An IBNR-RBNS insurance risk model with marked Poisson arrivals

Inspired by the claim reserving problem in non-life insurance, this paper proposes to study the insurer's surplus process under a micro-level framework, with particular focus on modeling the Incurred But Not Reported (IBNR) and the Reported But Not Settled (RBNS) claims. It is assumed that accidents occur according to a Poisson point process, and each accident is accompanied by a claim developmental mark that contains the reporting time, the settlement time, and the size of (possibly multiple) payments between these two times. Under exponential reporting and settlement delays, we show that our model can be represented as a Markovian risk process with countably infinite number of states. This can in turn be transformed to an equivalent fluid flow model when the payments are phase-type distributed. As a result, classical measures such as ruin probability or more generally the Gerber-Shiu expected discounted penalty function follow directly. The joint Laplace transform and the pairwise joint moments involving the ruin time and the aggregate payments of different types (with and without claim settlement) are further derived. Numerical illustrations are given at the end, including the use of a real insurance dataset.