An enlargement of filtration formula with applications to multiple non-ordered default times

In this work, for a reference filtration F$\mathbb {F}$, we develop a method for computing the semimartingale decomposition of F$\mathbb {F}$-martingales in a specific type of enlargement of F$\mathbb {F}$. As an application, we study the progressive enlargement of F$\mathbb {F}$ with a sequence of non-ordered default times and show how to deduce results concerning the first-to-default, k$k$th-to-default, k-out-of-n-to-default or all-to-default events. In particular, using this method, we compute explicitly the semimartingale decomposition of F$\mathbb {F}$-martingales under the absolute continuity condition of Jacod.