Parametrization in the progressively enlarged filtration

In this paper, we assume that the filtration $\text{bb}F$ is generated by a $d$-dimensional Brownian motion $W=(W_1,\cdots ,W_d{)}^{\mathrm{\prime }}$ as well as an integer-valued random measure $\mu (du,dy)$. The random variable $\text{ttau}$ is the default time and $L$ is the default loss. Let $\mathbb{G}=\{\text{scr}{G}_t;t\ge 0\}$ be the progressive enlargement of $\text{bb}F$ by $(\text{ttau},L)$, i.e, $\text{bb}G$ is the smallest filtration including $\text{bb}F$ such that $\text{ttau}$ is a $\text{bb}G$-stopping time and $L$ is $\text{scr}{G}_{\text{ttau}}$-measurable. We parameterize the conditional density process, which allows us to describe the survival process $G$ explicitly. We also obtain the explicit $\text{bb}G$-decomposition of a $\text{bb}F$ 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.