Optimization of Convex Risk Functions

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 with convex risk functions.

[1]  Wlodzimierz Ogryczak,et al.  On consistency of stochastic dominance and mean–semideviation models , 2001, Math. Program..

[2]  Howard Raiffa,et al.  Applied Statistical Decision Theory. , 1961 .

[3]  D. Varberg.  Convex Functions , 1973 .

[4]  E. Beckenbach.  CONVEX FUNCTIONS , 2007 .

[5]  F. Delbaen.  Coherent Risk Measures on General Probability Spaces , 2002 .

[6]  F. Delbaen,et al.  Coherent and convex monetary risk measures for bounded càdlàg processes , 2004 .

[7]  A. Ruszczynski,et al.  Optimization of Risk Measures , 2006 .

[8]  R. Rockafellar,et al.  Optimization of conditional value-at risk , 2000 .

[9]  R. Rockafellar,et al.  Deviation Measures in Risk Analysis and Optimization , 2002 .

[10]  R. Rockafellar,et al.  Generalized Deviations in Risk Analysis , 2004 .

[11]  J. Aubin,et al.  Applied Nonlinear Analysis , 1984 .

[12]  Howard Raiffa,et al.  Applied Statistical Decision Theory. , 1961 .

[13]  D. A. Edwards.  On the existence of probability measures with given marginals , 1978 .

[14]  C. Castaing,et al.  Convex analysis and measurable multifunctions , 1977 .

[15]  H. Markowitz.  Mean—Variance Analysis , 1989 .

[16]  Wlodzimierz Ogryczak,et al.  Dual Stochastic Dominance and Related Mean-Risk Models , 2002, SIAM J. Optim..

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

[18]  A. Ioffe,et al.  Theory of extremal problems , 1979 .

[19]  R. Tyrrell Rockafellar.  Conjugate Duality and Optimization , 1974 .

[20]  W. Sharpe,et al.  Mean-Variance Analysis in Portfolio Choice and Capital Markets , 1987 .

[21]  D. Blackwell.  Equivalent Comparisons of Experiments , 1953 .

[22]  R. Rockafellar,et al.  Stochastic convex programming: basic duality. , 1976 .

[23]  G. Pflug.  Some Remarks on the Value-at-Risk and the Conditional Value-at-Risk , 2000 .

[24]  R. Phelps.  Convex Functions, Monotone Operators and Differentiability , 1989 .

[25]  Alexander Schied,et al.  Convex measures of risk and trading constraints , 2002, Finance Stochastics.

[26]  F. Delbaen,et al.  Coherent and convex risk measures for bounded cadlag processes , 2003 .

[27]  Kurt Marti.  Duality in stochastic linear and dynamic programming , 1987 .

[28]  H. Raiffa,et al.  Applied Statistical Decision Theory. , 1961 .

[29]  M. O. Lorenz,et al.  Methods of Measuring the Concentration of Wealth , 1905, Publications of the American Statistical Association.

[30]  Mark A. McComb.  Comparison Methods for Stochastic Models and Risks , 2003, Technometrics.

[31]  D. Bertsimas,et al.  Shortfall as a risk measure: properties, optimization and applications , 2004 .

[32]  Alexander Shapiro,et al.  On a Class of Minimax Stochastic Programs , 2004, SIAM J. Optim..

[33]  Patrick Billingsley,et al.  Probability and Measure. , 1986 .

[34]  Wlodzimierz Ogryczak,et al.  From stochastic dominance to mean-risk models: Semideviations as risk measures , 1999, Eur. J. Oper. Res..