Convexity and smoothness of scale functions and de Finetti’s control problem∗

We continue the recent work of [2] and [25] 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. AMS 2000 Mathematics Subject Classification: Primary 60J99; secondary 93E20, 60G51.