IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v152y2022icp424-451.html
   My bibliography  Save this article

A central limit theorem for sets of probability measures

Author

Listed:
  • Chen, Zengjing
  • Epstein, Larry G.

Abstract

We prove a central limit theorem for a sequence of random variables whose means are ambiguous and vary in an unstructured way. Their joint distribution is described by a set of (suitably equivalent) probability measures. The limit is defined by a backward stochastic differential equation that can be interpreted as modeling an ambiguous continuous-time random walk.

Suggested Citation

  • Chen, Zengjing & Epstein, Larry G., 2022. "A central limit theorem for sets of probability measures," Stochastic Processes and their Applications, Elsevier, vol. 152(C), pages 424-451.
    • Handle: RePEc:eee:spapps:v:152:y:2022:i:c:p:424-451
    DOI: 10.1016/j.spa.2022.07.003
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304414922001624
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spa.2022.07.003?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. Larry G. Epstein & Martin Schneider, 2010. "Ambiguity and Asset Markets," Annual Review of Financial Economics, Annual Reviews, vol. 2(1), pages 315-346, December.
    2. Zengjing Chen & Larry Epstein, 2002. "Ambiguity, Risk, and Asset Returns in Continuous Time," Econometrica, Econometric Society, vol. 70(4), pages 1403-1443, July.
    3. Larry G. Epstein & Hiroaki Kaido & Kyoungwon Seo, 2016. "Robust Confidence Regions for Incomplete Models," Econometrica, Econometric Society, vol. 84, pages 1799-1838, September.
    4. Tamer, Elie, 2010. "Partial Identification in Econometrics," Scholarly Articles 34728615, Harvard University Department of Economics.
    5. N. El Karoui & S. Peng & M. C. Quenez, 1997. "Backward Stochastic Differential Equations in Finance," Mathematical Finance, Wiley Blackwell, vol. 7(1), pages 1-71, January.
    6. Epstein, Larry G. & Schneider, Martin, 2003. "Recursive multiple-priors," Journal of Economic Theory, Elsevier, vol. 113(1), pages 1-31, November.
    7. Epstein, Larry G. & Schneider, Martin, 2003. "IID: independently and indistinguishably distributed," Journal of Economic Theory, Elsevier, vol. 113(1), pages 32-50, November.
    8. Gilboa, Itzhak & Schmeidler, David, 1989. "Maxmin expected utility with non-unique prior," Journal of Mathematical Economics, Elsevier, vol. 18(2), pages 141-153, April.
    9. Gilboa,Itzhak, 2009. "Theory of Decision under Uncertainty," Cambridge Books, Cambridge University Press, number 9780521517324, October.
    10. Riedel, Frank, 2004. "Dynamic coherent risk measures," Stochastic Processes and their Applications, Elsevier, vol. 112(2), pages 185-200, August.
    11. Chen, Zengjing & Peng, Shige, 2000. "A general downcrossing inequality for g-martingales," Statistics & Probability Letters, Elsevier, vol. 46(2), pages 169-175, January.
    12. Chen, Pingyan & Sung, Soo Hak, 2016. "On the strong laws of large numbers for weighted sums of random variables," Statistics & Probability Letters, Elsevier, vol. 118(C), pages 87-93.
    13. Garud N. Iyengar, 2005. "Robust Dynamic Programming," Mathematics of Operations Research, INFORMS, vol. 30(2), pages 257-280, May.
    14. Elie Tamer, 2010. "Partial Identification in Econometrics," Annual Review of Economics, Annual Reviews, vol. 2(1), pages 167-195, September.
    15. Alexander Shapiro, 2016. "Rectangular Sets of Probability Measures," Operations Research, INFORMS, vol. 64(2), pages 528-541, April.
    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. Zengjing Chen & Larry G. Epstein & Guodong Zhang, 2022. "Approximate optimality and the risk/reward tradeoff in a class of bandit problems," Papers 2210.08077, arXiv.org, revised Dec 2023.
    2. Vladimir V. Ulyanov, 2024. "From Classical to Modern Nonlinear Central Limit Theorems," Mathematics, MDPI, vol. 12(14), pages 1-17, July.
    3. Chen, Zengjing & Epstein, Larry G. & Zhang, Guodong, 2023. "A central limit theorem, loss aversion and multi-armed bandits," Journal of Economic Theory, Elsevier, vol. 209(C).
    4. Shuhui Liu, 2024. "The Maximal and Minimal Distributions of Wealth Processes in Black–Scholes Markets," Mathematics, MDPI, vol. 12(10), pages 1-18, May.
    5. Zengjing Chen & Huaijin Liang & Wei Wang & Xiaodong Yan, 2022. "Long bet will lose: demystifying seemingly fair gambling via two-armed Futurity bandit," Papers 2212.11766, arXiv.org.
    6. Huaijin Liang & Jin Ma & Wei Wang & Xiaodong Yan, 2024. "Demystifying the Two-Armed Futurity Bandit’s Unfairness and Apparent Fairness," Mathematics, MDPI, vol. 12(11), pages 1-14, May.

    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. Chen, Zengjing & Epstein, Larry G. & Zhang, Guodong, 2023. "A central limit theorem, loss aversion and multi-armed bandits," Journal of Economic Theory, Elsevier, vol. 209(C).
    2. Maximilian Blesch & Philipp Eisenhauer, 2021. "Robust decision-making under risk and ambiguity," Papers 2104.12573, arXiv.org, revised Oct 2021.
    3. Martin Schneider, 2010. "The Research Agenda: Martin Schneider on Multiple Priors Preferences and Financial Markets," EconomicDynamics Newsletter, Review of Economic Dynamics, vol. 11(2), April.
    4. Paul Viefers, 2012. "Should I Stay or Should I Go?: A Laboratory Analysis of Investment Opportunities under Ambiguity," Discussion Papers of DIW Berlin 1228, DIW Berlin, German Institute for Economic Research.
    5. Larry G. Epstein & Martin Schneider, 2008. "Ambiguity, Information Quality, and Asset Pricing," Journal of Finance, American Finance Association, vol. 63(1), pages 197-228, February.
    6. Xiaoyu Cheng, 2022. "Robust Data-Driven Decisions Under Model Uncertainty," Papers 2205.04573, arXiv.org.
    7. Dan A. Iancu & Marek Petrik & Dharmashankar Subramanian, 2015. "Tight Approximations of Dynamic Risk Measures," Mathematics of Operations Research, INFORMS, vol. 40(3), pages 655-682, March.
    8. Carole Bernard & Shaolin Ji & Weidong Tian, 2013. "An optimal insurance design problem under Knightian uncertainty," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 36(2), pages 99-124, November.
    9. Beißner, Patrick, 2013. "Coherent Price Systems and Uncertainty-Neutral Valuation," VfS Annual Conference 2013 (Duesseldorf): Competition Policy and Regulation in a Global Economic Order 80010, Verein für Socialpolitik / German Economic Association.
    10. Klibanoff, Peter & Marinacci, Massimo & Mukerji, Sujoy, 2009. "Recursive smooth ambiguity preferences," Journal of Economic Theory, Elsevier, vol. 144(3), pages 930-976, May.
    11. Berend Roorda & J. M. Schumacher & Jacob Engwerda, 2005. "Coherent Acceptability Measures In Multiperiod Models," Mathematical Finance, Wiley Blackwell, vol. 15(4), pages 589-612, October.
    12. Vorbrink, Jörg, 2014. "Financial markets with volatility uncertainty," Journal of Mathematical Economics, Elsevier, vol. 53(C), pages 64-78.
    13. Rosazza Gianin, Emanuela, 2006. "Risk measures via g-expectations," Insurance: Mathematics and Economics, Elsevier, vol. 39(1), pages 19-34, August.
    14. Roger J. A. Laeven & Mitja Stadje, 2014. "Robust Portfolio Choice and Indifference Valuation," Mathematics of Operations Research, INFORMS, vol. 39(4), pages 1109-1141, November.
    15. Shi, Zhan, 2019. "Time-varying ambiguity, credit spreads, and the levered equity premium," Journal of Financial Economics, Elsevier, vol. 134(3), pages 617-646.
    16. Massimo Guidolin & Francesca Rinaldi, 2013. "Ambiguity in asset pricing and portfolio choice: a review of the literature," Theory and Decision, Springer, vol. 74(2), pages 183-217, February.
    17. Ariel M. Viale & Antoine Giannetti & Luis Garcia-Feijoó, 2020. "The stock market’s reaction to macroeconomic news under ambiguity," Financial Markets and Portfolio Management, Springer;Swiss Society for Financial Market Research, vol. 34(1), pages 65-97, March.
    18. Daniele Pennesi, 2013. "Asset Prices in an Ambiguous Economy," Carlo Alberto Notebooks 315, Collegio Carlo Alberto.
    19. Larry G. Epstein & Martin Schneider, 2007. "Learning Under Ambiguity," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 74(4), pages 1275-1303.
    20. Larry G Epstein & Yoram Halevy, 2019. "Ambiguous Correlation," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 86(2), pages 668-693.

    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:eee:spapps:v:152:y:2022:i:c:p:424-451. 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: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/505572/description#description .

    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.