Mathematical Programming

, Volume 153, Issue 2, pp 459–493 | Cite as

Time-consistent approximations of risk-averse multistage stochastic optimization problems

Full Length Paper Series A

Abstract

In this paper we study the concept of time consistency as it relates to multistage risk-averse stochastic optimization problems on finite scenario trees. We use dynamic time-consistent formulations to approximate problems having a single coherent risk measure applied to the aggregated costs over all time periods. The dual representation of coherent risk measures is used to create a time-consistent cutting plane algorithm. Additionally, we also develop methods for the construction of universal time-consistent upper bounds, when the objective function is the mean-semideviation measure of risk. Our numerical results indicate that the resulting dynamic formulations yield close approximations to the original problem.

Keywords

Dynamic measures of risk Time consistency Decomposition 

Mathematics Subject Classification

90C15 90C25 49M27 

References

  1. 1.
    Artzner, P., Delbaen, F., Eber, J.M., Heath, D.: Coherent measures of risk. Math. Financ. 9(3), 203–228 (1999)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Föllmer, H., Schied, A.: Stochastic Finance: An Introduction in Discrete Time, de Gruyter Studies in Mathematics, vol. 27, 2nd edn. Walter de Gruyter & Co., Berlin (2004)Google Scholar
  3. 3.
    Fritelli, M., Rosazza Gianin, E.: Putting order in risk measures. J. Bank. Financ. 26, 1473–1486 (2002)CrossRefGoogle Scholar
  4. 4.
    Ruszczyński, A., Shapiro, A.: Optimization of convex risk functions. Math. Oper. Res. 31(3), 433–452 (2006)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    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)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Cheridito, P., Delbaen, F., Kupper, M.: Dynamic monetary risk measures for bounded discrete-time processes. Electron. J. Probab. 11, 57–106 (2006)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Fritelli, M., Scandolo, G.: Risk measures and capital requirements for processes. Math. Financ. 16, 589–612 (2006)CrossRefGoogle Scholar
  8. 8.
    Pflug, G.C., Römisch, W.: Modeling, Measuring and Managing Risk. World Scientific, Singapore (2007)CrossRefMATHGoogle Scholar
  9. 9.
    Riedel, F.: Dynamic coherent risk measures. Stoch. Process. Appl. 112, 185–200 (2004)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Ruszczyński, A., Shapiro, A.: Conditional risk mappings. Math. Oper. Res. 31(3), 544–561 (2006)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Wüthrich, M.V., Embrechts, P., Tsanakas, A.: Statistics & Risk Modeling. Issues 30 (2013)Google Scholar
  12. 12.
    A. Shapiro D. Dentcheva, A.R.: Lectures on Stochastic Programming. SIAM, Philadelphia (2009)Google Scholar
  13. 13.
    Miller, N., Ruszczyński, A.: Risk-averse two-stage stochastic linear programming: modeling and decomposition. Oper. Res. 59(1), 125–132 (2011)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Collado, R.A., Papp, D., Ruszczyński, A.: Scenario decomposition of risk-averse multistage stochastic programming problems. Ann. Oper. Res. 200(1), 147–170 (2012)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Pflug, G.C., Pichler, A.: Time consistency and temporal decomposition of positively homogeneous risk functionals (2012)Google Scholar
  16. 16.
    Xin, L., Shapiro, A.: Bounds for nested law invariant coherent risk measures. Oper. Res. Lett. (2012)Google Scholar
  17. 17.
    Iancu, D.A., Petrik, M., Subramanian, D.: Tight approximations of dynamic risk measuresGoogle Scholar
  18. 18.
    Roorda, B., Schumacher, J.M., Engwerda, J.: Coherent acceptability measures in multiperiod models. Math. Financ. 15(4), 589–612 (2005)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Aubin, J.P., Frankowska, H.: Set-Valued Analysis. Birkhäuser, Boston (1990)MATHGoogle Scholar
  20. 20.
    Ruszczyński, A.: Risk-averse dynamic programming for Markov decision processes. Math. Program. 125(2), 235–261 (2010)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Ruszczynski, A.: Nonlinear Optimization, vol. 13. Princeton University Press, Princeton (2006)MATHGoogle Scholar
  22. 22.
    Rockafellar, R.T.: Convex Analysis, vol. 28. Princeton University Press, Princeton (1997)Google Scholar
  23. 23.
    Rockafellar, R.T.: Conjugate Duality and Optimization, vol. 14. SIAM, Philadelphia (1974)Google Scholar
  24. 24.
    Roorda, B., Schumacher, J.M.: Time consistency conditions for acceptability measures, with an application to Tail Value at Risk. Insur. Math. Econ. 40(2), 209–230 (2007)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Pereira, M.V.F., Pinto, L.M.V.G.: Multi-stage stochastic optimization applied to energy planning. Math. Program. 52, 359–375 (1991)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Kozmik, V., Morton, D.P.: Risk-averse stochastic dual dynamic programming. Optimization Online (2013)Google Scholar
  27. 27.
    Philpott, A.B., De Matos, V.L.: Dynamic Sampling Algorithms for Multistage Stochastic Programs with Risk Aversion. Electric Power Optimization Centre, University of Auckland, Technical report (2011)Google Scholar
  28. 28.
    Shapiro, A.: Analysis of stochastic dual dynamic programming method. Eur. J. Oper. Res. 209(1), 63–72 (2011)CrossRefMATHGoogle Scholar
  29. 29.
    Birge, J.R., Louveaux, F.: Introduction to Stochastic Programming. Springer, New York (1997)MATHGoogle Scholar
  30. 30.
    Ruszczyński, A.: Decomposition methods. In: Ruszczyński, A., Shapiro, A. (eds.) Stochastic Programming, Handbooks Oper. Res. Management Sci., pp. 141–211. Elsevier, Amsterdam (2003)Google Scholar
  31. 31.
    Gülten, S.: Two-stage portfolio optimization with higher-order conditional measures of risk. Ph.D. thesis, Rutgers University (2014)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2014

Authors and Affiliations

  1. 1.Department of Operations Research and Financial EngineeringPrinceton UniversityPrincetonUSA
  2. 2.Department of Management Science and Information SystemsRutgers UniversityPiscatawayUSA

Personalised recommendations