On optimal dividends in the dual model

TL;DR

This paper analyzes the optimal dividend payment strategies within the dual model using spectrally positive Lévy processes, establishing barrier strategies' optimality and characterizing the optimal barrier explicitly.

Contribution

It provides a concise exposition proving the optimality of barrier strategies for all spectrally positive Lévy processes in the dual model.

Findings

Barrier strategies are optimal for all spectrally positive Lévy processes.

The optimal barrier can be explicitly characterized via the inverse of a scale function.

The value function for the capital injection problem has a similar form to the ruin time horizon case.

Abstract

We revisit the dividend payment problem in the dual model of Avanzi et al. ([2], [1], and [3]). Using the fluctuation theory of spectrally positive L\'{e}vy processes, we give a short exposition in which we show the optimality of barrier strategies for all such L\'{e}vy processes. Moreover, we characterize the optimal barrier using the functional inverse of a scale function. We also consider the capital injection problem of [3] and show that its value function has a very similar form to the one in which the horizon is the time of ruin.