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Time Consistency

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Multistage Stochastic Optimization

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Abstract

In a multistage problem decisions x t have to be made at several stages, say at times \(t = 0,1,\ldots T\). The solution of the problem at time 0 consists of a complete plan for all future decisions at later times. If it turns out that it is preferable to change the initial plan at later stages, then the decision problem is calledinconsistent in time. As will be shown, time inconsistency may appear quite naturally in risk-averse stochastic multistage decision problems.

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Notes

  1. 1.

    More precisely: among all solutions of the problem for remaining times is also the solution of the original problem.

  2. 2.

    In the finite case, u can be just the node number.

  3. 3.

    \(\left (x_{0:t-1}^{{\ast}},x_{t:T}\right )\) is the concatenated vector \(\left (x_{0:t-1}^{{\ast}},\ldots,x_{t-1}^{{\ast}},\,x_{t},\ldots x_{T}\right )\).

  4. 4.

    Notice that ordinary version-independence means that the functional depends only on the distribution of the random variable. However, the composed functionals are all dependent only on thenested distribution and in this extended sense, they are version independent.

  5. 5.

    Cf. Williams [142] for independence of two sigma algebras.

  6. 6.

    We are grateful to Jochen Gönsch and Michael Hassler (University Augsburg) for pointing out a shortcoming in a previous version of the algorithm.

Bibliography

  1. Acerbi, C.: Spectral measures of risk: A coherent representation of subjective risk aversion. J. Bank. Finance 26, 1505–1518 (2002)

    Google Scholar 

  2. Acerbi, C., Simonetti, P.: Portfolio optimization with spectral measures of risk. EconPapers (2002)

    Google Scholar 

  3. Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd edn. Birkhäuser Verlag AG, Basel (2005)

    Google Scholar 

  4. Analui, B., Pflug, G.: On distributionnally robust mulistage stochastic optimization. Comput. Manag. Sci. 11(3), 197–220 (2014). DOI 10. 1007/s10287-014-0213-y

    Google Scholar 

  5. Arrow, K., Gould, F., Howe, S.: General saddle point results for constrained optimization. Math. Programm. 5, 225–234 (1973)

    Google Scholar 

  6. Artzner, P., Delbaen, F., Eber, J.M., Heath, D.: Coherent measures of risk. Math. Finance 9, 203–228 (1999)

    Google Scholar 

  7. Artzner, P., Delbaen, F., Eber, J.M., Heath, D., Ku, H.: Coherent multiperiod risk adjusted values and Bellman’s principle. Ann. Oper. Res. 152, 5–22 (2007). DOI 10.1007/s10479-006-0132-6

    Google Scholar 

  8. Bally, V., Pagés, G., Printems, J.: A quantization tree method for pricing and hedging multidimensional american options. Math. Finance 15(1), 119–168 (2005)

    Google Scholar 

  9. Bellman, R.E.: Dynamic Programming. Princeton University Press, Princton (1957)

    Google Scholar 

  10. Ben-Tal, A., Nemirovski, A.: Robust solution of uncertain linear programs. Oper. Res. Lett. 25, 1–13 (1999)

    Google Scholar 

  11. Birge, J.R., Louveaux, F.: Introduction to Stochastic Programming. Springer, New York (1997)

    Google Scholar 

  12. Bisschop, J.: AIMMS Optimization Modeling. Lulu Enterprises Incorporated (2006)

    Google Scholar 

  13. Bolley, F.: Separability and completeness for the Wasserstein distance. In: Donati-Martin, C., Émery, M., Rouault, A., Stricker, C. (eds.) Séminaire de Probabilités XLI, Lecture Notes in Mathematics, vol. 1934, pp. 371–377. Springer, Berlin/Heidelberg (2008)

    Google Scholar 

  14. Bolley, F., Guillin, A., Villani, C.: Quantitative concentration inequalities for empirical measures on non-compact spaces. Probab. Theory Relat. Fields 137(3-4), 541–593 (2007). DOI 10.1007/s00440-006-0004-7. URL http://dx.doi.org/10.1007/s00440-006-0004-7

  15. Calafiore, G.: Ambiguous risk measures and optimal robust portfolios. SIAM J. Control Optim. 18 (3), 853–877 (2007)

    Google Scholar 

  16. Carpentier, P., Chancelier, J.P., Cohen, G., de Lara, M., Girardeau, P.: Dynamic consistency for stochastic optimal control problems. Ann. Oper. Res. 200(1), 247–263 (2012). DOI 10.1007/s10479-011-1027-8

    Google Scholar 

  17. Cheridito, P., Kupper, M.: Recursiveness of indifference prices and translation-invariant preferences. Math. Financ. Econ. 2(3), 173–188 (2009). 10.1007/s11579-009-0020-3

    Google Scholar 

  18. Cheridito, P., Kupper, M.: Composition of time-consistent dynamic monetary risk measures in discrete time. Int. J. Theor. Appl. Finance (IJTAF) 14(1), 137–162 (2011). DOI http://dx.doi.org/10.1142/S0219024911006292

  19. Collado, R.A., Papp, D., Ruszczyński, A.P.: Scenario decomposition of risk-averse multistage stochastic programming problems. Ann. Oper. Res. 200(1), 147–170 (2012). http://dx.doi.org/10.1007/s10479-011-0935-y

  20. Danilin, Y.M., Panin, V.M.: Methods for searching saddle points. Kibernetika 3, 119–124 (1974)

    Google Scholar 

  21. Delage, E., Ye, Y.: Distributionally robust optimization under moment uncertainty with application to data-driven problems. Oper. Res. 58, 596–612 (2010)

    Google Scholar 

  22. Demyanov, V.F., Malomezov, V.N.: Introduction to Minimax. Wiley, New York (1974)

    Google Scholar 

  23. Demyanov, V.F., Pevnyi, A.B.: Numerical methods for finding saddle points. USSR Comput. Math. Math. Phys. 12, 1099–1127 (1972)

    Google Scholar 

  24. Denneberg, D.: Distorted probabilities and insurance premiums. In: Proceedings of the 14th SOR, Ulm. Athenäum, Frankfurt (1989)

    Google Scholar 

  25. Denneberg, D.: Distorted probabilities and insurance premiums. Meth. Oper. Res. 63, 21–42 (1990)

    Google Scholar 

  26. Deprez, O., Gerber, H.U.: On convex principles of premium calculation. Insur. Math. Econ. 4(3), 179–189 (1985). DOI http://dx.doi.org/10.1016/0167-6687(85)90014-9

  27. Dick, J., Pillichshammer, F.: Digital Nets and Sequences. Cambridge University Press, Cambridge (2010)

    Google Scholar 

  28. Dobrushin, R.L.: Central limit theorem for non-standard Markov chains. Dokl. Akad. Nauk SSSR 102(1), 5–8 (1956)

    Google Scholar 

  29. Dobrushin, R.L.: Prescribing a system of random variables by conditional distributions. Theor. Probab. Appl. 15, 458–486 (1970)

    Google Scholar 

  30. Dudley, R.M.: The speed of mean Glivenko-Cantelli convergence. Ann. Math. Stat. 40(1), 40–50 (1969)

    Google Scholar 

  31. Dunford, N., Schwartz, J.T.: Linear Operators. Part I. General Theory. Wiley-Interscience, New York (1957)

    Google Scholar 

  32. Dupačová, J.: On minimax decision rule in stochastic linear programing. Stud. Math. Program. 47–60 (1980)

    Google Scholar 

  33. Dupačová, J.: The minimax approach to stochastic programming and an illustrative application. Stochastics 20, 73–88 (1987)

    Google Scholar 

  34. Dupačová, J.: Stability and sensitivity-analysis for stochastic programming. Ann. Oper. Res. 27, 115–142 (1990)

    Google Scholar 

  35. Dupačová, J.: Uncertainties in minimax stochastic programs. Optimization 1, 191–220 (2010)

    Google Scholar 

  36. Dupačová, J., Gröwe-Kuska, N., Römisch, W.: Scenario reduction in stochastic programming. Math. Programm. A 95(3), 493–511 (2003). DOI 10.1007/s10107-002-0331-0

    Google Scholar 

  37. Dupačová, J., Hurt, J., Štěpán, J.: Stochastic Modeling in Economics and Finance. Applied Optimization. Kluwer Academic, Dordrecht (2003)

    Google Scholar 

  38. Durrett, R.: Probability: Theory and Examples. Duxbury Advanced Series. Thompson, Belmont (2005)

    Google Scholar 

  39. Edirishinge, N.C.P.: Stochastic Programming: The state of the Art in Honor of George B. Dantzig, chap. Stochastic Programming Approximations using Limited Moment Information with Application to Asset Allocation, pp. 97–138. International Series in Operations Research and Management Science. Springer (2011)

    Google Scholar 

  40. Eichhorn, A., Römisch, W.: Polyhedral risk measures in stochastic programming. SIAM J. Optim. 16(1), 69–95 (2005)

    Google Scholar 

  41. Eichhorn, A., Römisch, W.: Dynamic risk management in electricity portfolio optimization via polyhedral risk functionals. In: Proc. of the IEEE Power Engineering Society (PES) General Meeting, Pittsburgh, PA, USA (2008)

    Google Scholar 

  42. Ellsberg, D.: Risk, ambiguity and Savage axioms. Q. J. Econ. 75(4), 643–669 (1961)

    Google Scholar 

  43. Fan, K.: Minimax theorems. Proc. N.A.S 39, 42–47 (1953)

    Google Scholar 

  44. Fleming, W.H., Soner, H.M.: Controlled Markov Processes and Viscosity Solutions. Springer, New York (2006)

    Google Scholar 

  45. Gibbs, A.L., Su, F.E.: On choosing and bounding probability metrics. Int. Stat. Rev. 70(3), 419–435 (2002)

    Google Scholar 

  46. Glasserman, P.: Monte Carlo Methods in Financial Engineering, vol. 53. Springer, New York (2004)

    Google Scholar 

  47. Goh, J., Sim, M.: Distributionally robust optimization and its tractable approximations. Oper. Res. 58, 902–917 (2010)

    Google Scholar 

  48. Graf, S., Luschgy, H.: Foundations of Quantization for Probability Distributions, Lecture Notes in Mathematics, vol. 1730. Springer, Berlin/Heidelberg (2000)

    Google Scholar 

  49. Gutjahr, W.J., Pichler, A.: Stochastic multi-objective optimization: a survey on non-scalarizing method. Ann. Oper. Res. 1–25 (2013). DOI 10.1007/s10479-013-1369-5

    Google Scholar 

  50. Hansen, L.P., Sargent, T.J.: Robustness. Princeton University Press, Princeton (2007)

    Google Scholar 

  51. Hartigan, J.A.: Clustering Algorithms. Wiley, New York (1975)

    Google Scholar 

  52. Heitsch, H., Römisch, W.: Scenario reduction algorithms in stochastic programming. Comput. Optim. Appl. Stoch. Programm. 24(2-3), 187–206 (2003)

    Google Scholar 

  53. Heitsch, H., Römisch, W.: Scenario tree modeling for multistage stochastic programs. Math. Program. A 118, 371–406 (2009)

    Google Scholar 

  54. Heitsch, H., Römisch, W.: Scenario tree reduction for multistage stochastic programs. Comput. Manag. Sci. 2, 117–133 (2009)

    Google Scholar 

  55. Heitsch, H., Römisch, W., Strugarek, C.: Stability of multistage stochastic programs. SIAM J. Optim. 17(2), 511–525 (2006)

    Google Scholar 

  56. Henrion, R., Strugarek, C.: Convexity of chance constraints with independent random variables. Comput. Optim. Appl. 41, 263–276 (2008)

    Google Scholar 

  57. Henrion, R., Strugarek, C.: Convexity of chance constraints with dependent random variables: the use of copulae. In: Bertocchi, M., Consigli, G., Dempster, M. (eds.), Stochastic Optimization Methods in Finance and Energy, International Series in Operations Research and Management Science, Vol. 163, pp. 427-439. Springer, New York (2011)

    Google Scholar 

  58. Heyde, C.C.: On a property of the lognormal distribution. J. Roy. Stat. Soc. B 25(2), 392–393 (1963)

    Google Scholar 

  59. Hochreiter, R., Pflug, G.Ch.: Financial scenario generation for stochastic multi-stage decision processes as facility location problems. Ann. Oper. Res. 152(1), 257–272 (2007)

    Google Scholar 

  60. Hoffman, A.J., Kruskal, J.B.: Integral Boundary Points of Convex Polyhedra, chap. 3, pp. 49–76. Springer, Berlin/Heidelberg (2010)

    Google Scholar 

  61. Jagannathan, R.: Minimax procedure for a class of linear programs under uncertainty. Oper. Res. 25, 173–177 (1977)

    Google Scholar 

  62. Jobert, A., Rogers, L.C.G.: Valuations and dynamic convex risk measures. Math. Finance 18(1), 1–22 (2008). DOI http://dx.doi.org/10.1111/j.1467-9965.2007.00320.x

  63. Jouini, E., Schachermayer, W., Touzi, N.: Law invariant risk measures have the Fatou property. In: Kusuoka, S., Yamazaki, A. (eds.) Advances in Mathematical Economics, vol. 9, pp. 49–71. Springer, Tokyo (2006). DOI 10.1007/4-431-34342-3_4

    Google Scholar 

  64. Kantorovich, L.: On the translocation of masses. C.R. Acad. Sci. URSS 37, 199–201 (1942)

    Google Scholar 

  65. Karatzas, I., Shreve, S.E.: Methods of Mathematical Finance. Stochastic Modelling and Applied Probability. Springer, New York (1998)

    Google Scholar 

  66. Kersting, G.: Die Geschwindigkeit der Glivenko-Cantelli-Konvergenz gemessen in der Prohorov Metrik. Math. Z. 163, 65–102 (1978). In German

    Google Scholar 

  67. King, A.J., Wallace, S.W.: Modeling with Stochastic Programming, Springer Series in Operations Research and Financial Engineering, vol. XVI. Springer, New York (2013)

    Google Scholar 

  68. Knight, F.: Risk, Uncertainty and Profit. Houghton Mifflin, Boston (1921)

    Google Scholar 

  69. Komiya, H.: Elementary proof for Sion’s minimax theorem. Kodai Math. J. 11, 5–7 (1988). Sion

    Google Scholar 

  70. Kovacevic, R., Pflug, G.Ch.: Are time consistent valuations information-monotone? Int. J. Theor. Appl. Finan. 17(1) (2014). DOI 10.1142/S0219024914500034

    Google Scholar 

  71. Kovacevic, R., Pflug, G.Ch.: Time consistency and information monotonicity of multiperiod acceptability functionals. No. 8 in Radon Series on Computational and Applied Mathematics, pp. 347–369. de Gruyter, Berlin (2009)

    Google Scholar 

  72. Kudō, H.: A note on the strong convergence of σ-algebras. Ann. Probab. 2(1), 76–83 (1974). URL http://dx.doi.org/10.1214/aop/1176996752

  73. Kusuoka, S.: On law invariant coherent risk measures. Adv. Math. Econ. 3, 83–95 (2001)

    Google Scholar 

  74. Lemieux, C.: Monte Carlo and Quasi Monte Carlo Sampling. Springer Series in Statistics. Springer, New York (2009)

    Google Scholar 

  75. MacQueen, J.: Some methods for classification and analysis of multivariate observations. In: Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, Volume 1: Statistics, pp. 281–297. University of California Press, Berkeley, California (1967). URL http://projecteuclid.org/euclid.bsmsp/1200512992

  76. Maggioni, F., Allevi, E., Bertocchi, M.: Measures of information in multistage stochastic programming. SPEPS Ser. 2 (2012)

    Google Scholar 

  77. Maggioni, F., Pflug, G.: Bounds and approximations for multistage stochastic optimization. Manuscript, U Bergamo

    Google Scholar 

  78. Mirkov, R., Pflug, G.Ch.: Tree approximations of dynamic stochastic programs. SIAM J. Optim. 18(3), 1082–1105 (2007)

    Google Scholar 

  79. Monge, G.: Mémoire sue la théorie des déblais et de remblais. Histoire de l’Académie Royale des Sciences de Paris, avec les Mémoires de Mathématique et de Physique pour la même année, pp. 666–704 (1781)

    Google Scholar 

  80. Müller, A., Stoyan, D.: Comparison Methods for Stochastic Models and Risks. Wiley Series in Probability and Statistics. Wiley, Chichester (2002)

    Google Scholar 

  81. Na, S., Neuhoff, D.L.: Bennett’s integral for vector quantizers. IEEE Trans. Inform. Theory 41(4), 886–900 (1995)

    Google Scholar 

  82. von Neumann, J.: Zur Theorie der Gesellschaftsspiele. Math. Ann. (100), 295–320 (1928). In German

    Google Scholar 

  83. Niederreiter, H.: Random Number Generation and Quasi Monte Carlo Methods. SIAM, Philadelphia (1992)

    Google Scholar 

  84. Papamichail, D.M., Georgiou, P.E.: Seasonal ARIMA inflow models for reservoir sizing. J. Am. Water Res. Assoc. 37(4), 877–885 (2001)

    Google Scholar 

  85. Parthasarathy, K., Kalyanapuram, R.: Probability Measures on Metric Spaces. Academic press, New York (1972)

    Google Scholar 

  86. Penner, I.: Dynamic convex risk measures: Time consistency, prudence, and sustainability. Ph.D. thesis, Humboldt University of Berlin (2009)

    Google Scholar 

  87. Pflug, G.Ch., Pichler, A., Wozabal, D.: The 1/N investment strategy is optimal under high model ambiguity. J. Bank. Finance 36, 410–417 (2012)

    Google Scholar 

  88. Pflug, G.Ch.: Optimization of Stochastic Models, The Kluwer International Series in Engineering and Computer Science, vol. 373. Kluwer Academic, Dordrecht (1996). URL http://link.springer.com/book/10.10072F978-1-4613-1449-3

  89. Pflug, G.Ch.: Some remarks on the value-at-risk and the conditional value-at-risk. In: Uryasev, S. (ed.) Probabilistic Constrained Optimization: Methodology and Applications, pp. 272–281. Kluwer Academic, Dordrecht (2000)

    Google Scholar 

  90. Pflug, G.Ch.: Scenario tree generation for multiperiod financial optimization by optimal discretization. Math. Programm. 89, 251–271 (2001). DOI 10.1007/s101070000202

    Google Scholar 

  91. Pflug, G.Ch.: On distortion functionals. Stat. Risk Model. 24, 45–60 (2006). DOI dx.doi.org/10.1524/stnd.2006.24.1.45

    Google Scholar 

  92. Pflug, G.Ch.: Version-independence and nested distribution in multistage stochastic optimization. SIAM J. Optim. 20, 1406–1420 (2009). DOI http://dx.doi.org/10.1137/080718401

    Google Scholar 

  93. Pflug, G.Ch., Pichler, A.: Approximations for Probability Distributions and Stochastic Optimization Problems, International Series in Operations Research & Management Science, vol. 163, chap. 15, pp. 343–387. Springer, New York (2011). DOI 10.1007/978-1-4419-9586-5_ 15

    Google Scholar 

  94. Pflug, G.Ch., Pichler, A.: A distance for multistage stochastic optimization models. SIAM J. Optim. 22(1), 1–23 (2012). DOI http: //dx.doi.org/10.1137/110825054

    Google Scholar 

  95. Pflug, G.Ch., Pichler, A.: Time consistent decisions and temporal decomposition of coherent risk functionals. Optimization online (2012)

    Google Scholar 

  96. Pflug, G.Ch., Pichler, A.: Time-inconsistent multistage stochastic programs: martingale bounds. No. 3 in Stochastic Programming E-Print Series. Humboldt Universität, Institut für Mathematik (2012)

    Google Scholar 

  97. Pflug, G.Ch., Römisch, W.: Modeling, Measuring and Managing Risk. World Scientific, River Edge (2007)

    Google Scholar 

  98. Pflug, G.Ch., Wozabal, D.: Ambiguity in portfolio selection. Quant. Finance 7(4), 435–442 (2007). DOI 10.1080/14697680701455410

    Google Scholar 

  99. Pichler, A.: Distance of probability measures and respective continuity properties of acceptability functionals. Ph.D. thesis, University of Vienna, Vienna, Austria (2010)

    Google Scholar 

  100. Pichler, A.: Evaluations of risk measures for different probability measures. SIAM J. Optim. 23(1), 530–551 (2013). DOI http://dx.doi. org/10.1137/110857088

    Google Scholar 

  101. Pichler, A.: The natural Banach space for version independent risk measures. Insur. Math. Econ. 53(2), 405–415 (2013). DOI http://dx.doi.org/10.1016/j.insmatheco.2013.07.005. URL http://www.sciencedirect.com/science/article/pii/S0167668713001054

  102. Pichler, A.: Premiums and reserves, adjusted by distortions. Scand. Actuarial J. (2013). DOI 10.1080/03461238.2013.830228

    Google Scholar 

  103. Prekopa, A.: On logarithmic concave measures and functions. Acta Sci. Math. 34, 335–343 (1973)

    Google Scholar 

  104. Qi, L., Sun, W.: An iterative method for the minimax problem. Minimax and Applications. Kluwer Academic, Boston (1995)

    Google Scholar 

  105. Rachev, S.T.: Probability Metrics and the Stability of Stochastic Models. Wiley, West Sussex (1991)

    Google Scholar 

  106. Rachev, S.T., Römisch, W.: Quantitative stability in stochastic programming: The method of probability method. Math. Oper. Res. 27(4), 792–818 (2002)

    Google Scholar 

  107. Rachev, S.T., Rüschendorf, L.: Mass Transportation Problems Vol. I: Theory, Vol. II: Applications, Probability and its applications, vol. XXV. Springer, New York (1998)

    Google Scholar 

  108. Rachev, S.T., Stoyanov, S.V., Fabozzi, F.J.: A Probability Metrics Approach to Financial Risk Measures. Wiley, London (2011).

    Google Scholar 

  109. Robinson, W., Wets, R.J.B.: Stability in two stage stochastic programming. SIAM J. Control Optim. 25, 1409–1416 (1987)

    Google Scholar 

  110. Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)

    Google Scholar 

  111. Rockafellar, R.T., Uryasev, S.: Optimization of conditional value-at-risk. J. Risk 2, 21–41 (2000)

    Google Scholar 

  112. Rockafellar, R.T., Wets, R.J.B.: Nonanticipativity and L 1-martingales in stochastic optimization problems. Math. Programm. Study 6, 170–187 (1976)

    Google Scholar 

  113. Rockafellar, R.T., Wets, R.J.B.: The optimal recourse problem in discrete time: L 1-multipliers for inequality constraints. SIAM J. Control Optim. 16(1), 16–36 (1978)

    Google Scholar 

  114. Rockafellar, R.T., Wets, R.J.B.: Variational Analysis. Springer, New York (1997)

    Google Scholar 

  115. Römisch, W., Schultz, R.: Stability analysis for stochastic programs. Ann. Oper. Res. 30, 241–266 (1991)

    Google Scholar 

  116. Rüschendorf, L.: The Wasserstein distance and approximtion theorems. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 70, 117–129 (1985)

    Google Scholar 

  117. Rustem, B., Howe, M.: Algorithms for Worst-Case Design and Applications to Risk Management. Princeton University Press, Princeton (2002)

    Google Scholar 

  118. Ruszczyński, A., Shapiro, A.: Conditional risk mappings. Math. Oper. Res. 31, 544–561 (2006)

    Google Scholar 

  119. Ruszczyński, A.: Nonlinear Optimization. Princeton University Press, Princeton (2006)

    Google Scholar 

  120. Sasai, H.: An interior penalty method for minimax for problems with constraints. SIAM J. Control Optim. 12, 643–649 (1974)

    Google Scholar 

  121. Scarf, H.E.: A min-max solution of an inventory problem. In: Arrow, K.J., Karlin, S., Scarf, H. (eds.). Studies in the Mathematical Theory of Inventory and Production. Stanford University Press, Stanford (1958)

    Google Scholar 

  122. Schachermayer, W., Kupper, M.: Representation results for law invariant time consistent functions. Math. Financ. Econ. 2, 189–210 (2009). DOI 10.1007/s11579-009-0019-9

    Google Scholar 

  123. Shaked, M., Shanthikumar, J.G.: Stochastic Order. Springer Series in Statistics. Springer, New York (2007)

    Google Scholar 

  124. Shapiro, A.: On complexity of multistage stochastic programs. Oper. Res. Lett. 34, 1–8 (2006)

    Google Scholar 

  125. Shapiro, A.: On a time consistency concept in risk averse multistage stochastic programming. Oper. Res. Lett. 37(37), 143–147 (2009)

    Google Scholar 

  126. Shapiro, A.: On Kusuoka representations of law invariant risk measures. Math. Oper. Res. 38(1), 142–152 (2013). http://dx.doi.org/10.1287/moor.1120.0563

  127. Shapiro, A.: Time consistency of dynamic risk measures. Oper. Res. Lett. 40(6), 436–439 (2012). DOI 10.1016/j.orl.2012.08.007. URL http://www.sciencedirect.com/science/article/pii/S0167637712001010

  128. Shapiro, A., Ahmded, Sh.: On a class of minimax stochastic programs. SIAM J. Optim. 14, 1237–1249 (2004)

    Google Scholar 

  129. Shapiro, A., Dentcheva, D., Ruszczyński, A.: Lectures on Stochastic Programming. MOS-SIAM Series on Optimization. MQS-SIAM Series on Optimization 9 (2009). URL http://epubs.siam.org/doi/book/10.1137/1.9780898718751

  130. Shapiro, A., Kleywegt, A.J.: Minimax analysis of stochastic problems. Optim. Meth. Software 17, 523–542 (2002)

    Google Scholar 

  131. Shapiro, A., Nemirovski, A.: On complexity of stochastic programming problems. In: Jeyakumar, V., Rubinov, A. (eds.) Continuous Optimization: Current Trends and Applications, pp. 111–144. Springer, New York (2005)

    Google Scholar 

  132. Sion, M.: On general minimax theorems. Pac. J. Math. 8(1), 171–176 (1958)

    Google Scholar 

  133. Thiele, A.: Robust stochastic programming with uncertain probabilities. IMA J. Manag. Math. 19, 289–321 (2008)

    Google Scholar 

  134. van der Vaart, A.W.: Asymptotic Statistics. Cambridge University Press, Cambridge (1998)

    Google Scholar 

  135. Vallander, S.S.: Calculation of the Wasserstein distance between probability distributions on the line. Theory Probab. Appl. 18, 784–786 (1973)

    Google Scholar 

  136. Vershik, A.M.: Kantorovich metric: Initial history and little-known applications. J. Math. Sci. 133(4), 1410–1417 (2006). DOI 10.1007/s10958-006-0056-3. URL http://dx.doi.org/10.1007/s10958-006-0056-3

  137. Villani, C.: Topics in Optimal Transportation, Graduate Studies in Mathematics, vol. 58. American Mathematical Society, Providence (2003)

    Google Scholar 

  138. Žáčková, J.: On minimax solutions of stochastic linear programming problems. Časopis pro pěstování Mathematiky 91, 423–430 (1966)

    Google Scholar 

  139. Wang, S.S., Young, V.R., Panjer, H.H.: Axiomatic characterization of insurance prices. Insur. Math. Econ. 21, 173–183 (1997). DOI http://dx.doi.org/10.1016/S0167-6687(97)00031-0

  140. Weber, S.: Distribution-invariant risk measures, information, and dynamic consistency. Math. Finance 16(2), 419–441 (2006)

    Google Scholar 

  141. Werner, A.S., Pichler, A., Midthun, K.T., Hellemo, L., Tomasgard, A.: Risk measures in multi-horizon scenario trees. In: Kovacevic, R., Pflug, G.Ch., Vespucci, M.T. (eds.) Handbook of Risk Management in Energy Production and Trading, International Series in Operations Research & Management Science, vol. 199, chap. 8, pp. 183–208. Springer, New York (2013). URL http://www.springer.com/business+26+management/operations+research/book/978-1-4614-9034-0

  142. Williams, D.: Probability with Martingales. Cambridge University Press, Cambridge (1991)

    Google Scholar 

  143. Wozabal., D.: A framework for optimization under ambiguity. Ann. Oper. Res. 193(1), 21–47 (2012)

    Google Scholar 

  144. Zador, P.L.: Asymptotic quantization error of continuous signals and the quantization dimension. IEEE Trans. Inform. Theory 28, 139–149 (1982)

    Google Scholar 

  145. Zillober, Ch., Schittkowski, K., Moritzen, K.: Very large scale optimization by sequential convex programming. Optim. Math. Software 19 (1), 103–120 (2004)

    Google Scholar 

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Authors and Affiliations

  1. Department of Statistics and Operations Research, University of Vienna, Vienna, Austria

    Georg Ch. Pflug

  2. Department of Industrial Economics and Technology Management, Norwegian University of Science and Technology, Trondheim, Norway

    Alois Pichler

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  1. Georg Ch. Pflug
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  2. Alois Pichler
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Pflug, G.C., Pichler, A. (2014). Time Consistency. In: Multistage Stochastic Optimization. Springer Series in Operations Research and Financial Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-08843-3_5

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