On optimal dividends with exponential and linear penalty payments

We study the optimal dividend problem where the surplus process of an insurance company is modelled by a diffusion process. The insurer is not ruined when the surplus becomes negative, but penalty payments occur, depending on the level of the surplus. The penalty payments shall avoid that losses can rise above any number and can be seen as a preference measure or costs for negative capital. As examples, exponential and linear penalty payments are considered. It turns out that a barrier dividend strategy is optimal.