×

An explicit model of default time with given survival probability. (English) Zbl 1298.91176

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.

MSC:

91G40 Credit risk
60G40 Stopping times; optimal stopping problems; gambling theory
60G44 Martingales with continuous parameter
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Barlow, M. T., Study of a filtration expanded to include an honest time, Probability Theory and Related Fields, 44, 4, 307-323 (1978) · Zbl 0369.60047
[2] Bielecki, T. R.; Jeanblanc, M.; Rutkowski, M., Credit Risk Modelling (2009), Osaka University Press
[3] El Karoui, N.; Jeanblanc, M.; Jiao, Y., What happens after a default: the conditional density approach, Stochastic Processes and their Applications, 120, 7, 1011-1032 (2009) · Zbl 1194.91187
[4] Gapeev, P.; Jeanblanc, M.; Li, L.; Rutkowski, M., Constructing random times with given survival processes and applications to valuation of credit derivatives, (Chiarella, C.; Novikov, A., Contemporary Quantitative Finance (2010), Springer-Verlag: Springer-Verlag Berlin, Heidelberg, New York) · Zbl 1228.91070
[5] Jeanblanc, M.; Le Cam, Y., Progressive enlargement of filtration with initial times, Stochastic Processes and their Applications, 119, 8, 2443-2523 (2009) · Zbl 1175.60041
[6] M. Jeanblanc, S. Song, Random times with given survival probability and their \(\mathbb{F} \); M. Jeanblanc, S. Song, Random times with given survival probability and their \(\mathbb{F} \)
[7] Jeulin, T., (Semi-Martingales et Grossissement d’une Filtration. Semi-Martingales et Grossissement d’une Filtration, Lecture Notes in Mathematics, vol. 833 (1980), Springer) · Zbl 0444.60002
[8] Jeulin, T.; Yor, M., Grossissement d’une filtration et semi-martingales: formules explicites, (Séminaire de Probabilités XII. Séminaire de Probabilités XII, Lecture Notes in Mathematics, vol. 649 (1978), Springer-Verlag), 78-97 · Zbl 0411.60045
[9] Meyer, P. A., La mesure de H. Föllmer en théorie des surmartingales, (Séminaire de Probabilités VI (1972)), 118-129 · Zbl 0231.60034
[10] Nikeghbali, A., An essay on the general theory of stochastic processes, Probability Surveys, 3, 345-412 (2006) · Zbl 1189.60076
[11] Nikeghbali, A.; Yor, M., Doob’s maximal identity, multiplicative decompositions and enlargements of filtrations, (Burkholder, D., Joseph Doob: A Collection of Mathematical Articles in his Memory. Joseph Doob: A Collection of Mathematical Articles in his Memory, Illinois Journal of Mathematics, vol. 50 (2007)), 791-814 · Zbl 1101.60059
[12] Protter, P., Stochastic integration and differential equations (2004), Springer · Zbl 1041.60005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.