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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.