An explicit model of default time with given survival probability
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.
scientific article; zbMATH DE number 5930906
scientific article