An enlargement of filtration formula with applications to multiple non-ordered default times
type: Item
instance of: scholarly article
title: An enlargement of filtration formula with applications to multiple non-ordered default times
zbMATH Open document ID: 1379.60040
author: Shiqi Song
author: Monique Jeanblanc
author: Li-Bo Li
published in: Finance and Stochastics
publication date: 2018-01-16T00:00:00Z
review text: 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.
reviewed by: Claudio Fontana
Mathematics Subject Classification ID: 60G07
Mathematics Subject Classification ID: 91G40
Mathematics Subject Classification ID: 60G44
zbMATH DE Number: 6826642
zbMATH Keywords: enlargement of filtration
zbMATH Keywords: reduced form credit risk modeling
zbMATH Keywords: non-ordered default times
zbMATH Keywords: semimartingale decomposition
MaRDI profile type: Publication
full work available at URL: s00780-017-0349-z
OpenAlex ID: W2768647238
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DOI: 10.1007/S00780-017-0349-Z
description: scientific article; zbMATH DE number 6826642