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

TL;DR

This paper investigates the smoothness and convexity properties of q-scale functions for spectrally negative Lévy processes, applying recent potential analysis results to extend previous work and solve de Finetti's control problem with barrier strategies.

Contribution

It establishes new convexity and smoothness results for scale functions under certain conditions, advancing the theoretical understanding and application to actuarial control problems.

Findings

Scale functions are convex on a half-line when the Lévy measure density is log convex.

The barrier strategy solves de Finetti's control problem with the barrier at the point of minimum derivative.

The approach leverages recent developments in potential analysis of subordinators.

Abstract

Under appropriate conditions, we obtain smoothness and convexity properties of $q$-scale functions for spectrally negative L\'evy 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 L\'evy 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 $a^*$.