The optimal dividend problem in the compound Poisson model with covering the deficit at ruin

In this paper, We consider the optimal dividend problem in the compound Poisson model with covering the deficit at ruin which is restrained to strategies with restricted densities. As explained by Dickson and Waters (2004), the shareholders should be liable to cover the deficit at ruin. Therefore, we want to maximize the expectation of the difference between the accumulated discounted dividends until ruin and the discounted deficit at ruin, and find out the optimal dividend strategy. We obtain explicit solutions of V(x) when the claim amount distribution is exponential.