Smoothness of scale functions for spectrally negative Lévy processes

T. Chan, A. E. Kyprianou, M. Savov

Research output: Contribution to journalArticlepeer-review

68 Citations (Scopus)


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. © 2010 Springer-Verlag.

Original languageEnglish
Pages (from-to)691-708
Number of pages18
JournalProbability Theory and Related Fields
Issue number3-4
Publication statusPublished - Aug 2011


  • Mathematics Subject Classification (2000): 60G51, 91B30, 60J45

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