On the analysis of a general class of dependent risk processes

A generalized Sparre Andersen risk process is examined, whereby the joint distribution of the interclaim time and the ensuing claim amount is assumed to have a particular mathematical structure. This structure is present in various dependency models which have previously been proposed and analyzed. It is then shown that this structure in turn often implies particular functional forms for joint discounted densities of ruin related variables including some or all of the deficit at ruin, the surplus immediately prior to ruin, and the surplus after the second last claim. Then, employing a fairly general interclaim time structure which involves a combination of Erlang type densities, a complete identification of a generalized Gerber–Shiu function is provided. An application is given applying these results to a situation involving a mixed Erlang type of claim amount assumption. Various examples and special cases of the model are then considered, including one involving a bivariate Erlang mixture model.