On the optimality of double barrier strategies for Lévy processes

This paper studies de Finetti’s optimal dividend problem with capital injection. We confirm the optimality of a double barrier strategy when underlying risk model follows Lévy process that may have positive and negative jumps. In contrast spectrally one-sided cases, strategies cannot be handled by using scale functions to obtain some properties expected net present values (NPVs) dividends injections. Instead, these properties, we observe changes in sample path (and associated NPV) there is slight change initial value or value.