This paper investigates the inverse problem of bi-revealed utilities in a defaultable universe, defined as a standard universe (represented by a filtration $\mathbb{F}$) 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 $\mathbb{G}$ up to time $\tau $ ($\tau $ included), where $\mathbb{G}$ stands for the progressive enlargement of $\mathbb{F}$ by $\tau $. The basic assumption in force is that $\tau $ avoids $\mathbb{F}$ -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 $\mathbb{G}$ -universe and then are interpreted in the $\mathbb{F}$ -universe. The decomposition of $\mathbb{G}$ -adapted processes ${X}^{\mathbb{G}}$ provides an interpretation of a $\mathbb{G}$ -characteristic ${X}_{\tau }^{\mathbb{G}}$ stopped at $\tau $ as a reserve process. Thanks to the characterization of $\mathbb{G}$ -martingales stopped at $\tau $ in terms of $\mathbb{F}$ -martingales, we establish a correspondence between $\mathbb{G}$ -bi-revealed utilities from characteristic and $\mathbb{F}$ -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.
Acknowledgements
This work is with the financial support of the “ Chaire Risque Financier ” of the “ Fondation du Risque ”, the Labex MME-DII. The authors’s research is part of the ANR project DREAMeS (ANR-21-CE46-0002).
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