  In this paper, we assume that the filtration $\bb F$ is generated by a
$d$-dimensional Brownian motion $W=(W_1,\cdots,W_d)'$ as well as an
integer-valued random measure $\mu(du,dy)$. The random variable $\ttau$ is the
default time and $L$ is the default loss. Let $\mathbb G=\{\scr G_t;t\geq 0\}$
be the progressive enlargement of $\bb F$ by $(\ttau,L)$, i.e, $\bb G$ is the
smallest filtration including $\bb F$ such that $\ttau$ is a $\bb G$-stopping
time and $L$ is $\scr G_\ttau$-measurable. We parameterize the conditional
density process, which allows us to describe the survival process $G$
explicitly. We also obtain the explicit $\bb G$-decomposition of a $\bb F$
martingale and the predictable representation theorem for a $(P,\bb
G)$-martingale by all known parameters. Formula parametrization in the enlarged
filtration is a useful quality in financial modeling.
