Martingale characterizations of risk-averse stochastic optimization problems

  • Alois PichlerEmail author
  • Ruben Schlotter
Full Length Paper Series B


This paper addresses risk awareness of stochastic optimization problems. Nested risk measures appear naturally in this context, as they allow beneficial reformulations for algorithmic treatments. The reformulations presented extend usual dynamic equations by involving risk awareness in the problem formulation. Nested risk measures are built on risk measures, which originate by conditioning on the history of a stochastic process. We derive martingale properties of these risk measures and use them to prove continuity. It is demonstrated that stochastic optimization problems, which incorporate risk awareness via nesting risk measures, are continuous with respect to the natural distance governing these optimization problems, the nested distance.


Risk measures Stochastic optimization Stochastic processes 

Mathematics Subject Classification

90C15 60B05 62P05 



We would like to thank Prof. Shapiro for proposing to elaborate the continuity relations of nested risk measures with respect to the nested distance.


  1. 1.
    Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd edn. Birkhäuser Verlag, Basel (2005). zbMATHGoogle Scholar
  2. 2.
    Artzner, P., Delbaen, F., Heath, D.: Thinking coherently. Risk 10, 68–71 (1997)Google Scholar
  3. 3.
    Brenier, Y.: Décomposition polaire et réarrangement monotone des champs de vecteurs. Comptes Rendus l’Acad. Sci. Paris Sér. I Math. 305(19), 805–808 (1987)zbMATHGoogle Scholar
  4. 4.
    Brenier, Y.: Polar factorization and monotone rearrangement of vector-valued functions. Commun. Pure Appl. Math. 44(4), 375–417 (1991). MathSciNetzbMATHGoogle Scholar
  5. 5.
    Dellacherie, C., Meyer, P.-A.: Probabilities and Potential. North-Holland Publishing Co., Amsterdam (1988).
  6. 6.
    Dentcheva, D., Ruszczyński, A.: Time-coherent risk measures for continuous-time Markov chains. SIAM J. Financ. Math. 9(2), 690–715 (2018a). MathSciNetzbMATHGoogle Scholar
  7. 7.
    Dentcheva, D., Ruszczyński, A.: Risk forms: representation, disintegration, and application to partially observable two-stage systems, unpublished (2018b).
  8. 8.
    Fleming, W.H., Soner, H.M.: Controlled Markov Processes and Viscosity Solutions, 2nd edn. Springer, Berlin (2006). zbMATHGoogle Scholar
  9. 9.
    Föllmer, H., Schied, A.: Stochastic Finance: An Introduction in Discrete Time. de Gruyter Studies in Mathematics 27. Berlin, Boston, De Gruyter (2004). ISBN 978-3-11-046345-3. 10.1515/9783110218053.
  10. 10.
    Girardeau, P., Leclère, V., Philpott, A.B.: On the convergence of decomposition methods for multistage stochastic convex programs. Math. Oper. Res. 40(1), 1–16 (2014). MathSciNetzbMATHGoogle Scholar
  11. 11.
    Goulart, F.C., da Costa, B.F.P.: Nested distance for stagewise-independent processes, unpublished (2017).
  12. 12.
    Jouini E, Schachermayer W, Touzi N (2006) Law invariant risk measures have the Fatou property. In S. Kusuoka, A. Yamazaki(ed) Advances in Mathematical Economics, volume 9 of Kusuoka, Shigeo and Yamazaki, Akira chapter 4, pp 49–71. Springer, Japan. Scholar
  13. 13.
    Kallenberg, O.: Foundations of Modern Probability. Springer, New York (2002). zbMATHGoogle Scholar
  14. 14.
    Karatzas, I., Shreve, S.E.: Methods of Mathematical Finance. Stochastic Modelling and Applied Probability. Springer, Berlin (1998). zbMATHGoogle Scholar
  15. 15.
    Kusuoka, S.: On law invariant coherent risk measures. In: Kusuoka, S., Maruyama, T. (eds.) Advances in Mathematical Economics. Springer, Tokyo (2001). Google Scholar
  16. 16.
    Maggioni, F., Allevi, E., Bertocchi, M.: Measures of information in multistage stochastic programming. STOPROG (2012). zbMATHGoogle Scholar
  17. 17.
    McCann, R.J.: Polar factorization of maps on Riemannian manifolds. Geom. Funct. Anal. 11(3), 589–608 (2001). MathSciNetzbMATHGoogle Scholar
  18. 18.
    Peng, S.: Nonlinear expectations, nonlinear evaluations and risk measures. In: Lecture Notes in Mathematics, pp. 165–253. Springer, Berlin Heidelberg, (2004).
  19. 19.
    Pflug, GCh.: Version-independence and nested distributions in multistage stochastic optimization. SIAM J. Optim. 20, 1406–1420 (2009). MathSciNetzbMATHGoogle Scholar
  20. 20.
    Pflug, G. Ch., Pichler, A.: Multistage Stochastic Optimization. Springer Series in Operations Research and Financial Engineering. Springer, Berlin (2014). ISBN 978-3-319-08842-6.
  21. 21.
    Pflug, GCh., Römisch, W.: Modeling, Measuring and Managing Risk. World Scientific, NJ (2007). zbMATHGoogle Scholar
  22. 22.
    Philpott, A.B., de Matos, V.L.: Dynamic sampling algorithms for multi-stage stochastic programs with risk aversion. Eur. J. Oper. Res. 218(2), 470–483 (2012). MathSciNetzbMATHGoogle Scholar
  23. 23.
    Philpott, A.B., de Matos, V.L., Finardi, E.: On solving multistage stochastic programs with coherent risk measures. Oper. Res. 61(4), 957–970 (2013). MathSciNetzbMATHGoogle Scholar
  24. 24.
    Pichler, A.: The natural Banach space for version independent risk measures. Insur. Math. Econ. 53(2), 405–415 (2003). MathSciNetzbMATHGoogle Scholar
  25. 25.
    Pichler, A., Shapiro, A.: Minimal representations of insurance prices. Insur. Math. Econ. 62, 184–193 (2015). MathSciNetzbMATHGoogle Scholar
  26. 26.
    Pichler, A., Shapiro, A.: Risk averse stochastic programming: time consistency and optimal stopping (2018). arXiv:1808.10807
  27. 27.
    Riedel, F.: Dynamic coherent risk measures. Stoch. Process. Appl. 112(2), 185–200 (2004). MathSciNetzbMATHGoogle Scholar
  28. 28.
    Rockafellar, R.T., Wets, R.J.-B.: Nonanticipativity and \({L}^1\)-martingales in stochastic optimization problems. Math. Program. Study 6, 170–187 (1976)zbMATHGoogle Scholar
  29. 29.
    Römisch, W., Guigues, V.: Sampling-based decomposition methods for multistage stochastic programs based on extended polyhedral risk measures. SIAM J. Optim. 22(2), 286–312 (2012). MathSciNetzbMATHGoogle Scholar
  30. 30.
    Ruszczyński, A.: Risk-averse dynamic programming for Markov decision processes. Math. Program. Ser. B 125, 235–261 (2010). MathSciNetzbMATHGoogle Scholar
  31. 31.
    Ruszczyński, A., Shapiro, A.: Conditional risk mappings. Math. Oper. Res. 31(3), 544–561 (2006). MathSciNetzbMATHGoogle Scholar
  32. 32.
    Shapiro, A.: On Kusuoka representation of law invariant risk measures. Math. Oper. Res. 38(1), 142–152 (2013). MathSciNetzbMATHGoogle Scholar
  33. 33.
    Shapiro, A.: Rectangular sets of probability measures. Oper. Res. 64(2), 528–541 (2016). MathSciNetzbMATHGoogle Scholar
  34. 34.
    Shapiro, A.: Interchangeability principle and dynamic equations in risk averse stochastic programming. Oper. Res. Lett. 45(4), 377–381 (2017). MathSciNetzbMATHGoogle Scholar
  35. 35.
    Shapiro, A., Dentcheva, D., Ruszczyński, A.: In: Lectures on Stochastic Programming. MOS-SIAM Series on Optimization. SIAM, second edition (2014).
  36. 36.
    Shiryaev, A.N.: Probability. Springer, New York (1996). zbMATHGoogle Scholar
  37. 37.
    Villani, C.: Topics in Optimal Transportation, vol. 58 of Graduate Studies in Mathematics. American Mathematical Society, Providence (2003). ISBN 0-821-83312-X.
  38. 38.
    Xin, L., Shapiro, A.: Bounds for nested law invariant coherent risk measures. Oper. Res. Lett. 40, 431–435 (2012). MathSciNetzbMATHGoogle Scholar
  39. 39.
    Zhang, J.: Backward Stochastic Differential Equations. Springer, New York (2017). Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2019

Authors and Affiliations

  1. 1.Technische Universität ChemnitzChemnitzGermany

Personalised recommendations