On optimal dividends with penalty payments in the Cramer-Lundberg model

In this paper we study the optimal dividend problem where the surplus process of an insurance company is modeled by a Cramer-Lundberg model. As distinguished from classical models, we assume that the insurer can continue doing business although the surplus becomes negative, but penalty payments occur, depending on the level of the surplus. The higher the surplus level, the lower the penalty payments. The penalty payments are rather technical and necessary to avoid that losses can rise above any number. Nevertheless, the concept can also be reasonable in practice. For example, penalty payments can occur if the insurer needs to borrow money. The aim is to determine a dividend strategy that maximizes the difference between the expected discounted dividend and penalty payments. We show that the optimal strategy is a barrier strategy. As examples, exponential and linear penalty payments are considered