  We consider the optimal dividend problem for the insurance risk process in a
general Levy 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 Levy 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) [Annals of Applied
Probability 17, 156-180], Loeffen (2008) [Annals of Applied Probability 18,
1669-1680] and Kyprianou et al. (2010) [Journal of Theoretical Probability 23,
547-564] in which the same problem was considered under the spectrally negative
Levy processes setting.
