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A Potential Theoretic Approach to Tanaka Formula for Asymmetric Lévy Processes

  • Hiroshi Tsukada
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, 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).

Keywords

Local time Lévy process Resolvent Excessive function 

MSC 2010

60J55 60G51 60J45 

Notes

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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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Hiroshi Tsukada
    • 1
  1. 1.Graduate School of ScienceOsaka City UniversityOsakaJapan

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