Optimal control with absolutely continuous strategies for spectrally negative Levy processes

TL;DR

This paper investigates optimal control strategies for spectrally negative Levy processes, establishing explicit solutions under conditions on the Levy measure, extending previous results to a broader class of processes with absolutely continuous strategies.

Contribution

It extends de Finetti's control problem to spectrally negative Levy processes with absolutely continuous strategies, providing explicit optimal strategies under completely monotone Levy measure densities.

Findings

Explicit optimal refraction strategy derived

Robustness of the completely monotone Levy measure condition

Extension of previous diffusive and Cramer-Lundberg results

Abstract

In the last few years there has been renewed interest in the classical control problem of de Finetti for the case that underlying source of randomness is a spectrally negative Levy process. In particular a significant step forward is made in an article of Loeffen where it is shown that a natural and very general condition on the underlying Levy process which allows one to proceed with the analysis of the associated Hamilton-Jacobi-Bellman equation is that its Levy measure is absolutely continuous, having completely monotone density. In this paper we consider de Finetti's control problem but now with the restriction that control strategies are absolutely continuous with respect to Lebesgue measure. This problem has been considered by Asmussen and Taksar, Jeanblanc and Shiryaev and Boguslavskaya in the diffusive case and Gerber and Shiu for the case of a Cramer-Lundberg process with exponentially distributed jumps. We show the robustness of the condition that the underlying Levy measure has a completely monotone density and establish an explicit optimal strategy for this case that envelopes the aforementioned existing results. The explicit optimal strategy in question is the so-called refraction strategy.