An explicit model of default time with given survival probability
type: Item
instance of: scholarly article
title: An explicit model of default time with given survival probability
zbMATH Open document ID: 1298.91176
author: Shiqi Song
author: Monique Jeanblanc
published in: Stochastic Processes and their Applications
publication date: 2011-07-22T00:00:00Z
review text: The considered problem arises in credit risk modeling: Given an increasing continuous process \(\Lambda\), one wants to construct a random time \(\tau\) (the default time) and a probability measure \(Q\), such that \(1_{\tau \leq t} - \Lambda_{t \wedge \tau}\) is a \(Q\)-local martingale in the filtration enlarged by \(\tau\), i.e. \(\mathcal{G}_t = \mathcal{F}_t \vee \sigma(\tau \wedge t)\), and such that \(Q|_{\mathcal{F}_\infty} = P|_{\mathcal{F}_\infty}\). It is known that then the conditional survival probability will be of the form \[ Q(\tau > t | \mathcal{F}_t) = N_t e^{-\Lambda_t} \] for a positive \((\mathcal{F}_t)\)-local martingale \(N\). In case \(N \equiv 1\), a canonical solution is given by the Cox model. In this article, the authors start with a given positive supermartingale of the form \(N e^{- \Lambda}\) which is bounded away from 1 (from below). They construct \(Q\) and \(\tau\) such that \(Q|_{\mathcal{F}_\infty} = P|_{\mathcal{F}_\infty}\) and such that \(Q(\tau > t | \mathcal{F}_t) = N_t e^{-\Lambda_t}\). The constructed \(Q\) is absolutely continuous with respect to the Cox measure, and they give the explicit density. They give several characterizations of the constructed measure. They prove that under \(Q\), \((\mathcal{F}_t)\)-semimartingales stay \((\mathcal{G}_t)\)-semimartingales, and they provide an explicit decomposition.
reviewed by: Nicolas Perkowski
Mathematics Subject Classification ID: 91G40
Mathematics Subject Classification ID: 60G40
Mathematics Subject Classification ID: 60G44
zbMATH DE Number: 5930906
zbMATH Keywords: credit risk modeling
zbMATH Keywords: default time
zbMATH Keywords: survival probability
zbMATH Keywords: explicit construction
zbMATH Keywords: Cox measure
zbMATH Keywords: enlargement of filtration
zbMATH Keywords: Föllmer measure
MaRDI profile type: Publication
full work available at URL: j.spa.2011.04.002
OpenAlex ID: W2056866308
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DOI: 10.1016/J.SPA.2011.04.002
description: scientific article; zbMATH DE number 5930906