The dual risk model with dividends taken at arrival

We consider the dual risk model with special dividend or tax payments: If an arriving gain finds the surplus above a barrier b or if it would bring the surplus above that level, a certain part of the gain is paid as dividends or taxes. We obtain expressions for the joint Laplace-Stieltjes transform of the time to ruin and the amount of dividends paid until ruin, and for the expected discounted dividend paid until ruin. We consider the case where the dividend paid from each gain is a general function of the gain. More explicit results are obtained when the dividend is a given percentage of the gain amount.