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

Author

Listed:
  • El Karoui

    (LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation - SU - Sorbonne Université - CNRS - Centre National de la Recherche Scientifique - UPCité - Université Paris Cité)

  • Mrad Mohamed

    (LAGA - Laboratoire Analyse, Géométrie et Applications - UP8 - Université Paris 8 Vincennes-Saint-Denis - CNRS - Centre National de la Recherche Scientifique - Université Sorbonne Paris Nord)

  • Caroline Hillairet

    (ENSAE Paris - École Nationale de la Statistique et de l'Administration Économique)

Abstract
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.

Suggested Citation

  • El Karoui & Mrad Mohamed & Caroline Hillairet, 2022. "Bi-revealed utilities in a defaultable universe : a new point of view on consumption," Working Papers hal-03919186, HAL.
    • Handle: RePEc:hal:wpaper:hal-03919186
    Note: View the original document on HAL open archive server: https://hal.science/hal-03919186v1
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    References listed on IDEAS

    as
    1. El Karoui, Nicole & Jeanblanc, Monique & Jiao, Ying, 2010. "What happens after a default: The conditional density approach," Stochastic Processes and their Applications, Elsevier, vol. 120(7), pages 1011-1032, July.
    2. Chambers,Christopher P. & Echenique,Federico, 2016. "Revealed Preference Theory," Cambridge Books, Cambridge University Press, number 9781107087804, October.
    3. Tahir Choulli & Catherine Daveloose & Michèle Vanmaele, 2020. "A martingale representation theorem and valuation of defaultable securities," Mathematical Finance, Wiley Blackwell, vol. 30(4), pages 1527-1564, October.
    4. M. Musiela & T. Zariphopoulou, 2009. "Portfolio choice under dynamic investment performance criteria," Quantitative Finance, Taylor & Francis Journals, vol. 9(2), pages 161-170.
    Full references (including those not matched with items on IDEAS)

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