An enlargement of filtration formula with applications to multiple non-ordered default times
This paper provides a new and general decomposition formula in the theory of enlargement of filtrations. More specifically, the authors consider a setting with a reference filtration \(\mathbb{F}=(\mathcal{F}_t)_{t\geq0}\) and an enlarged filtration \(\mathbb{G}=(\mathcal{G}_t)_{t\geq0}\), such that \(\mathcal{F}_t\subseteq\mathcal{G}_t\), for all \(t\geq0\). The key assumption of the paper is that there exist \(k\in\mathbb{N}\), a family of filtrations \((\mathbb{F}^1,\ldots,\mathbb{F}^k)\) with \(\mathbb{F}^i\subseteq\mathbb{F}\), for all \(i=1,\ldots,k\), and a partition \(\mathcal{D}=(D_1,\ldots,D_k)\) of \(\Omega\) such that \(\mathcal{G}_t\cap D_i=\mathcal{F}^i_t\cap D_i\) and the \(H'\)-hypothesis is satisfied between \(\mathbb{F}\) and \(\mathbb{F}'\) for all \(i=1,\ldots,k\). We recall that the \(H'\)-hypothesis between \(\mathbb{F}\) and \(\mathbb{F}^i\) means that every \(\mathbb{F}\)-semimartingale is an \(\mathbb{F}^i\)-semimartingale. This structure of enlargement arises naturally in the context of progressive enlargement with respect to multiple random times (see Example 1.2 in the paper). The main result is an explicit description of the finite variation term (the so-called drift operator) appearing in the \(\mathbb{G}\)-canonical decomposition of an arbitrary \(\mathbb{F}\)-martingale. This result is obtained in two steps: in the first step (aggregation), classical results of enlargements of filtration are used to aggregate the \(\mathbb{F}^i\)-drift operators into a a drift operator with respect to an aggregate filtration \(\widehat{\mathbb{F}}\), which satisfies \(\mathbb{G}\subseteq\widehat{\mathbb{F}}\). In the second step (projection), the \(\mathbb{G}\)-drift operator is computed by taking the optional projection of the \(\widehat{\mathbb{F}}\)-drift operator onto the filtration \(\mathbb{G}\), taking care of several delicate technical issues.
scientific article; zbMATH DE number 6826642
scientific article