Modeling and filtering credit merit in a set of firms

We analyze the credit risk of a set of firms, which are considered as an heterogeneous population of identical firms, divided into a finite number of classes in relation to their credit quality. The partition changes during the time to let the firms to dynamically change their own credit merit. Under a suitable exchangeability assumption, the model can be stated in terms of the cardinality and the cumulated defaults of every credit class. This allows to specify the model in order to catch self exciting and clustering behaviour of the default process. In a partial information setting, in which the observations is just the cardinality of a the class of the defaulted firms, the problem of finding the conditional distribution of firms distance to defaults, given the observation of the firms already defaulted, is discussed, using stochastic filtering and particle filtering techniques