Alternative approach to the optimality of the threshold strategy for spectrally negative Levy processes

TL;DR

This paper explores an alternative method for establishing the optimality of threshold dividend strategies in spectrally negative Levy process models for insurance surplus, expanding beyond previous conditions on Levy measures.

Contribution

It introduces a new approach to prove the optimality of threshold strategies without requiring the Levy measure to have a completely monotone density.

Findings

Threshold strategies are confirmed to be optimal under broader conditions.

The new approach provides a different proof technique for the optimal dividend problem.

Potential for applying the method to other stochastic control problems.

Abstract

Consider the optimal dividend problem for an insurance company whose uncontrolled surplus precess evolves as a spectrally negative Levy process. We assume that dividends are paid to the shareholders according to admissible strategies whose dividend rate is bounded by a constant. The objective is to find a dividend policy so as to maximize the expected discounted value of dividends which are paid to the shareholders until the company is ruined. Kyprianou, Loeffen and Perez [28] have shown that a refraction strategy (also called threshold strategy) forms an optimal strategy under the condition that the Levy measure has a completely monotone density. In this paper, we propose an alternative approach to this optimal problem.