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Conditional nonlinear expectations

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  • Bartl, Daniel

Abstract

Let Ω be a Polish space with Borel σ-field F and countably generated sub σ-field G⊂F. Denote by L(F) the set of all bounded F-upper semianalytic functions from Ω to the reals and by L(G) the subset of G-upper semianalytic functions. Let E(⋅|G):L(F)→L(G) be a sublinear increasing functional which leaves L(G) invariant. It is shown that there exists a G-analytic set-valued mapping PG from Ω to the set of probabilities which are concentrated on atoms of G with compact convex values such that E(X|G)(ω)=supP∈PG(ω)EP[X] if and only if E(⋅|G) is pointwise continuous from below and continuous from above on the continuous functions. Further, given another sublinear increasing functional E(⋅):L(F)→R which leaves the constants invariant, the tower property E(⋅)=E(E(⋅|G)) is characterized via a pasting property of the representing sets of probabilities, and the importance of analytic functions is explained. Finally, it is characterized when a nonlinear version of Fubini’s theorem holds true and when the product of a set of probabilities and a set of kernels is compact.

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  • Bartl, Daniel, 2020. "Conditional nonlinear expectations," Stochastic Processes and their Applications, Elsevier, vol. 130(2), pages 785-805.
    • Handle: RePEc:eee:spapps:v:130:y:2020:i:2:p:785-805
    DOI: 10.1016/j.spa.2019.03.014
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    References listed on IDEAS

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    1. Daniel Bartl & Samuel Drapeau & Ludovic Tangpi, 2017. "Computational aspects of robust optimized certainty equivalents and option pricing," Papers 1706.10186, arXiv.org, revised Mar 2019.
    2. Daniel Bartl, 2016. "Exponential utility maximization under model uncertainty for unbounded endowments," Papers 1610.00999, arXiv.org, revised Feb 2019.
    3. Nutz, Marcel & van Handel, Ramon, 2013. "Constructing sublinear expectations on path space," Stochastic Processes and their Applications, Elsevier, vol. 123(8), pages 3100-3121.
    4. Romain Blanchard & Laurence Carassus, 2018. "Multiple-Priors Optimal Investment In Discrete Time For Unbounded Utility Function," Working Papers hal-01883787, HAL.
    5. Bruno Bouchard & Shuoqing Deng & Xiaolu Tan, 2019. "Superreplication with proportional transaction cost under model uncertainty," Mathematical Finance, Wiley Blackwell, vol. 29(3), pages 837-860, July.
    6. Marcel Nutz, 2016. "Utility Maximization Under Model Uncertainty In Discrete Time," Mathematical Finance, Wiley Blackwell, vol. 26(2), pages 252-268, April.
    7. Marcel Nutz & Ramon van Handel, 2012. "Constructing Sublinear Expectations on Path Space," Papers 1205.2415, arXiv.org, revised Apr 2013.
    8. Patrick Cheridito & Michael Kupper, 2011. "Composition Of Time-Consistent Dynamic Monetary Risk Measures In Discrete Time," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 14(01), pages 137-162.
    9. Bruno Bouchard & Marcel Nutz, 2013. "Arbitrage and duality in nondominated discrete-time models," Papers 1305.6008, arXiv.org, revised Mar 2015.
    10. Matteo Burzoni & Marco Frittelli & Zhaoxu Hou & Marco Maggis & Jan Obłój, 2019. "Pointwise Arbitrage Pricing Theory in Discrete Time," Mathematics of Operations Research, INFORMS, vol. 44(3), pages 1034-1057, August.
    11. Peng, Shige, 2008. "Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation," Stochastic Processes and their Applications, Elsevier, vol. 118(12), pages 2223-2253, December.
    12. Shuoqing Deng & Xiaolu Tan & Xiang Yu, 2018. "Utility maximization with proportional transaction costs under model uncertainty," Papers 1805.06498, arXiv.org, revised Aug 2019.
    13. Ariel Neufeld & Mario Sikic, 2016. "Robust Utility Maximization in Discrete-Time Markets with Friction," Papers 1610.09230, arXiv.org, revised May 2018.
    14. Anna Aksamit & Shuoqing Deng & Jan Obłój & Xiaolu Tan, 2019. "The robust pricing–hedging duality for American options in discrete time financial markets," Mathematical Finance, Wiley Blackwell, vol. 29(3), pages 861-897, July.
    15. Charalambos D. Aliprantis & Kim C. Border, 2006. "Infinite Dimensional Analysis," Springer Books, Springer, edition 0, number 978-3-540-29587-7, December.
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    2. Tomasz Kosmala & Randall Martyr & John Moriarty, 2023. "Markov risk mappings and risk-sensitive optimal prediction," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 97(1), pages 91-116, February.

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