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Optimization of Convex Risk Functions


Author Info

  • Andrzej Ruszczynski

    (Rutgers University)

  • Alexander Shapiro

    (Georgia Institute of Technology)


We consider optimization problems involving convex risk functions. By employing techniques of convex analysis and optimization theory in vector spaces of measurable functions we develop new representation theorems for risk models, and optimality and duality theory for problems involving risk functions.

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Bibliographic Info

Paper provided by EconWPA in its series Risk and Insurance with number 0404001.

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Length: 26 pages
Date of creation: 12 Apr 2004
Date of revision: 08 Oct 2005
Handle: RePEc:wpa:wuwpri:0404001

Note: Type of Document - pdf; pages: 26
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Related research

Keywords: Convex analysis; stochastic optimization; risk measures; mean- variance models; duality;

This paper has been announced in the following NEP Reports:


References listed on IDEAS
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  1. Ogryczak, Wlodzimierz & Ruszczynski, Andrzej, 1999. "From stochastic dominance to mean-risk models: Semideviations as risk measures," European Journal of Operational Research, Elsevier, vol. 116(1), pages 33-50, July.
  2. Hans Föllmer & Alexander Schied, 2002. "Convex measures of risk and trading constraints," Finance and Stochastics, Springer, vol. 6(4), pages 429-447.
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Cited by:
  1. Fertis, Apostolos & Baes, Michel & Lüthi, Hans-Jakob, 2012. "Robust risk management," European Journal of Operational Research, Elsevier, vol. 222(3), pages 663-672.
  2. Ludger Rüschendorf, 2012. "Worst case portfolio vectors and diversification effects," Finance and Stochastics, Springer, vol. 16(1), pages 155-175, January.
  3. Dentcheva, Darinka & Penev, Spiridon, 2010. "Shape-restricted inference for Lorenz curves using duality theory," Statistics & Probability Letters, Elsevier, vol. 80(5-6), pages 403-412, March.
  4. Stadje, Mitja, 2010. "Extending dynamic convex risk measures from discrete time to continuous time: A convergence approach," Insurance: Mathematics and Economics, Elsevier, vol. 47(3), pages 391-404, December.
  5. Karl-Theodor Eisele & Sonia Taieb, 2013. "Lattice Modules Over Rings Of Bounded Random Variables," Working Papers of LaRGE Research Center 2013-06, Laboratoire de Recherche en Gestion et Economie (LaRGE), Université de Strasbourg (France).
  6. Patrick Cheridito & Tianhui Li, 2009. "Risk Measures On Orlicz Hearts," Mathematical Finance, Wiley Blackwell, vol. 19(2), pages 189-214.
  7. Alejandro Balbas, 2008. "Capital requirements: Are they the best solution?," Business Economics Working Papers wb087114, Universidad Carlos III, Departamento de Economía de la Empresa.
  8. Volker Kr\"atschmer & Alexander Schied & Henryk Z\"ahle, 2012. "Comparative and qualitative robustness for law-invariant risk measures," Papers 1204.2458,, revised Jan 2014.
  9. Keita Owari, 2013. "On the Lebesgue Property of Monotone Convex Functions," CARF F-Series CARF-F-317, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
  10. Eduard Kromer & Ludger Overbeck, 2013. "Suitability of Capital Allocations for Performance Measurement," Papers 1301.5497,, revised Nov 2013.
  11. Shapiro, Alexander, 2012. "Minimax and risk averse multistage stochastic programming," European Journal of Operational Research, Elsevier, vol. 219(3), pages 719-726.
  12. Eskandarzadeh, Saman & Eshghi, Kourosh, 2013. "Decision tree analysis for a risk averse decision maker: CVaR Criterion," European Journal of Operational Research, Elsevier, vol. 231(1), pages 131-140.
  13. Alejandro Balbás & Raquel Balbás, 2009. "Compatibility between pricing rules and risk measures: The CCVaR," Business Economics Working Papers wb090201, Universidad Carlos III, Departamento de Economía de la Empresa.
  14. Bellini, Fabio & Rosazza Gianin, Emanuela, 2008. "On Haezendonck risk measures," Journal of Banking & Finance, Elsevier, vol. 32(6), pages 986-994, June.
  15. Mustafa Pınar, 2011. "Gain–loss based convex risk limits in discrete-time trading," Computational Management Science, Springer, vol. 8(3), pages 299-321, August.
  16. Alois Pichler & Alexander Shapiro, 2012. "Uniqueness of Kusuoka Representations," Papers 1210.7257,, revised Feb 2013.
  17. Walter Farkas & Pablo Koch-Medina & Cosimo Munari, 2012. "Beyond cash-additive risk measures: when changing the num\'{e}raire fails," Papers 1206.0478,, revised Feb 2014.
  18. Andrzej Ruszczynski & Alexander Shapiro, 2004. "Optimization of Risk Measures," Risk and Insurance 0407002, EconWPA.
  19. Shapiro, Alexander, 2011. "Analysis of stochastic dual dynamic programming method," European Journal of Operational Research, Elsevier, vol. 209(1), pages 63-72, February.
  20. Andrzej Ruszczynski & Alexander Shapiro, 2004. "Conditional Risk Mappings," Risk and Insurance 0404002, EconWPA, revised 08 Oct 2005.
  21. Volker Krätschmer, 2007. "On s-additive robust representation of convex risk measures for unbounded financial positions in the presence of uncertainty about the market model," SFB 649 Discussion Papers SFB649DP2007-010, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
  22. Tiexin Guo, 2010. "Recent progress in random metric theory and its applications to conditional risk measures," Papers 1006.0697,, revised Mar 2011.


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