On optimal periodic dividend strategies for L\'evy risk processes

In this paper, we revisit the optimal periodic dividend problem, in which dividend payments can only be made at the jump times of an independent Poisson process. In the dual (spectrally positive L\'evy) model, recent results have shown the optimality of a periodic barrier strategy, which pays dividends at Poissonian dividend-decision times, if and only if the surplus is above some level. In this paper, we show the optimality of this strategy for a spectrally negative L\'evy process whose dual has a completely monotone L\'evy density. The optimal strategies and value functions are concisely written in terms of the scale functions. Numerical results are also provided.