Default Risk and Hazard Process
The so-called intensity-based approach to the modelling and valuation of de-faultable securities has attracted a considerable attention of both practitioners and academics in recent years; to mention a few papers in this vein: Duffie [8], Duffie and Lando
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M.J. Equipe d'Analyse et Probabilits, Universit d'Evry Val d'Essonne Boulevard Franois Mitterrand, 91025 Evry Cedex, France [email protected] M.R. Faculty of Mathematics and Information Science Warsaw University of Technology 00-661 Warszawa, Poland [email protected]
Introduction
The so-called intensity-based approach to the modelling and valuation of defaultable securities has attracted a considerable attention of both practitioners and academics in recent years to mention a few papers in this vein: Due 8, Due and Lando 9, Due et al. 10, Jarrow and Turnbull 13, Jarrow et al. 14, Jarrow and Yu 15, Lando 21, Madan and Unal 23. In the context of nancial modelling, there was also a renewed interest in the detailed analysis of the properties of random times we refer to the recent papers by Elliott et al. 12 and Kusuoka 20 in this regard. In fact, the systematic study of stopping times and the associated enlargements of ltrations, motivated by a purely mathematical interest, was initiated in the 1970s by the French school, including: Brmaud and Yor 4, Dellacherie 5, Dellacherie and Meyer 7, Jeulin 16, and Jeulin and Yor 17. On the other hand, the classic concept of the intensity or the hazard rate of a random time was also studied in some detail in the context of the theory of Cox processes, as well as in relation to the theory of martingales. The interested reader may consult, in particular, the monograph by Last and Brandt 22 for the former approach, and by Brmaud 3 for the latter. It seems to us that no single comprehensive source focused on the issues related to default risk modelling is available to nancial researchers, though. Furthermore, it is worth noting that some challenging mathematical problems associated with the modelling of default risk remain still open. The aim of this text is thus to ll the gap by furnishing a relatively concise and self-contained exposition of the most relevant from the viewpoint of nancial modelling results related to the analysis of random times and their ltrations. We also present some recent developments and we indicate the directions for a further research. Due to the limited space, the proofs of some results were omitted a full version of the working paper 19 is available from the authors upon request. H. Geman et al. (eds.), Mathematical Finance — Bachelier Congress 2000 © Springer-Verlag Berlin Heidelberg 2002
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Monique Jeanblanc, Marek Rutkowski
2 Hazard Process Γ of a Random Time Let τ be a non-negative random variable on a probability space (Ω, G, P), such that P(τ = 0) = 0 and P(τ > t) > 0 for any t ≥ 0. We introduce a right-continuous process D by setting Dt = 11{τ ≤t} , and we denote by D the ltration generated by D that is, Dt = σ(Du : u ≤ t). Setup 1. Suppose that F = (Ft ) t≥0 is a given but arbitrary ltration1 on (Ω, G, P). Let us consider the joint ltration G := D ∨ F that is, we set Gt = Dt ∨ Ft for every t ∈ IR+ . Our rst goal is to nd a representation of the conditiona
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