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Convex risk measures and the dynamics of their penalty functions

Author

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  • Föllmer Hans
  • Penner Irina

Abstract

We study various properties of a dynamic convex risk measure for bounded random variables which describe the discounted terminal values of financial positions. In particular we characterize time-consistency by a joint supermartingale property of the risk measure and its penalty function. Moreover we discuss the limit behavior of the risk measure in terms of asymptotic safety and of asymptotic precision, a property which may be viewed as a non-linear analogue of martingale convergence. These results are illustrated by the entropic dynamic risk measure.

Suggested Citation

  • Föllmer Hans & Penner Irina, 2006. "Convex risk measures and the dynamics of their penalty functions," Statistics & Risk Modeling, De Gruyter, vol. 24(1), pages 61-96, July.
    • Handle: RePEc:bpj:strimo:v:24:y:2006:i:1:p:61-96:n:3
    DOI: 10.1524/stnd.2006.24.1.61
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    Citations

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    Cited by:

    1. Bier, Monika & Engelage, Daniel, 2011. "Merging of opinions under uncertainty," Center for Mathematical Economics Working Papers 433, Center for Mathematical Economics, Bielefeld University.
    2. Zachary Feinstein & Birgit Rudloff, 2018. "Time consistency for scalar multivariate risk measures," Papers 1810.04978, arXiv.org, revised Nov 2021.
    3. Zachary Feinstein & Birgit Rudloff, 2018. "Scalar multivariate risk measures with a single eligible asset," Papers 1807.10694, arXiv.org, revised Feb 2021.
    4. Zachary Feinstein & Birgit Rudloff, 2013. "Time consistency of dynamic risk measures in markets with transaction costs," Quantitative Finance, Taylor & Francis Journals, vol. 13(9), pages 1473-1489, September.
    5. Acciaio, Beatrice & Föllmer, Hans & Penner, Irina, 2012. "Risk assessment for uncertain cash flows: model ambiguity, discounting ambiguity, and the role of bubbles," LSE Research Online Documents on Economics 50118, London School of Economics and Political Science, LSE Library.
    6. Zachary Feinstein & Birgit Rudloff, 2012. "Multiportfolio time consistency for set-valued convex and coherent risk measures," Papers 1212.5563, arXiv.org, revised Oct 2014.
    7. Zachary Feinstein & Birgit Rudloff, 2015. "A Supermartingale Relation for Multivariate Risk Measures," Papers 1510.05561, arXiv.org, revised Jan 2018.
    8. c{C}au{g}{i}n Ararat & Bar{i}c{s} Bilir & Elisa Mastrogiacomo, 2022. "Decomposable sums and their implications on naturally quasiconvex risk measures," Papers 2201.05686, arXiv.org.
    9. Zachary Feinstein & Birgit Rudloff, 2013. "A comparison of techniques for dynamic multivariate risk measures," Papers 1305.2151, arXiv.org, revised Jan 2015.

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