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On convex risk measures on L p-spaces

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Abstract

Much of the recent literature on risk measures is concerned with essentially bounded risks in L ∞. In this paper we investigate in detail continuity and representation properties of convex risk measures on L p spaces. This frame for risks is natural from the point of view of applications since risks are typically modelled by unbounded random variables. The various continuity properties of risk measures can be interpreted as robustness properties and are useful tools for approximations. As particular examples of risk measures on L p we discuss the expected shortfall and the shortfall risk. In the final part of the paper we consider the optimal risk allocation problem for L p risks.

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References

  • Acerbi C, Tasche D (2002) On the coherence of expected shortfall. J Bank Financ 26: 1487–1503

    Article  Google Scholar 

  • Aliprantis CD, Border KC (1994) Infinite dimensional analysis: a Hitchhikers’s guide. Springer, Heidelberg

    Google Scholar 

  • Barrieu P, Karoui NE (2005) Inf-convolution of risk measures and optimal risk transfer. Financ Stoch 9: 269–298

    Article  MATH  Google Scholar 

  • Biagini S, Frittelli M (2006) On continuity properties and dual representation of convex and monotone functionals on Fréchet lattices. Preprint

  • Brezis H (1999) Analyse fonctionnelle. Théorie et application. Dunod

  • Burgert C, Rüschendorf L (2006a) Consistent risk measures for portfolio vectors. Insur Math Econ 38: 289–297

    Article  MATH  Google Scholar 

  • Burgert C, Rüschendorf L (2006b) On the optimal risk allocation problem. Stat Decis 24: 153–171

    Article  MATH  Google Scholar 

  • Cheridito P, Li T (2006) Monetary risk measures on maximal subsets of Orlicz classes. Preprint

  • Dana RA (2005) A representation result for concave Schur concave functions. Math Financ 14: 613–634

    Article  MathSciNet  Google Scholar 

  • Delbaen F (2002) Coherent risk measures on general probability spaces. In: Sandmann K, Schönbucher P(eds) Advances in finance and stochastics: essays in honor of Dieter Sondermann. Springer, Heidelberg, pp 1–37

    Google Scholar 

  • Ekeland I, Teman R (1974) Analyse convexe et problèmes variationnels. Dunod

  • Filipovic D, Kupper M (2006) Equilibrium prices for monetary utility functions. Int J Theor Appl Financ (to appear)

  • Filipovic D, Kupper M (2007) Monotone and cash-invariant convex functions and hulls. Insur Math Econ 41: 1–16

    Article  MATH  MathSciNet  Google Scholar 

  • Föllmer H, Schied A (2004) Stochastic finance: an introduction in discrete time. de Gruyter

  • Frittelli M, Gianin E (2002) Putting order in risk measures. J Bank Finance 26: 1473–1486

    Article  Google Scholar 

  • Frittelli M, Scandolo G (2006) Risk measures and capital requirements for processes. Math Financ 16: 589–612

    MATH  MathSciNet  Google Scholar 

  • Heath D, Ku H (2004) Pareto equilibria with coherent measures of risk. Math Financ 14: 163–172

    Article  MATH  MathSciNet  Google Scholar 

  • Inoue A (2003) On the worst conditional expectation. J Math Anal Appl 286: 237–247

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • Kaina M (2007) Darstellung und Stetigkeitseigenschaften von Risikomaßen. Diplomarbeit, Universität Freiburg

  • Lo A (1999) The three P’s of total risk management. Finan Anal J 55: 13–26

    Article  MathSciNet  Google Scholar 

  • Nakano Y (2004) Efficient hedging with coherent risk measure. J Math Anal Appl 293: 345–354

    Article  MATH  MathSciNet  Google Scholar 

  • Rachev ST, Menn C, Fabozzi F (2005) Fat-tailed and skewed asset return distributions. Wiley Finance, New York

    Google Scholar 

  • Rockafellar RT, Uryasev S (2002) Conditional value-at-risk for general loss distributions. J Bank Financ 26: 1443–1471

    Article  Google Scholar 

  • Rüschendorf L (2006) Risk measures for portfolio vectors and allocation of risks. In: Proceedings of Karlsruhe econometric workshop, April 2006 (to appear)

  • Scandolo G (2004) Models of capital requirements in static and dynamic settings. Econ Notes 33: 415–435

    Article  Google Scholar 

  • Tasche D (2002) Expected shortfall and beyond. J Bank Financ 26: 1519–1533

    Article  Google Scholar 

Download references

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Authors and Affiliations

  1. Department of Mathematical Stochastics, University of Freiburg, Eckerstr. 1, 79104, Freiburg, Germany

    M. Kaina & L. Rüschendorf

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  1. M. Kaina
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  2. L. Rüschendorf
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Correspondence to L. Rüschendorf.

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Kaina, M., Rüschendorf, L. On convex risk measures on L p-spaces. Math Meth Oper Res 69, 475–495 (2009). https://doi.org/10.1007/s00186-008-0248-3

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  • DOI: https://doi.org/10.1007/s00186-008-0248-3

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