Abstract
We provide some extensions of Norros’ lemma for a model with several default times and nontrivial reference filtrations. These results allow a characterization of the filtration immersion properties in terms of the terminal values of compensators of the associated default processes. The method of proof is based on the analysis of properties of exponential martingales associated with the default times.
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Azéma, J.: Quelques applications de la théorie générale des processus. Invent. Math. 18, 293–336 (1972)
Bielecki, T.R., Rutkowski, M.: Credit Risk: Modeling, Valuation and Hedging. Springer, Berlin (2002)
Brémaud, P., Yor, M.: Changes of filtrations and of probability measures. Z. Wahrscheinlichkeitstheor. Verw. Geb. 45, 269–295 (1978)
Dellacherie, C., Meyer, P.A.: Probabilités et Potentiel. Hermann, Paris (1975). Chapitres I–IV; English translation: Probabilities and Potential, Chapters I–IV. North-Holland (1978)
El Karoui, N.: Modélisation de l’information. CEA-EDF-INRIA, École d’été (1999)
Elliott, R.J., Jeanblanc, M., Yor, M.: On models of default risk. Math. Finance 10, 179–195 (2000)
Jeulin, T.: Semi-Martingales et Grossissement d’une Filtration. Lecture Notes in Mathematics, vol. 833. Springer, Berlin (1980)
Kusuoka, S.: A remark on default risk models. Adv. Math. Econ. 1, 69–82 (1999)
Lando, D.: On Cox processes and credit risky securities. Rev. Deriv. Res. 2, 99–120 (1998)
Mansuy, R., Yor, M.: Random Times and Enlargements of Filtrations in a Brownian Setting. Lecture Notes in Mathematics, vol. 1873. Springer, Berlin (2004)
Norros, I.: A compensator representation of multivariate life length distributions, with applications. Scand. J. Stat. 13, 99–112 (1986)
Rogers, L.C.G., Williams, D.: Diffusions, Markov Processes and Martingales II. Itô Calculus. Wiley, New York (1987)
Acknowledgements
The paper was initiated when the author was visiting the Université d’Evry-Val-d’Essonne in November 2008. He is grateful to Monique Jeanblanc and the Département de Mathématiques for helpful discussions and warm hospitality. Financial support from the Europlace Institute of Finance and the European Science Foundation (ESF) through the grant number 2500 of the program Advanced Mathematical Methods for Finance (AMaMeF) are gratefully acknowledged. The author thanks Ashkan Nikeghbali for his comments and references to the literature. It is also a pleasure to thank Ilkka Norros for his encouragement to extend his result to the case of credit risk models.
This research also benefited from the support of the ’Chaire Risque de Crédit’, Fédération Bancaire Française.
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Gapeev, P.V. (2014). Some Extensions of Norros’ Lemma in Models with Several Defaults. In: Kabanov, Y., Rutkowski, M., Zariphopoulou, T. (eds) Inspired by Finance. Springer, Cham. https://doi.org/10.1007/978-3-319-02069-3_12
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DOI: https://doi.org/10.1007/978-3-319-02069-3_12
Publisher Name: Springer, Cham
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