Optimality of multi-refraction dividend strategies in the dual model

We consider the multi-refraction strategies in two equivalent versions of the optimal dividend problem in the dual (spectrally positive L\'evy) model. The first problem is a variant of the bail-out case where both dividend payments and capital injections must be absolutely continuous with respect to the Lebesgue measure. The second is an extension of Avanzi et al. [4] where a strategy is a combination of two absolutely continuous dividend payments with different upper bounds and different transaction costs. In both problems, it is shown to be optimal to refract the process at two thresholds, with the optimally controlled process being the multi-refracted L\'evy process recently studied by Czarna et al. [9]. The optimal strategy and the value function are succinctly written in terms of a version of the scale function. Numerical results are also given.