Modeling and optimization of risk

Surveys in Operations Research and Management Science

Volumn 16, Issue 2, 2011, Pages 49-66

  Krokhmal, Pavlo ^a   Zabarankin, Michael ^b   Uryasev, Stan ^c  

^a IA United States   (United States)

^b Stevens Institute of Technology   (United States)

^c University of Florida   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 79955592002     PISSN: 18767354     EISSN: None     Source Type: Book Series    
DOI: 10.1016/j.sorms.2010.08.001     Document Type: Article

References (130)

1. Prékopa A. Stochastic Programming 1995, Kluwer Academic Publishers.
2. Bernoulli D. Specimen teoriae novae de mensura sortis. Commentarii Academiae Scientarium Imperialis Petropolitanae 1738, V:175-192. English translation: exposition of a new theory on the measurement of risk, Econometrica, 22 (1954) 23-36.
3. von Neumann J., Morgenstern O. Theory of Games and Economic Behavior 1953, Princeton University Press, Princeton, NJ, edition.
4. Markowitz H.M. Portfolio selection. Journal of Finance 1952, 7:77-91.
5. Birge J.R., Louveaux F. Introduction to Stochastic Programming 1997, Springer, New York.
6. Kall P., Mayer J. Stochastic Linear Programming: Models, Theory, and Computation 2005, Springer.
7. Shapiro A., Dentcheva D., Ruszczyński A. Lectures on Stochastic Programming: Modeling and Theory 2009, SIAM, Philadelphia, PA.
8. Fishburn P.C. Utility Theory for Decision-Making 1970, Wiley, New York.
9. Fishburn P.C. Non-Linear Preference and Utility Theory 1988, Johns Hopkins University Press, Baltimore.
10. Karni E., Schmeidler D. Utility theory with uncertainty. Handbook of Mathematical Economics 1991, vol. IV. North-Holland, Amsterdam.
11. Föllmer H., Schied A. Stochastic Finance: An Introduction in Discrete Time 2004, Walter de Gruyter, Berlin. 2nd ed.
12. Quirk J.P., Saposnik R. Admissibility and measurable utility functions. The Review of Economic Studies 1962, 29:140-146.
13. Fishburn P.C. Decision and Value Theory 1964, John Wiley & Sons, New York.
14. Hadar J., Russell W.R. Rules for ordering uncertain prospects. The American Economic Review 1969, 59:25-34.
15. Levy H. Stochastic dominance and expected utility: survey and analysis. Management Science 1992, 38:555-593.
16. Müller A., Stoyan D. Comparison Methods for Stochastic Models and Risks 2002, John Wiley & Sons, Chichester.
17. Rothschild M., Stiglitz J. Increasing risk I: a definition. Journal of Economic Theory 1970, 2:225-243.
18. Quiggin J. A theory of anticipated utility. Journal of Economic Behavior and Organization 1982, 3:225-243.
19. Quiggin J. Generalized Expected Utility Theory-The Rank-Dependent Expected Utility Model 1993, Kluwer Academic Publishers, Dordrecht.
20. Yaari M.E. The dual theory of choice under risk. Econometrica 1987, 55:95-115.
21. Allais M. Le comportement de l'homme rationnel devant le risque: critique des postulats et axiomes de l'école américaine. Econometrica 1953, 21:503-546.
22. Ellsberg D. Risk, ambiguity, and the savage axioms. Quarterly Journal of Economics 1961, 75:643-669.
23. Choquet G. Theory of capacities. Annales de l'Institut Fourier Grenoble 1955, 5:131-295.
24. Schmeidler D. Integral representation without additivity. Proceedings of the Anerican Mathematical Society 1986, 97:255-261.
25. Schmeidler D. Subjective probability and expected utility without additivity. Econometrica 1989, 57:571-587.
26. Dentcheva D., Ruszczyński A. Optimization with stochastic dominance constraints. SIAM Journal on Optimization 2003, 14:548-566.
27. Dentcheva D., Ruszczyński A. Semi-infinite probabilistic optimization: first order stochastic dominance constraints. Optimization 2004, 53:583-601.
28. Luedtke J. New formulations for optimization under stochastic dominance constraints. SIAM Journal on Optimization 2008, 19:1433-1450.
29. Klein Haneveld W.K., van der Vlerk M.H. Integrated chance constraints: reduced forms and an algorithm. Computational Management Science 2006, 3:245-269.
30. Fábián C.I., Mitra G., Roman D. Processing second-order stochastic dominance models using cutting-plane representations. Mathematical Programming 2009.
31. Ogryczak W., Ruszczyński A. Dual stochastic dominance and related mean-risk models. SIAM Journal on Optimization 2002, 13:60-78.
32. Dentcheva D., Ruszczyński A. Inverse stochastic dominance constraints and rank dependent expected utility theory. Mathematical Programming 2006, 108:297-311.
33. Dentcheva D., Ruszczyński A. Duality between coherent risk measures and stochastic dominance constraints in risk-averse optimization. Pacific Journal of Optimization 2008, 4:433-446.
34. Dentcheva D., Ruszczyński A. Optimality and duality theory for stochastic optimization problems with nonlinear dominance constraints. Mathematical Programming 2004, 99:329-350.
35. Dentcheva D., Ruszczyński A. Robust stochastic dominance and its application to risk-averse optimization. Mathematical Programming 2010, 123:85-100.
36. Dentcheva D., Ruszczyński A. Portfolio optimization with stochastic dominance constraints. Journal of Banking & Finance 2006, 30:433-451.
37. Roman D., Darby-Dowman K., Mitra G. Portfolio construction based on stochastic dominance and target return distributions. Mathematical Programming 2006, 108:541-569.
38. Markowitz H.M. Portfolio Selection 1959, Wiley and Sons, New York.
39. Steinbach M.C. Markowitz revisited: mean-variance models in financial portfolio analysis. SIAM Review 2001, 43:31-85.
40. Krokhmal P., Uryasev S., Zrazhevsky G. Risk management for hedge fund portfolios: a comparative analysis of linear rebalancing strategies. Journal of Alternative Investments 2002, 5:10-29.
41. Dembo R.S., Rosen D. The practice of portfolio replication: a practical overview of forward and inverse problems. Annals of Operations Research 1999, 85:267-284.
42. Markowitz H.M. Mean-Variance Analysis in Portfolio Choice and Capital Markets 1987, Blackwell, Oxford.
43. Ogryczak W., Ruszczyński A. From stochastic dominance to mean-risk models: semideviations as risk measures. European Journal of Operational Research 1999, 116:33-50.
44. Ogryczak W., Ruszczyński A. On consistency of stochastic dominance and mean-semideviation models. Mathematical Programming 2001, 89:217-232.
45. Testuri C., Uryasev S. On relation between expected regret and conditional value-at-risk. Handbook of Numerical Methods in Finance 2004, 361-373. Birkhauser. Z. Rachev (Ed.).
46. Bawa V.S. Optimal rules for ordering uncertain prospects. Journal of Financial Economics 1975, 2:95-121.
47. Fishburn P.C. Mean-risk analysis with risk associated with below-target returns. The American Economic Review 1977, 67:116-126.
48. Porter R.B. Semivariance and stochastic dominance: a comparison. American Economic Review 1974, 64:200-204.
49. Klein Haneveld W.K. Duality in Stochastic Linear and Dynamic Programming. Lecture Notes in Economics and Mathematical Systems 1986, vol. 274. Springer, Berlin.
50. Young M.R. A minimax portfolio selection rule with linear programming solution. Management Science 1998, 44:673-683.
51. Kouvelis P., Yu G. Robust Discrete Optimization and its Applications 1997, Kluwer Academic Publishers, Dodrecht.
52. Morgan JP Riskmetrics 1994, JP Morgan, New York.
53. Jorion P. Value at Risk: The New Benchmark for Controlling Market Risk 1997, McGraw-Hill.
54. Duffie D., Pan J. An overview of value-at-risk. Journal of Derivatives 1997, 4:7-49.
55. Rockafellar R.T., Uryasev S. Conditional value-at-risk for general loss distributions. Journal of Banking & Finance 2002, 26:1443-1471.
56. Tasche D. Expected shortfall and beyond. Journal of Banking & Finance 2002, 26:1519-1533.
57. Inui K., Kijima M. On the significance of expected shortfall as a coherent risk measure. Journal of Banking & Finance 2005, 29:853-864.
58. Pflug G. Some remarks on the value-at-risk and the conditional value-at-risk. Probabilistic Constrained Optimization: Methodology and Applications 2000, 272-281. Kluwer Academic Publishers, Dordrecht. S. Uryasev (Ed.).
59. Wang S.S., Young V.R., Panjer H.H. Axiomatic characterization of insurance prices. Insurance: Mathematics and Economics 1997, 21:173-183. In Honor of Prof. J.A. Beekman.
60. Extremes and Integrated Risk Management 2000, Risk Books, London. P. Embrechts (Ed.).
61. Cooper A.C.W.W., Symonds G.H. Cost horizons and certainty equivalents: an approach to stochastic programming of heating oil. Management Science 1958, 4:235-263.
62. Rausand M., Høyland A. System Reliability Theory: Models, Statistical Methods, and Applications 2004, Wiley, Hoboken, NJ. 2nd ed.
63. Epstein B., Weissman I. Mathematical Models for Systems Reliability 2008, CRC Press, Boca Raton, FL.
64. Choi S.K., Grandhi R.V., Canfield R.A. Reliability-Based Structural Design 2007, Springer, London.
65. Dentcheva D. Optimization models with probabilistic constraints. Probabilistic and Randomized Methods for Design under Uncertainty 2006, 49-98. Springer, London. G. Calafiore, F. Dabbene (Eds.).
66. M.H. van der Vlerk, Integrated chance constraints in an alm model for pension funds. Working paper, 2003.
67. Nemirovski A., Shapiro A. Convex approximations of chance constrained programs. SIAM Journal on Optimization 2006, 17:969-996.
68. Artzner P., Delbaen F., Eber J.M., Heath D. Coherent measures of risk. Mathematical Finance 1999, 9:203-228.
69. Föllmer H., Schied A. Robust preferences and convex measures of risk. Advances in Finance and Stochastics: Essays in Honour of Dieter Sondermann 2002, 39-56. Springer. K. Sandmann, P.J. Schönbucher (Eds.).
70. Ruszczyński A., Shapiro A. Optimization of convex risk functions. Mathematics of Operations Research 2006, 31:433-452.
71. Rockafellar R.T., Uryasev S., Zabarankin M. Generalized deviations in risk analysis. Finance and Stochastics 2006, 10:51-74.
72. Delbaen F. Coherent risk measures on general probability spaces. Advances in Finance and Stochastics: Essays in Honour of Dieter Sondermann 2002, 1-37. Springer. K. Sandmann, P.J. Schönbucher (Eds.).
73. Cheridito P., Delbaen F., Kupper M. Coherent and convex monetary risk measures for bounded càdlàg processes. Stochastic Processes and their Applications 2004, 112:1-22.
74. De Giorgi E. Reward-risk portfolio selection and stochastic dominance. Journal of Banking & Finance 2005, 29:895-926.
75. Pflug G.C. Subdifferential representations of risk measures. Mathematical Programming 2006, 108:339-354.
76. Krokhmal P. Higher moment coherent risk measures. Quantitative Finance 2007, 7:373-387.
77. Frittelli M., Gianin E.R. Putting order in risk measures. Journal of Banking & Finance 2002, 26:1473-1486.
78. Rockafellar R.T. Convex Analysis. Princeton Mathematics 1970, vol. 28. Princeton University Press.
79. Zǎlinescu C. Convex Analysis in General Vector Spaces 2002, World Scientific, Singapore.
80. Kusuoka S. On law invariant risk measures. Advances in Mathematical Economics 2001, 3:83-95.
81. Kusuoka S. A remark on law invariant convex risk measures. Advances in Mathematical Economics 2007, 10:91-100.
82. Frittelli M., Gianin E.R. Law invariant convex risk measures. Advances in Mathematical Economics 2005, 7:33-46.
83. Dana R.A. A representation result for concave Schur concave functions. Mathematical Finance 2005, 15:613-634.
84. Acerbi C. Spectral measures of risk: a coherent representation of subjective risk aversion. Journal of Banking & Finance 2002, 26:1487-1503.
85. Jouini E., Meddeb M., Touzi N. Vector-valued coherent risk measures. Finance and Stochastics 2004, 8:531-552.
86. Hamel A.H. A duality theory for set-valued functions I: Fenchel conjugation theory. Set-Valued and Variational Analysis 2009, 17:153-182.
87. P. Artzner, F. Delbaen, J.M. Eber, D. Heath, H. Ku, (2002). Coherent multiperiod risk measurement, prprint.
88. Artzner P., Delbaen F., Eber J.M., Heath D., Ku H. Coherent multiperiod risk adjusted values and bellman's principle. Annals of Operations Research 2007, 152:5-22.
89. Ruszczyński A., Shapiro A. Conditional risk mappings. Mathematics of Operations Research 2006, 31:544-561.
90. Rockafellar R.T., Uryasev S. Optimization of conditional value-at-risk. Journal of Risk 2000, 2:21-41.
91. Acerbi C., Tasche D. On the coherence of expected shortfall. Journal of Banking & Finance 2002, 26:1487-1503.
92. Cherny A.S. Weighted V@R and its properties. Finance and Stochastics 2006, 10:367-393.
93. Krokhmal P., Palmquist J., Uryasev S. Portfolio optimization with conditional value-at-risk objective and constraints. Journal of Risk 2002, 4:43-68.
94. Künzi-Bay A., Mayer J. Computational aspects of minimizing conditional value-at-risk. Computational Management Science 2006, 3:3-27.
95. Alexander S., Coleman T., Li Y. Minimizing cvar and var for a portfolio of derivatives. Journal of Banking & Finance 2006, 30:583-605.
96. Lim C., Sherali H.D., Uryasev S. Portfolio optimization by minimizing conditional value-at-risk via nondifferentiable optimization. Computation Optimization and Applications 2008.
97. Rockafellar R.T., Royset J. On buffered failure probability in design and optimization of structures. Reliability Engineering & System Safety 2010, 95:499-510.
98. Chen G., Daskin M., Shen M., Uryasev S. The α-reliable mean-excess regret model for stochastic facility location modeling. Naval Research Logistics 2006, 53:617-626.
99. Ben-Tal A., Teboulle M. An old-new concept of convex risk measures: an optimized certainty equivalent. Mathematical Finance 2007, 17:449-476.
100. Ben-Tal A., Teboulle M. Expected utility, penalty functions, and duality in stochastic nonlinear programming. Management Science 1986, 32:1145-1466.
101. Cheridito P., Li T. Risk measures on Orlicz hearts. Mathematical Finance 2009, 19:189-214.
102. Fischer T. Risk capital allocation by coherent risk measures based on one-sided moments. Insurance: Mathematics and Economics 2003, 32:135-146.
103. Krokhmal P., Soberanis P. Risk optimization with p-order conic constraints: a linear programming approach. European Journal of Operational Research 2010, 201:653-671.
104. Ben-Tal A., Teboulle M. Portfolio theory for the recourse certainty equivalent maximizing investor. Annals of Operations Research 1991, 31:479-499.
105. Csiszar I. Information-type measures of difference of probability distributions and indirect observations. Studia Scientiarum Mathematicarum Hungarica 1967, 2:299-318.
106. Kullback S., Leibler R.A. On information and sufficiency. The Annals of Mathematical Statistics 1951, 22:79-86.
107. Föllmer H., Schied A. Convex measures of risk and trading constraints. Finance and Stochastics 2002, 6:429-447.
108. Eichhorn A., Römisch W. Polyhedral measures of risk in stochastic programming. SIAM Journal on Optimization 2005, 16:69-95.
109. Rockafellar R.T., Uryasev S., Zabarankin M. Risk tuning with generalized linear regression. Mathematics of Operations Research 2008, 33:712-729.
110. Koenker R., Bassett G. Regression quantiles. Econometrica 1978, 46:33-50.
111. R.T. Rockafellar, S. Uryasev, M. Zabarankin, 2002. Deviation measures in risk analysis and optimization. Technical Report 2002-7. ISE Department, University of Florida. Gainesville, FL.
112. Rockafellar R.T. Coherent approaches to risk in optimization under uncertainty. Tutorials in Operations Research INFORMS 2007 2007, 38-61. INFORMS.
113. Rockafellar R.T., Uryasev S., Zabarankin M. Master funds in portfolio analysis with general deviation measures. Journal of Banking & Finance 2006, 30:743-778.
114. Rockafellar R.T., Uryasev S., Zabarankin M. Optimality conditions in portfolio analysis with general deviation measures. Mathematical Programming 2006, 108:515-540.
115. Rockafellar R.T., Uryasev S., Zabarankin M. Equilibrium with investors using a diversity of deviation measures. Journal of Banking & Finance 2007, 31:3251-3268.
116. Grechuk B., Molyboha A., Zabarankin M. Chebyshev's inequalities with law invariant deviation measures. Probability in the Engineering and Informational Sciences 2010, 24:145-170.
117. Grechuk B., Molyboha A., Zabarankin M. Maximum entropy principle with general deviation measures. Mathematics of Operations Research 2009, 34:445-467.
118. Roell A. Risk aversion in Quiggin and Yaari's rank-order model of choice under uncertainty. The Economic Journal, Issue Supplement: Conference papers 1987, 97:143-159.
119. Markowitz H.M. Foundations of portfolio theory. Journal of Finance 1991, 46:469-477.
120. Tobin J. Liquidity preference as behavior towards risk. The Review of Economic Studies 1958, 25:65-86.
121. Sharpe W.F. Capital asset prices: a theory of market equilibrium under conditions of risk. Journal of Finance 1964, 19:425-442.
122. Sharpe W.F. Capital asset prices with and without negative holdings. Journal of Finance 1991, 46:489-509.
123. Grauer R.R. Introduction to asset pricing theory and tests. The International Library of Critical Writings in Financial Economics 2001, Edward Elgar Publishing Inc. R. Roll (Ed.).
124. Roy A.D. Safety first and the holding of assets. Econometrica 1952, 20:431-449.
125. Bonami P., Lejeune M.A. An exact solution approach for portfolio optimization problems under stochastic and integer constraints. Operations Research 2009, 57:650-670.
126. Jaynes E.T. Information theory and statistical mechanics. Physical Review 1957, 106:620-630.
127. Cozzolino J.M., Zahner M.J. The maximum-entropy distribution of the future market price of a stock. Operations Research 1973, 21:1200-1211.
128. Thomas M.U. A generalized maximum entropy principle. Operations Research 1979, 27:1188-1196.
129. Buckley J.J. Entropy principles in decision making under risk. Risk Analysis 1979, 5:303-313.
130. Cover T.M., Thomas J.A. Elements of Information Theory 2006, Wiley, New York. 2nd ed.