Parametrization in the progressively enlarged filtration

TL;DR

This paper develops a parametrization framework for the survival process and martingale decompositions in a progressively enlarged filtration, enhancing the mathematical tools for financial modeling involving default times.

Contribution

It introduces explicit parametrizations of the survival process and martingale decompositions in the enlarged filtration setting, providing new formulas for financial applications.

Findings

Explicit description of the survival process G

Derivation of the G-decomposition of F-martingales

Predictable representation theorem in the 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.