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Set-valued dynamic risk measures for processes and for vectors

Abstract

The relationship between set-valued risk measures for processes and vectors on the optional filtration is investigated. The equivalence of risk measures for processes and vectors and the equivalence of their penalty function formulations are provided. In contrast to scalar risk measures, this equivalence requires an augmentation of the set-valued risk measures for processes. We utilise this result to deduce a new dual representation for risk measures for processes in the set-valued framework. Finally, the equivalence of multi-portfolio time-consistency between set-valued risk measures for processes and vectors is provided. To accomplish this, an augmented definition for multi-portfolio time-consistency of set-valued risk measures for processes is proposed.

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Acknowledgements

The authors are very grateful to the Editors and the anonymous referees for their comments and suggestions which led to the present greatly improved version of the manuscript. Yanhong Chen’s research is partially supported by National Natural Science Foundation of China (No. 11901184) and Natural Science Foundation of Hunan Province (No. 2020JJ5025).

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Correspondence to Zachary Feinstein.

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Chen, Y., Feinstein, Z. Set-valued dynamic risk measures for processes and for vectors. Finance Stoch (2022). https://doi.org/10.1007/s00780-022-00476-9

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  • DOI: https://doi.org/10.1007/s00780-022-00476-9

Keywords

  • Set-valued risk measure
  • Dynamic risk measure
  • Time-consistency
  • Optional filtration

Mathematics Subject Classification (2020)

  • 26E25
  • 46A20
  • 46N10
  • 91B05
  • 91G05

JEL Classification

  • C61
  • D81
  • G32