On optimality of the barrier strategy for a general Levy risk process

TL;DR

This paper establishes conditions under which the optimal dividend payout strategy for a general Levy risk process is a barrier strategy, extending previous results from spectrally negative Levy processes to more general Levy processes.

Contribution

It provides sufficient conditions, specifically the complete monotonicity of the Levy density, for the optimal dividend strategy to be of barrier type in a general Levy process setting.

Findings

Barrier strategy is optimal when Levy density is completely monotone.

Extends previous spectrally negative Levy process results to general Levy processes.

Provides theoretical conditions for optimal dividend strategies.

Abstract

We consider the optimal dividend problem for the insurance risk process in a general Levy process setting. The objective is to find a strategy which maximizes the expected total discounted dividends until the time of ruin. We give sufficient conditions under which the optimal strategy is of barrier type. In particular, we show that if the Levy density is a completely monotone function, then the optimal dividend strategy is a barrier strategy. This approach was inspired by the work of Avram et al. (2007) [Annals of Applied Probability 17, 156-180], Loeffen (2008) [Annals of Applied Probability 18, 1669-1680] and Kyprianou et al. (2010) [Journal of Theoretical Probability 23, 547-564] in which the same problem was considered under the spectrally negative Levy processes setting.