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Continuity properties of law-invariant (quasi-)convex risk functions on L

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We study continuity properties of law-invariant (quasi-)convex functions \({f:L^\infty(\Omega, \mathcal{F}, \mathbb{P}) \to (-\infty,\infty]}\) over a non-atomic probability space \({(\Omega, \mathcal{F}, \mathbb{P})}\). This is a supplementary note to Jouini et al. (Adv Math Econ 9:49–71, 2006).

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References

  1. Acciaio B.: Optimal risk sharing with non-monotone monetary functionals. Finan. Stoch. 11, 267–289 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  2. Aliprantis C.D., Border K.C.: Infinite Dimensional Analysis. Springer, NY (1999)

    MATH  Google Scholar 

  3. Cerreia-Vioglio, S., Maccheroni, F., Marinacci, M., Montrucchio, L.: Risk Measures: Rationality and Diversification. Collegio Carlo Alberto, Working Paper No. 100 (2008)

  4. Cerreia-Vioglio, S., Maccheroni, F., Marinacci, M., Montrucchio, L.: Complete Monotone Quasiconcave Duality, Collegio Carlo Alberto, Working Paper No. 80 (2008)

  5. Drapeau, S., Kupper, M.: Risk preferences and their robust representations, preprint (2009)

  6. Ekeland, I., Témam, R.: Convex Analysis and Variational Problems, Chap. I. SIAM (1999)

  7. Filipović D., Svindland G.: Optimal capital and risk allocations for law-invariant convex cash-invariant functions. Finan. Stoch. 12, 423–439 (2008)

    Article  MATH  Google Scholar 

  8. Filipović, D., Svindland, G.: The canonical model space for law-invariant convex risk measures is L 1. Math. Finan., forthcoming (2009)

  9. Föllmer H., Schied A.: Stochastic Finance, An Introduction in Discrete Time, De Gruyter Studies in Mathematics 27. 2nd edn. Walter de Gruyter, NY (2004)

    Book  Google Scholar 

  10. Föllmer H., Schied A., Weber S.: Robust preferences and robust portfolio choice. In: Bensoussan, A, Zhang, Q (eds) Handbook of Numerical Analysis. Mathematical Modeling and Numerical Methods in Finance, pp. 29–89. Springer, NY (2009)

    Google Scholar 

  11. Jouini E., Schachermayer W., Touzi N.: Law-invariant risk measures have the Fatou property. Adv. Math. Econ. 9, 49–71 (2006)

    Article  MathSciNet  Google Scholar 

  12. Jouini E., Schachermayer W., Touzi N.: Optimal risk sharing for law invariant monetary utility functions. Math. Finan. 18, 269–292 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. Kaina M., Rüschendorf L.: On convex risk measures on L p-spaces. Math. Methods Oper. Res. 69(3), 475–495 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Rockafellar, R.T.: Conjugate Duality and Optimization. SIAM (1974)

  15. Rohlin V.A.: On the fundamental ideas of measure theory. Am. Math. Soc: Transl. 1 10, 1–54 (1962)

    Google Scholar 

  16. Royden H.L.: Real Analysis. vol. XV, 2nd edn. Macmillan Publishing, New York (1968)

    Google Scholar 

  17. Weber S.: Distrxibution-invariant risk measures, entropy, and large deviations. J. Appl. Probab. 44(1), 16–40 (2007)

    Article  MathSciNet  MATH  Google Scholar 

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Correspondence to Gregor Svindland.

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Svindland, G. Continuity properties of law-invariant (quasi-)convex risk functions on L . Math Finan Econ 3, 39–43 (2010). https://doi.org/10.1007/s11579-010-0026-x

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