Optimal Control with Absolutely Continuous Strategies for Spectrally Negative Lévy Processes

In the last few years there has been renewed interest in the classical control problem of de Finetti [10] for the case that underlying source of randomness is a spectrally negative Lévy process. In particular a significant step forward is made in [25] where it is shown that a natural and very general condition on the underlying Lévy process which allows one to proceed with the analysis of the associated Hamilton-Jacobi-Bellman equation is that its Lévy 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 [2], Jeanblanc and Shiryaev [19] and Boguslavskaya [9] in the diffusive case and Gerber and Shiu [16] for the case of a Cramér-Lundberg process with exponentially distributed jumps. We show the robustness of the condition that the underlying Lévy 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. AMS 2000 Mathematics Subject Classification: Primary 60J99; secondary 93E20, 60G51.