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Nonlinear expectations of random sets

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  • Ilya Molchanov
  • Anja Muhlemann

Abstract

Sublinear functionals of random variables are known as sublinear expectations; they are convex homogeneous functionals on infinite-dimensional linear spaces. We extend this concept for set-valued functionals defined on measurable set-valued functions (which form a nonlinear space), equivalently, on random closed sets. This calls for a separate study of sublinear and superlinear expectations, since a change of sign does not convert one to the other in the set-valued setting. We identify the extremal expectations as those arising from the primal and dual representations of them. Several general construction methods for nonlinear expectations are presented and the corresponding duality representation results are obtained. On the application side, sublinear expectations are naturally related to depth trimming of multivariate samples, while superlinear ones can be used to assess utilities of multiasset portfolios.

Suggested Citation

  • Ilya Molchanov & Anja Muhlemann, 2019. "Nonlinear expectations of random sets," Papers 1903.04901, arXiv.org.
    • Handle: RePEc:arx:papers:1903.04901
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    References listed on IDEAS

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    1. Dilip Madan, 2015. "Asset pricing theory for two price economies," Annals of Finance, Springer, vol. 11(1), pages 1-35, February.
    2. M. Kaina & L. Rüschendorf, 2009. "On convex risk measures on L p -spaces," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 69(3), pages 475-495, July.
    3. Ignacio Cascos & Ilya Molchanov, 2007. "Multivariate risks and depth-trimmed regions," Finance and Stochastics, Springer, vol. 11(3), pages 373-397, July.
    4. Ilya Molchanov & Ignacio Cascos, 2016. "Multivariate Risk Measures: A Constructive Approach Based On Selections," Mathematical Finance, Wiley Blackwell, vol. 26(4), pages 867-900, October.
    5. Emmanuel Lepinette & Ilya Molchanov, 2017. "Conditional cores and conditional convex hulls of random sets," Papers 1711.10303, arXiv.org.
    6. Molchanov,Ilya & Molinari,Francesca, 2018. "Random Sets in Econometrics," Cambridge Books, Cambridge University Press, number 9781107121201, October.
    7. Ignacio Cascos & Ilya Molchanov, 2013. "Multivariate risk measures: a constructive approach based on selections," Papers 1301.1496, arXiv.org, revised Jul 2016.
    8. Hiai, Fumio & Umegaki, Hisaharu, 1977. "Integrals, conditional expectations, and martingales of multivalued functions," Journal of Multivariate Analysis, Elsevier, vol. 7(1), pages 149-182, March.
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