On the distribution of IBNR reserves

Starting from a basic IBNR model, for which variations in the three dimensions of the problem are considered such as e.g. given in Doray, 1996 (Insurance: Mathematics and Economics 18 (1), 43–58) the random fluctuations in the direction of the calendar years is modelized, taking into account the apparatus of financial mathematics. The method can be extended to take into account random fluctuations in the other directions of the problem (such as development and year of origin direction). The results are based on supermodularity order, such that, in the framework of stop-loss ordering one obtains the distribution of the IBNR reserve corresponding to an extremal element in this ordering, when some marginals are fixed. The results obtained in this way are general in that sense that all of the possible dependencies between the variables are allowed.