Progressive Filtration Enlargement in the Generalized Cox Model

We assume that there exist two kinds of investors in the market, the first kind investors, have the market information , which is given by a -dimensional Brownian motion as well as an integer-valued random measure . The second kind, however, have the information from the progressive enlargement filtration of by the default time modeled by the so called the generalized Cox model. We characterize this model with a triplet and describe main properties such as the survival process and the conditional density of . The -decomposition of a -martingale is not trivial in contrast to the class Cox model.