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

Author

Listed:
  • Andreas E. Kyprianou

    (University of Bath)

  • Víctor Rivero

    (University of Bath
    Centro de Investigación en Matemáticas (CIMAT A.C.))

  • Renming Song

    (University of Illinois)

Abstract

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.

Suggested Citation

  • Andreas E. Kyprianou & Víctor Rivero & Renming Song, 2010. "Convexity and Smoothness of Scale Functions and de Finetti’s Control Problem," Journal of Theoretical Probability, Springer, vol. 23(2), pages 547-564, June.
    • Handle: RePEc:spr:jotpro:v:23:y:2010:i:2:d:10.1007_s10959-009-0220-z
    DOI: 10.1007/s10959-009-0220-z
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    References listed on IDEAS

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    1. Pablo Azcue & Nora Muler, 2005. "Optimal Reinsurance And Dividend Distribution Policies In The Cramér‐Lundberg Model," Mathematical Finance, Wiley Blackwell, vol. 15(2), pages 261-308, April.
    2. Renming Song & Zoran Vondraček, 2006. "Potential Theory of Special Subordinators and Subordinate Killed Stable Processes," Journal of Theoretical Probability, Springer, vol. 19(4), pages 817-847, December.
    3. Florin Avram & Zbigniew Palmowski & Martijn R. Pistorius, 2007. "On the optimal dividend problem for a spectrally negative L\'{e}vy process," Papers math/0702893, arXiv.org.
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    Cited by:

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    2. Amaury Lambert & Florian Simatos, 2015. "Asymptotic Behavior of Local Times of Compound Poisson Processes with Drift in the Infinite Variance Case," Journal of Theoretical Probability, Springer, vol. 28(1), pages 41-91, March.

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