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

Optimal control with absolutely continuous strategies for spectrally negative Levy processes