Journal of Theoretical Probability

, Volume 23, Issue 2, pp 547–564 | Cite as

Convexity and Smoothness of Scale Functions and de Finetti’s Control Problem

  • Andreas E. KyprianouEmail author
  • Víctor Rivero
  • Renming Song


We continue the recent work of Avram et al. (Ann. Appl. Probab. 17:156–180, 2007) and Loeffen (Ann. Appl. Probab., 2007) by showing that whenever the Lévy measure of a spectrally negative Lévy process has a density which is log-convex then the solution of the associated actuarial control problem of de Finetti is solved by a barrier strategy. Moreover, the level of the barrier can be identified in terms of the scale function of the underlying Lévy process. Our method appeals directly to very recent developments in the theory of potential analysis of subordinators and their application to convexity and smoothness properties of the relevant scale functions.


Potential analysis Special Bernstein function Scale functions for spectrally negative Lévy processes Control theory 

Mathematics Subject Classification (2000)

60J99 93E20 60G51 


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Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Andreas E. Kyprianou
    • 1
    Email author
  • Víctor Rivero
    • 1
    • 2
  • Renming Song
    • 3
  1. 1.Department of Mathematical SciencesUniversity of BathBathUK
  2. 2.Centro de Investigación en Matemáticas (CIMAT A.C.)GuanajuatoMexico
  3. 3.Department of MathematicsUniversity of IllinoisUrbanaUSA

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