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On the Range of Exponential Functionals of Lévy Processes

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Séminaire de Probabilités XLVIII

Part of the book series: Lecture Notes in Mathematics ((SEMPROBAB,volume 2168))

Abstract

We characterize the support of the law of the exponential functional \(\int _{0}^{\infty }e^{-\xi _{s-}}\,d\eta _{s}\) of two one-dimensional independent Lévy processes ξ and η. Further, we study the range of the mapping Φ ξ for a fixed Lévy process ξ, which maps the law of η 1 to the law of the corresponding exponential functional \(\int _{0}^{\infty }e^{-\xi _{s-}}\,d\eta _{s}\). It is shown that the range of this mapping is closed under weak convergence and in the special case of positive distributions several characterizations of laws in the range are given.

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Acknowledgements

We would like to thank the anonymous referee for valuable suggestions which helped to improve the exposition of the manuscript. Makoto Maejima’s research was partially supported by JSPS Grand-in-Aid for Science Research 22340021.

Author information

Authors and Affiliations

  1. Institut für Mathematische Stochastik, Technische Universität Dresden, 01062, Dresden, Germany

    Anita Behme

  2. Zentrum Mathematik, Technische Universität München, Boltzmannstraße 3, 85748, Garching bei München, Germany

    Anita Behme

  3. Institut für Mathematische Stochastik, Technische Universität Braunschweig, 38106, Braunschweig, Germany

    Alexander Lindner

  4. Institute of Mathematical Finance, Ulm University, 89081, Ulm, Germany

    Alexander Lindner

  5. Department of Mathematics, Keio University, Hiyoshi, Yokohama, 223-8522, Japan

    Makoto Maejima

Authors
  1. Anita Behme
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  2. Alexander Lindner
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  3. Makoto Maejima
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Corresponding author

Correspondence to Anita Behme .

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Editors and Affiliations

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

    Catherine Donati-Martin

  2. Campus scientifique, IECL, Vandoeuvre-les-Nancy CEDEX, France

    Antoine Lejay

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

    Alain Rouault

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Behme, A., Lindner, A., Maejima, M. (2016). On the Range of Exponential Functionals of Lévy Processes. In: Donati-Martin, C., Lejay, A., Rouault, A. (eds) Séminaire de Probabilités XLVIII. Lecture Notes in Mathematics(), vol 2168. Springer, Cham. https://doi.org/10.1007/978-3-319-44465-9_10

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