Bi-revealed utilities in a defaultable universe: A new point of view on consumption

This paper investigates the inverse problem of bi-revealed utilities in a defaultable universe, defined as a standard universe (represented by a filtration F) perturbed by an exogenous defaultable time τ. We assume that the standard universe does not take into account the possibility of the default, thus τ adds an additional source of risk. The defaultable universe is represented by the filtration G up to time τ (τ included), where G stands for the progressive enlargement of F by τ. The basic assumption in force is that τ avoids F-stopping times. The bi-revealed problem consists in recovering a consistent dynamic utility from the observable characteristic of an agent. The general results on bi-revealed utilities, first given in a general and abstract framework, are translated in the defaultable G-universe and then are interpreted in the F-universe. The decomposition of G-adapted processes X^G provides an interpretation of a G-characteristic Xτ^G stopped at τ as a reserve process. Thanks to the characterization of G-martingales stopped at τ in terms of F-martingales, we establish a correspondence between G-bi-revealed utilities from characteristic and F-bi-revealed pair of utilities from characteristic and reserves. In a financial framework, characteristic can be interpreted as wealth and reserves as consumption. This result sheds a new light on the consumption in utility criterion: the consumption process can be interpreted as a certain quantity of wealth, or reserves, that are accumulated for the financing of losses at the default time.