Parametrization in the progressively enlarged filtration

Abstract: 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.