IDEAS home Printed from https://ideas.repec.org/a/spr/finsto/v25y2021i1d10.1007_s00780-020-00442-3.html
   My bibliography  Save this article

Nonlinear expectations of random sets

Author

Listed:
  • Ilya Molchanov

    (University of Bern)

  • Anja Mühlemann

    (University of Bern)

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) or, equivalently, on random closed sets. This calls for a separate study of sublinear and superlinear expectations, since a change of sign does not alter the direction of the inclusion in the set-valued setting. We identify the extremal expectations as those arising from the primal and dual representations of nonlinear expectations. 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 Mühlemann, 2021. "Nonlinear expectations of random sets," Finance and Stochastics, Springer, vol. 25(1), pages 5-41, January.
    • Handle: RePEc:spr:finsto:v:25:y:2021:i:1:d:10.1007_s00780-020-00442-3
    DOI: 10.1007/s00780-020-00442-3
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s00780-020-00442-3
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s00780-020-00442-3?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. 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.
    2. Yuri Kabanov, 2009. "Markets with Transaction Costs. Mathematical Theory," Post-Print hal-00488168, HAL.
    3. Molchanov,Ilya & Molinari,Francesca, 2018. "Random Sets in Econometrics," Cambridge Books, Cambridge University Press, number 9781107121201.
    4. 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.
    5. Dilip Madan, 2015. "Asset pricing theory for two price economies," Annals of Finance, Springer, vol. 11(1), pages 1-35, February.
    6. Ignacio Cascos & Ilya Molchanov, 2007. "Multivariate risks and depth-trimmed regions," Finance and Stochastics, Springer, vol. 11(3), pages 373-397, July.
    7. Emmanuel Lépinette & Ilya Molchanov, 2021. "Risk arbitrage and hedging to acceptability under transaction costs," Finance and Stochastics, Springer, vol. 25(1), pages 101-132, January.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Hongjie Tang & Shicheng Zhang & Jinhui Li & Lingwei Kong & Baoqiang Zhang & Fei Xing & Huageng Luo, 2023. "Imprecise P-Box Sensitivity Analysis of an Aero-Engine Combustor Performance Simulation Model Considering Correlated Variables," Energies, MDPI, vol. 16(5), pages 1-22, March.
    2. Andreas H. Hamel & Frank Heyde, 2021. "Set-Valued T -Translative Functions and Their Applications in Finance," Mathematics, MDPI, vol. 9(18), pages 1-33, September.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Ilya Molchanov & Anja Muhlemann, 2019. "Nonlinear expectations of random sets," Papers 1903.04901, arXiv.org.
    2. Emmanuel Lepinette & Ilya Molchanov, 2016. "Risk Arbitrage and Hedging to Acceptability under Transaction Costs," Papers 1605.07884, arXiv.org, revised Apr 2020.
    3. Emmanuel Lépinette & Ilya Molchanov, 2021. "Risk arbitrage and hedging to acceptability under transaction costs," Finance and Stochastics, Springer, vol. 25(1), pages 101-132, January.
    4. Emmanuel Lepinette, 2020. "Random optimization on random sets," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 91(1), pages 159-173, February.
    5. Antonio Avilés López & José Miguel Zapata García, 2020. "Boolean Valued Representation of Random Sets and Markov Kernels with Application to Large Deviations," Mathematics, MDPI, vol. 8(10), pages 1-23, October.
    6. Andreas Haier & Ilya Molchanov & Michael Schmutz, 2015. "Intragroup transfers, intragroup diversification and their risk assessment," Papers 1511.06320, arXiv.org, revised Nov 2016.
    7. Haier Andreas & Molchanov Ilya, 2019. "Multivariate risk measures in the non-convex setting," Statistics & Risk Modeling, De Gruyter, vol. 36(1-4), pages 25-35, December.
    8. Jun Zhao & Emmanuel Lépinette & Peibiao Zhao, 2019. "Pricing under dynamic risk measures," Post-Print hal-02135232, HAL.
    9. Emmanuel Lepinette & Ilya Molchanov, 2017. "Conditional cores and conditional convex hulls of random sets," Papers 1711.10303, arXiv.org.
    10. Andreas Haier & Ilya Molchanov, 2019. "Multivariate risk measures in the non-convex setting," Papers 1902.00766, arXiv.org, revised Sep 2019.
    11. Cascos Fernández, Ignacio & Molchanov, Ilya, 2013. "Multivariate risk measures : a constructive approach based on selections," DES - Working Papers. Statistics and Econometrics. WS ws130101, Universidad Carlos III de Madrid. Departamento de Estadística.
    12. Asgar Jamneshan & Michael Kupper & José Miguel Zapata-García, 2020. "Parameter-Dependent Stochastic Optimal Control in Finite Discrete Time," Journal of Optimization Theory and Applications, Springer, vol. 186(2), pages 644-666, August.
    13. Dilip B. Madan, 2016. "Benchmarking in two price financial markets," Annals of Finance, Springer, vol. 12(2), pages 201-219, May.
    14. Andreas Haier & Ilya Molchanov & Michael Schmutz, 2016. "Intragroup transfers, intragroup diversification and their risk assessment," Annals of Finance, Springer, vol. 12(3), pages 363-392, December.
    15. Zachary Feinstein & Birgit Rudloff, 2013. "A comparison of techniques for dynamic multivariate risk measures," Papers 1305.2151, arXiv.org, revised Jan 2015.
    16. Meriam El Mansour & Emmanuel Lépinette, 2020. "Conditional Interior and Conditional Closure of Random Sets," Journal of Optimization Theory and Applications, Springer, vol. 187(2), pages 356-369, November.
    17. Zuo, Yijun & Lai, Shaoyong, 2011. "Exact computation of bivariate projection depth and the Stahel-Donoho estimator," Computational Statistics & Data Analysis, Elsevier, vol. 55(3), pages 1173-1179, March.
    18. Erhan Bayraktar & Yuchong Zhang, 2016. "Fundamental Theorem of Asset Pricing Under Transaction Costs and Model Uncertainty," Mathematics of Operations Research, INFORMS, vol. 41(3), pages 1039-1054, August.
    19. Claus, Matthias, 2022. "Existence of solutions for a class of bilevel stochastic linear programs," European Journal of Operational Research, Elsevier, vol. 299(2), pages 542-549.
    20. Tuan Nguyen Dinh, 2023. "Regularity of Multipliers for Multiobjective Optimal Control Problems Governed by Evolution Equations," Journal of Optimization Theory and Applications, Springer, vol. 196(2), pages 762-796, February.

    More about this item

    Keywords

    Multiasset portfolio; Random set; Selection expectation; Sublinear expectation; Superlinear expectation; Set-valued function; Transaction costs; Utility;
    All these keywords.

    JEL classification:

    • C18 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Methodolical Issues: General
    • C65 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Miscellaneous Mathematical Tools

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:finsto:v:25:y:2021:i:1:d:10.1007_s00780-020-00442-3. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.