On the optimality of the refraction--reflection strategy for L\'evy processes

In this paper, we study de Finetti's optimal dividend problem with capital injection under the assumption that the dividend strategies are absolutely continuous. In many previous studies, the process before being controlled was assumed to be a spectrally one-sided L\'evy process, however in this paper we use a L\'evy process that may have both positive and negative jumps. In the main theorem, we show that a refraction--reflection strategy is an optimal strategy. We also mention the existence and uniqueness of solutions of the stochastic differential equations that define refracted L\'evy processes.