Dividend problems in the dual risk model

We consider the compound Poisson dual risk model, dual to the well known classical risk model forinsurance applications, where premiums are regarded as costs and claims are viewed as profits. Thesurplus can be interpreted as a venture capital like the capital of an economic activity involved in researchand development. Like most authors, we consider an upper dividend barrier so that we model the gainsof the capital and its return to the capital holders.By establishing a proper and crucial connection between the two models we show and explain clearlythe dividends process dynamics for the dual risk model, properties for different random quantities involvedas well as their relations. Using our innovative approach we derive some already known resultsand go further by finding several new ones. We study different ruin and dividend probabilities, such asthe calculation of the probability of a dividend, distribution of the number of dividends, expected andamount of dividends as well as the time of getting a dividend.We obtain integro-differential equations for some of the above results and also Laplace transforms.From there we can get analytical results for cases where solutions and/or inversions are possible, in othercases we may only get numerical ones. We present examples under the two cases.