Conditional Default Probability and Density

by Nicole El Karoui of the Centre de Mathématiques Appliquées,
Monique Jeanblanc of the Laboratoire de Probabilités et Modèles Aléatoires & Université d'Évry Val d'Essonne,
Ying Jiao of the Université Paris 7, and
Behnaz Zargari of the Laboratoire de Probabilités et Modèles Aléatoires & Université d'Évry Val d'Essonne

January 6, 2011

Abstract: This paper proposes different methods to construct conditional survival processes, i.e, families of martingales decreasing with respect to a parameter. Conditional survival processes play a pivotal role in the density approach for default risk, introduced by El Karoui et al. Concrete examples will lead to the construction of dynamic copulae, in particular dynamic Gaussian copulae. It is shown that the change of probability measure methodology is a key tool for that construction. As in Kallianpur and Striebel, we apply this methodology in filtering theory to recover in a straightforward way, the classical results when the signal is a random variable.

JEL Classification: C02.

AMS Classification: 91G40, 60G35.

Keywords: density processes, filtering

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