Imperfect information, Default time, Hazard process, 60G35, 91B29, 91B26, G12, G13,

In this paper we give a financial justification, based on non arbitrage conditions, of the (H) hypothesis in default time modelling. We also show how the (H) hypothesis is affected by an equivalent change of probability measure. The main technique used here is the theory of progressive enlargements of filtrations.2000 Mathematics Subject Classification. 15A52.

In this paper, we build a bridge between different reduced-form approaches to pricing defaultable claims. In particular, we show how the well-known formulas by Duffie, Schroder, and Skiadas and by Elliott, Jeanblanc, and Yor are related. Moreover, in the spirit of Collin Dufresne, Hugonnier, and Goldstein, we propose a simple pricing formula under an equivalent change of measure.Two processes will play a central role: the hazard process and the martingale hazard process attached to a default time. The crucial step is to understand the difference between them, which has been an open question in the literature so far. We show that pseudo-stopping times appear as the most general class of random times for which these two processes are equal. We also show that these two processes always differ when τ is an honest time, providing an explicit expression for the difference. Eventually we provide a solution to another open problem: we show that if τ is an arbitrary random (default) time such that its Azéma's supermartingale is continuous, then τ avoids stopping times.

Under short sales prohibitions, no free lunch with vanishing risk (NFLVRS) is known to be equivalent to the existence of an equivalent supermartingale measure for the price processes (Pulido [26]).We give a necessary condition for the drift of a price process to satsify (NFLVRS). For two given price processes, we introduce the concept of fundamental supermartingale measure, and when a certain condition necessary to the construction of this fundamental supermartingale measure is not fulfilled, we provide the corresponding arbitrage portfolios. The motivation of our study lies in understanding the particular case of converging prices, i.e., two prices that coincide at a bounded random time.