Abstract
Motivated by credit risk modelling, we consider a type of default times whose probability law can have atoms, where standard intensity and density hypotheses in the enlargement of filtrations are not satisfied. We propose a generalized density approach in order to treat such random times in the framework of progressive enlargement of filtrations. We determine the compensator process of the random time and study the martingale and semimartingale processes in the enlarged filtration which are important for the change of probability measures and the evaluation of credit derivatives. The generalized density approach can also be applied to model simultaneous default events in the multi-default setting.
Highlights
In the credit risk analysis, the theory of enlargement of filtrations, which has been developed by the French school of probability since the 1970s, has been systematically adopted to model the default event
The generalized density model that we propose can be viewed as hybrid credit model
The main contribution of our work is to focus on the impact of the discontinuous part of the F-conditional law of τ and study the impact of the critical dates on the random time
Summary
In the credit risk analysis, the theory of enlargement of filtrations, which has been developed by the French school of probability since the 1970s (see e.g. Jacod [14], Jeulin [17], Jeulin and Yor [18]), has been systematically adopted to model the default event. The sovereign default time can coincide with some pre-determined dates In this case, the classical default modelling approaches, in particular, both intensity and density models are no longer adapted. We assume that the F-conditional law of τ contains a discontinuous part, besides the absolutely continuous part which has a density This generalized density approach allows to consider a random time τ which has positive probability to meet a finite family of F-stopping times. For applications of the generalized density approach, we study the immersion property, called the H-hypothesis in literature, i.e., any F-martingale is a G-martingale, which is commonly adopted in the default modelling. As one consequence of the characterization results of G-martingales, we study the change of probability and the associated Radon-Nikodym derivatives Another application consists of a model of two default times where the occurrence of simultaneous defaults is possible.
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