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Quasi-sure essential supremum and applications to finance

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  • Laurence Carassus

    (Université Paris-Saclay)

Abstract

When uncertainty is modelled by a non-dominated and non-compact set of probability measures, a notion of essential supremum for a family of real-valued functions is developed in terms of upper semi-analytic functions. We show how the properties postulated on the initial functions carry over to their quasi-sure essential supremum. We propose various applications to financial problems with frictions. We analyse superreplication and prove a bidual characterisation of the superhedging cost. We also study a weak no-arbitrage condition called absence of instantaneous profit ( AIP $\mathrm{AIP}$ ) under which prices are finite. This requires new results on the aggregation of quasi-sure statements.

Suggested Citation

  • Laurence Carassus, 2025. "Quasi-sure essential supremum and applications to finance," Finance and Stochastics, Springer, vol. 29(1), pages 219-260, January.
    • Handle: RePEc:spr:finsto:v:29:y:2025:i:1:d:10.1007_s00780-024-00553-1
    DOI: 10.1007/s00780-024-00553-1
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    References listed on IDEAS

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    1. repec:dau:papers:123456789/12268 is not listed on IDEAS
    2. Bruno Bouchard & Marcel Nutz, 2013. "Arbitrage and duality in nondominated discrete-time models," Papers 1305.6008, arXiv.org, revised Mar 2015.
    3. Kabanov, Yuri & Lépinette, Emmanuel, 2013. "Essential supremum with respect to a random partial order," Journal of Mathematical Economics, Elsevier, vol. 49(6), pages 478-487.
    4. Daniel Bartl, 2016. "Exponential utility maximization under model uncertainty for unbounded endowments," Papers 1610.00999, arXiv.org, revised Feb 2019.
    5. Teemu Pennanen, 2011. "Arbitrage and deflators in illiquid markets," Finance and Stochastics, Springer, vol. 15(1), pages 57-83, January.
    6. Laurence Carassus & Jan Obloj & Johannes Wiesel, 2018. "The robust superreplication problem: a dynamic approach," Papers 1812.11201, arXiv.org, revised Feb 2019.
    7. Bartl, Daniel, 2020. "Conditional nonlinear expectations," Stochastic Processes and their Applications, Elsevier, vol. 130(2), pages 785-805.
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    9. Kabanov, Yuri & Lépinette, Emmanuel, 2013. "Essential supremum and essential maximum with respect to random preference relations," Journal of Mathematical Economics, Elsevier, vol. 49(6), pages 488-495.
    10. Blanchard, Romain & Carassus, Laurence, 2020. "No-arbitrage with multiple-priors in discrete time," Stochastic Processes and their Applications, Elsevier, vol. 130(11), pages 6657-6688.
    11. Jan Obłój & Johannes Wiesel, 2021. "A unified framework for robust modelling of financial markets in discrete time," Finance and Stochastics, Springer, vol. 25(3), pages 427-468, July.
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    More about this item

    Keywords

    Quasi-sure essential supremum; Model uncertainty; Non-dominated model; Market with frictions; Superreplication; Absence of instantaneous profit ( AIP $mathrm{AIP}$ );
    All these keywords.

    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • C65 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Miscellaneous Mathematical Tools
    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates
    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

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