Finance and Stochastics

, Volume 22, Issue 2, pp 395–415 | Cite as

Fatou property, representations, and extensions of law-invariant risk measures on general Orlicz spaces

  • Niushan Gao
  • Denny Leung
  • Cosimo Munari
  • Foivos Xanthos


We provide a variety of results for quasiconvex, law-invariant functionals defined on a general Orlicz space, which extend well-known results from the setting of bounded random variables. First, we show that Delbaen’s representation of convex functionals with the Fatou property, which fails in a general Orlicz space, can always be achieved under the assumption of law-invariance. Second, we identify the class of Orlicz spaces where the characterization of the Fatou property in terms of norm-lower semicontinuity by Jouini, Schachermayer and Touzi continues to hold. Third, we extend Kusuoka’s representation to a general Orlicz space. Finally, we prove a version of the extension result by Filipović and Svindland by replacing norm-lower semicontinuity with the (generally non-equivalent) Fatou property. Our results have natural applications to the theory of risk measures.


Risk measures Law-invariance Fatou property Dual representations Conditional expectations Orlicz spaces 

Mathematics Subject Classification (2010)

91B30 60E05 46E30 46A20 

JEL Classification

C65 G32 



We are grateful for the comments of an anonymous referee that improved the clarity of some proofs in the paper. The first author is a PIMS Postdoctoral Fellow. The second author is supported by AcRF grant R-146-000-242-114. The fourth author acknowledges the support of an NSERC grant.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Niushan Gao
    • 1
  • Denny Leung
    • 2
  • Cosimo Munari
    • 3
  • Foivos Xanthos
    • 4
  1. 1.Department of Mathematics and Computer ScienceUniversity of LethbridgeLethbridgeCanada
  2. 2.Department of MathematicsNational University of SingaporeSingaporeSingapore
  3. 3.Center for Finance and InsuranceUniversity of ZurichZurichSwitzerland
  4. 4.Department of MathematicsRyerson UniversityTorontoCanada

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