Duality for Set-Valued Measures of Risk

Extending the approach of Jouini, Meddeb, and Touzi [Finance Stoch., 8 (2004), pp. 531-552] we define set-valued (convex) measures of risk and their acceptance sets, and we give dual representation theorems. A scalarization concept is introduced that has a meaning in terms of internal prices of portfolios of reference instruments. Using primal and dual descriptions, we introduce new examples for set-valued measures of risk, e.g., set-valued upper expectations, value at risk, average value at risk, and entropic risk measure.

[1]  Tetsuzo Tanino.  On supremum of a set in a multi-dimensional space , 1988 .

[2]  I. Ekeland,et al.  Convex analysis and variational problems , 1976 .

[3]  Andreas H. Hamel,et al.  A Duality Theory for Set-Valued Functions I: Fenchel Conjugation Theory , 2009 .

[4]  Philippe Artzner,et al.  Coherent Measures of Risk , 1999 .

[5]  Yuri Kabanov,et al.  Hedging and liquidation under transaction costs in currency markets , 1999, Finance Stochastics.

[6]  A. Göpfert.  Variational methods in partially ordered spaces , 2003 .

[7]  C. Zălinescu.  Convex analysis in general vector spaces , 2002 .

[8]  Uwe Küchler,et al.  Coherent risk measures and good-deal bounds , 2001, Finance Stochastics.

[9]  R. Tyrrell Rockafellar,et al.  Convex Analysis , 1970, Princeton Landmarks in Mathematics and Physics.

[10]  Rüschendorf Ludger,et al.  Law invariant convex risk measures for portfolio vectors , 2006 .

[11]  J. Aubin.  Set-valued analysis , 1990 .

[12]  Imen Bentahar,et al.  Tail Conditional Expectation for vector-valued Risks , 2006 .

[13]  Nizar Touzi,et al.  Vector-valued coherent risk measures , 2002, Finance Stochastics.

[14]  Ludger Rüschendorf,et al.  Consistent risk measures for portfolio vectors , 2006 .

[15]  Paul Embrechts,et al.  Bounds for functions of multivariate risks , 2006 .

[16]  A. V. Kulikov.  Multidimensional Coherent and Convex Risk Measures , 2008 .

[17]  Fabio Spizzichino,et al.  Kendall distributions and level sets in bivariate exchangeable survival models , 2009, Inf. Sci..

[18]  Christiane Tammer,et al.  A new approach to duality in vector optimization , 2007 .

[19]  Andreas Löhne,et al.  Solution concepts in vector optimization: a fresh look at an old story , 2011 .

[20]  A. Peressini.  Ordered topological vector spaces , 1967 .

[21]  W. Schachermayer.  The Fundamental Theorem of Asset Pricing under Proportional Transaction Costs in Finite Discrete Time , 2004 .

[22]  Ilya S. Molchanov,et al.  Multivariate risks and depth-trimmed regions , 2006, Finance Stochastics.