In the present paper we study some existing duality features between two very known models in Risk Theory. The classical Cramér–Lundberg risk model with application to insurance, and the dual risk model with (some) financial application. For simplicity the former will be referred as the primal model. The former has been of extensive treatment in the literature, it assumes that a given surplus process has constant deterministic gains (premiums) and random loses (claims) that come at random times. On the other hand, the latter, called as dual model, works in opposite direction, losses (costs) are constant and deterministic, and the gains (earnings) are random and come at random times. Sometimes this one is called the negative model. Similar quantities, with similar mathematical properties, work in opposite direction and have different meanings. There is however an important feature that makes the two models quite distinct, either in their application or in their nature: the loading condition, positive or negative, respectively.

The primal model has been worked extensively and focuses essentially in ruin problems (in many different aspects) whereas the dual model has developed more recently and focuses on dividend payments. I most cases, they have been worked apart, however they have connection points that allow us to use methods and results from one to another, basically from the first to the second. Identifying the right connection, or duality, is crucial so that we transport methods and results. In the work by Afonso et al. (2013) this connection is first addressed in the case when the times between claims/gains follow an exponential distribution.

We can easily understand that the ruin time in the primal has a correspondence to the dividend time in the latter. On the opposite side the time to hit an upper barrier in the primal model has a correspondence to the time to ruin in the dual model. Another interesting feature is the severity of ruin in the former and the size of the dividend payment in the latter.