On optimality of the barrier strategy in de Finetti's dividend problem for spectrally negative L\'{e}vy processes

TL;DR

This paper investigates the optimal dividend strategy for spectrally negative Lévy processes, providing conditions under which a barrier strategy is optimal and extending the class of processes where this strategy applies.

Contribution

It offers new sufficient conditions for the optimality of barrier strategies in de Finetti's dividend problem for spectrally negative Lévy processes, broadening previous results.

Findings

Identifies conditions for barrier strategy optimality.

Extends the class of processes with proven barrier strategy optimality.

Clarifies when barrier strategies are not optimal.

Abstract

We consider the classical optimal dividend control problem which was proposed by de Finetti [Trans. XVth Internat. Congress Actuaries 2 (1957) 433--443]. Recently Avram, Palmowski and Pistorius [Ann. Appl. Probab. 17 (2007) 156--180] studied the case when the risk process is modeled by a general spectrally negative L\'{e}vy process. We draw upon their results and give sufficient conditions under which the optimal strategy is of barrier type, thereby helping to explain the fact that this particular strategy is not optimal in general. As a consequence, we are able to extend considerably the class of processes for which the barrier strategy proves to be optimal.