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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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    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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