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Convexity and smoothness of scale functions and de Finetti’s control problem. (English) Zbl 1188.93115
Summary: We continue the recent work of F. Avram, A. E. Kyprianou and M. R. Pistorius [Ann. Appl. Probab. 14, No. 1, 215–238 (2004; Zbl 1042.60023)] and R. L. Loeffen [Ann. Appl. Probab. 18, No. 5, 1669–1680 (2008; Zbl 1152.60344)] 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.

MSC:
93E20 Optimal stochastic control
60J99 Markov processes
60G51 Processes with independent increments; Lévy processes
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