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Decision tree analysis for a risk averse decision maker: CVaR criterion. (English) Zbl 1317.91023
Summary: Risk aversion is a prevalent phenomenon when sufficiently large amounts are at risk. In this paper, we introduce a new prescriptive approach for coping with risk in sequential decision problems with discrete scenario space. We use Conditional Value-at-Risk (CVaR) risk measure as optimization criterion and prove that there is an explicit linear representation of the proposed model for the problem.

91B06 Decision theory
Full Text: DOI
[1] Ahmed, S., Convexity and decomposition of mean-risk stochastic programs, Mathematical Programming, 106, 433-446, (2006) · Zbl 1134.90025
[2] Ahmed, S.; Cakmak, U.; Shapiro, A., Coherent risk measures in inventory problems, European Journal of Operational Research, 182, 1, 226-238, (2007) · Zbl 1128.90002
[3] Anderson, S. P.; De Palma, A.; Thisse, J. F., Discrete choice theory of product differentiation, (1992), MIT Press Cambridge, Massachusetts · Zbl 0857.90018
[4] Arrow, K. J., Essays in the theory of risk-bearing, (1971), Markham Pub. Co. Chicago · Zbl 0215.58602
[5] Artzner, P., Coherent multiperiod risk adjusted values and bellmans principle, Annals of Operations Research, 5-22, 18, (2007) · Zbl 1132.91484
[6] Artzner, P.; Delbaen, F.; Eber, J.-M.; Heath, D., Coherent measures of risk, Mathematical Finance, 9, 3, 203-228, (1999) · Zbl 0980.91042
[7] Artzner, P., Delbaen, F., Eber, J. Marc, Heath, D., Ku, H., 2002. Coherent Multiperiod Risk Measurement. <www.math.ethz.ch/∼delbaen>. · Zbl 1132.91484
[8] Balas, E., Disjunctive programming: properties of the convex hull of feasible points, Discrete Applied Mathematics, 89, 1-3, 3-44, (1998) · Zbl 0921.90118
[9] Ben-Tal, A.; Teboulle, M., An old-new concept of convex risk measures: the optimized certainty equivalent, Mathematical Finance, 17, 3, 449-476, (2007) · Zbl 1186.91116
[10] Collado, R.A., Papp, D., Ruszczyǹski, A., 2010. Scenario Decomposition of Risk-Averse Multistage Stochastic Programming Problems (Working paper). RUTCOR, Rutgers University, Piscataway, NJ 08854. · Zbl 1255.90086
[11] Detlefsen, K.; Scandolo, G., Conditional and dynamic convex risk measures, Finance and Stochastics, 9, 4, 539-561, (2005) · Zbl 1092.91017
[12] Föllmer, H.; Schied, A., Convex measures of risk and trading constraints, Finance and Stochastics, 6, 429-447, (2002) · Zbl 1041.91039
[13] Föllmer, H.; Schied, A., Stochastic Finance, An Introduction in Discrete Time, (2004), Walter de Gruyter Berlin, New York · Zbl 1126.91028
[14] French, S., Mathematical programming approaches to sensitivity calculations in decision analysis, The Journal of the Operational Research Society, 43, 8, 813-819, (1992) · Zbl 0825.90579
[15] Frittelli, M.; Gianin, M. R., Dynamic convex risk measures, (Szegö, G. P., Risk Measures for the 21st Century, (2004), Wiley Chichester; Hoboken, NJ)
[16] Kreps, D. M., A course in microeconomic theory, (1990), Princeton University Princeton, NJ
[17] Laporte, G.; Louveaux, F. V., The integer L-shaped method for stochastic integer programs with complete recourse, Operations Research Letters, 13, 3, 133-142, (1993) · Zbl 0793.90043
[18] Miller, N.; Ruszczyński, A., Risk-averse two-stage stochastic linear programming: modeling and decomposition, Operations Research, 59, 1, 125-132, (2011) · Zbl 1218.90145
[19] Parija, G. R.; Ahmed, S.; King, A. J., On bridging the gap between stochastic integer programming and MIP solver technologies, Informs Journal on Computing, 16, 1, 73-83, (2004) · Zbl 1239.90078
[20] Raiffa, H.; Schlaifer, R., Applied statistical decision theory, (1961), Division of Research, Graduate School of Business Administration, Harvard University Boston · Zbl 0181.21801
[21] Rockafellar, R. T.; Uryasev, S., Optimization of conditional value-at-risk, Journal of Risk, 2, 21-41, (2000)
[22] Rockafellar, R. T.; Uryasev, S., Conditional value-at-risk for general loss distributions, Journal of Banking and Finance, 26, 7, 1443-1471, (2002)
[23] Rockafellar, R.T., Uryasev, S., Zabarankin, M., 2002. Deviation Measures in Risk Analysis and Optimization (Research Report 2002-7). Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL. · Zbl 1138.91474
[24] Rockafellar, R. T.; Uryasev, S.; Zabarankin, M., Generalized deviations in risk analysis, Finance and Stochastics, 10, 51-74, (2006) · Zbl 1150.90006
[25] Ruszczyński, A.; Shapiro, A., Optimization of convex risk functions, Mathematics of Operations Research, 31, 433-452, (2006) · Zbl 1278.90283
[26] Ruszczyński, A.; Shapiro, A., Optimization of risk measures, (Calafiore, G.; Dabbene, F., Probabilistic and Randomized Methods for Design under Uncertainty, (2006), Springer London), 119-157 · Zbl 1181.90281
[27] Schultz, R.; Tiedemann, S., Conditional value-at-risk in stochastic programs with mixed-integer recourse, Mathematical Programming, 105, 365-386, (2006) · Zbl 1085.90042
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