On optimality of the barrier strategy for a general Lévy risk process

We consider the optimal dividend problem for the insurance risk process in a general Lévy process setting. The objective is to find a strategy which maximizes the expected total discounted dividends until the time of ruin. We give sufficient conditions under which the optimal strategy is of barrier type. In particular, we show that if the Lévy density is a completely monotone function, then the optimal dividend strategy is a barrier strategy. This approach was inspired by the work of Avram et al. (2007), Loeffen (2008) and Kyprianou et al. (2010) in which the same problem was considered under the spectrally negative Lévy processes setting.