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

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 L,vy measure of a spectrally negative L,vy 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 L,vy 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.