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Smoothness of scale functions for spectrally negative Lévy processes
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  • Published: 14 April 2010

Smoothness of scale functions for spectrally negative Lévy processes

  • T. Chan1,
  • A. E. Kyprianou2 &
  • M. Savov3 

Probability Theory and Related Fields volume 150, pages 691–708 (2011)Cite this article

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Abstract

Scale functions play a central role in the fluctuation theory of spectrally negative Lévy processes and often appear in the context of martingale relations. These relations are often require excursion theory rather than Itô calculus. The reason for the latter is that standard Itô calculus is only applicable to functions with a sufficient degree of smoothness and knowledge of the precise degree of smoothness of scale functions is seemingly incomplete. The aim of this article is to offer new results concerning properties of scale functions in relation to the smoothness of the underlying Lévy measure. We place particular emphasis on spectrally negative Lévy processes with a Gaussian component and processes of bounded variation. An additional motivation is the very intimate relation of scale functions to renewal functions of subordinators. The results obtained for scale functions have direct implications offering new results concerning the smoothness of such renewal functions for which there seems to be very little existing literature on this topic.

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Author information

Authors and Affiliations

  1. School of Mathematical and Computer Sciences, Heriot-Watt University, Edinburgh, EH14 4AS, UK

    T. Chan

  2. Department of Mathematical Sciences, The University of Bath, Claverton Down, Bath, BA2 7AY, UK

    A. E. Kyprianou

  3. Department of Statistics, Oxford University, 1 South Parks Road, Oxford, OX1 3TG, UK

    M. Savov

Authors
  1. T. Chan
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  2. A. E. Kyprianou
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  3. M. Savov
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Corresponding author

Correspondence to A. E. Kyprianou.

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Chan, T., Kyprianou, A.E. & Savov, M. Smoothness of scale functions for spectrally negative Lévy processes. Probab. Theory Relat. Fields 150, 691–708 (2011). https://doi.org/10.1007/s00440-010-0289-4

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  • Received: 30 March 2009

  • Revised: 02 March 2010

  • Published: 14 April 2010

  • Issue Date: August 2011

  • DOI: https://doi.org/10.1007/s00440-010-0289-4

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Mathematics Subject Classification (2000)

  • 60G51
  • 91B30
  • 60J45
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