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

Motivated by a classical control problem from actuarial mathematics, we study smoothness and convexity properties of $q$-scale functions for spectrally negative L\'evy processes. Continuing from the very recent work of \cite{APP2007} and \cite{Loe} we strengthen their collective conclusions by showing, amongst other results, that whenever the L\'evy measure has a non-decreasing 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 $a^*$.