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 $\bF$) perturbed by an exogenous defaultable time $\tau$. We assume that the standard universe does not take into account the possibility of the default, thus $\tau$ adds an additional source of risk. The defaultable universe is represented by the filtration $\bG$ {\it up to time $\tau$}, where $\bG$ stands for the progressive enlargement of $\bF$ by $\tau$. The basic assumption in force is that $\tau$ avoids $\bF$-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 $\bG$-universe and then are interpreted in the $\bF$-universe. The decomposition of $\bG$-adapted processes provides an interpretation of a $\bG$-characteristic $X^\bG_\tau$ stopped at $\tau$ as a reserve process. Thanks to the characterization of $\bG$-martingales stopped at $\tau$ in terms of $\bF$-martingales, we establish a correspondence between $\bG$-bi-revealed utilities from characteristic and $\bF$-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.