The first goal of this article is to identify, for different defaultable claims, the fundamental processes which uniquely determine the pre-default price and therefore require to be modelled. The main message to the reader is that although the use of the default intensity or hazard process is ubiquitous, it may not uniquely characterise the price of some defaultable claims. The second goal is to better consolidate the reduced form approach with the structural approach, by extending the reduced form approach to allow for default times which can occur at stopping times and do not satisfy the immersion property.