Optimal dividend-payout in random discrete time

Assume that the surplus process of an insurance company is described by a general Lévy process and that possible dividend pay-outs to shareholders are restricted to random discrete times which are determined by an independent renewal process. Under this setting we show that the optimal dividend pay-out policy is a band-policy. If the renewal process is a Poisson process, it is further shown that for Cramér-Lundberg risk processes with exponential claim sizes and its diffusion limit the optimal policy collapses to a barrier-policy. Finally, a numerical example is given for which the optimal bands can be calculated explicitly. The random observation procedure studied in this paper also allows for an interpretation in terms of a random walk model with a certain type of random discounting.