Convexity and smoothness of scale functions and de Finetti's control problem

Abstract

Under appropriate conditions, we obtain smoothness and convexity properties of $q$-scale functions for spectrally negative Lévy processes. Our method appeals directly to very recent developments in the theory of potential analysis of subordinators. As an application of the latter results to scale functions, we are able to continue the very recent work of \cite{APP2007} and \cite{Loe}. We strengthen their collective conclusions by showing, amongst other results, that whenever the Lévy measure has a density which is log convex then for $q>0$ the scale function $W^{(q)}$ is convex on some half line $(a^*,\infty)$ where $a^*$ is the largest value at which $W^{(q)\prime}$ attains its global minimum. As a consequence we deduce that de Finetti's classical actuarial control problem is solved by a barrier strategy where the barrier is positioned at height $a^*$.