this paper, we shall introduce the Tanaka formula from viewpoint of the Doob–Meyer decomposition. For symmetric Lévy processes, if the local time exists, Salminen and Yor (Tanaka formula for symmetric Lévy processes. In: Séminaire de Probabilités XL. Lecture notes in mathematics, vol. 1899, Springer, Berlin, pp. 265–285, 2007) obtained the Tanaka formula by using the potential theoretic techniques. On the other hand, for strictly stable processes with index α ∈ (1, 2), we studied the Tanaka formula by using Itô’s stochastic calculus and the Fourier analysis. In this paper, we study the Tanaka formula for asymmetric Lévy processes via the potential theoretic approach. We give several examples for important processes. Our approach also gives the invariant excessive function with respect to the killed process in the case of asymmetric Lévy processes, and it generalized the result in Yano (J Math Ind 5(A):17–24, 2013).

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Acknowledgements

I would like to thank Professor Atsushi Takeuchi of Osaka City University and Professor Kouji Yano of Kyoto University for their valuable advice.

The author was partially supported by JSPS-MAEDI Sakura program.

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Authors and Affiliations

  1. Graduate School of Science, Osaka City University, Osaka, Japan

    Hiroshi Tsukada

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  1. Hiroshi Tsukada
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Editors and Affiliations

  1. Laboratoire de Mathématiques de Versailles, Université Versailles Saint-Quentin, Versailles, France

    Catherine Donati-Martin

  2. Institut Elie Cartan de Lorraine, Vandoeuvre-les-Nancy, France

    Antoine Lejay

  3. Laboratoire de Mathématiques de Versailles, Université Versailles Saint-Quentin, Versailles, France

    Alain Rouault

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Tsukada, H. (2018). A Potential Theoretic Approach to Tanaka Formula for Asymmetric Lévy Processes. In: Donati-Martin, C., Lejay, A., Rouault, A. (eds) Séminaire de Probabilités XLIX. Lecture Notes in Mathematics(), vol 2215. Springer, Cham. https://doi.org/10.1007/978-3-319-92420-5_15

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