Abstract
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.
Similar content being viewed by others
References
Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd edn. Birkhäuser Verlag, Basel (2005). https://doi.org/10.1007/978-3-7643-8722-8
Artzner, P., Delbaen, F., Heath, D.: Thinking coherently. Risk 10, 68–71 (1997)
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)
Brenier, Y.: Polar factorization and monotone rearrangement of vector-valued functions. Commun. Pure Appl. Math. 44(4), 375–417 (1991). https://doi.org/10.1002/cpa.3160440402
Dellacherie, C., Meyer, P.-A.: Probabilities and Potential. North-Holland Publishing Co., Amsterdam (1988). https://projecteuclid.org/euclid.bams/1183546371
Dentcheva, D., Ruszczyński, A.: Time-coherent risk measures for continuous-time Markov chains. SIAM J. Financ. Math. 9(2), 690–715 (2018a). https://doi.org/10.1137/16m1063794
Dentcheva, D., Ruszczyński, A.: Risk forms: representation, disintegration, and application to partially observable two-stage systems, unpublished (2018b). https://arxiv.org/abs/1807.02300
Fleming, W.H., Soner, H.M.: Controlled Markov Processes and Viscosity Solutions, 2nd edn. Springer, Berlin (2006). https://doi.org/10.1007/0-387-31071-1
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. http://books.google.com/books?id=cL-bZSOrqWoC
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). https://doi.org/10.1287/moor.2014.0664
Goulart, F.C., da Costa, B.F.P.: Nested distance for stagewise-independent processes, unpublished (2017). https://arxiv.org/pdf/1711.10633.pdf
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. https://doi.org/10.1007/4-431-34342-3
Kallenberg, O.: Foundations of Modern Probability. Springer, New York (2002). https://doi.org/10.1007/b98838
Karatzas, I., Shreve, S.E.: Methods of Mathematical Finance. Stochastic Modelling and Applied Probability. Springer, Berlin (1998). https://doi.org/10.1007/b98840
Kusuoka, S.: On law invariant coherent risk measures. In: Kusuoka, S., Maruyama, T. (eds.) Advances in Mathematical Economics. Springer, Tokyo (2001). https://doi.org/10.1007/978-4-431-67891-5
Maggioni, F., Allevi, E., Bertocchi, M.: Measures of information in multistage stochastic programming. STOPROG (2012). https://doi.org/10.5200/stoprog.2012.14
McCann, R.J.: Polar factorization of maps on Riemannian manifolds. Geom. Funct. Anal. 11(3), 589–608 (2001). https://doi.org/10.1007/PL00001679
Peng, S.: Nonlinear expectations, nonlinear evaluations and risk measures. In: Lecture Notes in Mathematics, pp. 165–253. Springer, Berlin Heidelberg, (2004). https://doi.org/10.1007/b100122
Pflug, GCh.: Version-independence and nested distributions in multistage stochastic optimization. SIAM J. Optim. 20, 1406–1420 (2009). https://doi.org/10.1137/080718401
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. https://doi.org/10.1007/978-3-319-08843-3. https://books.google.com/books?id=q_VWBQAAQBAJ
Pflug, GCh., Römisch, W.: Modeling, Measuring and Managing Risk. World Scientific, NJ (2007). https://doi.org/10.1142/9789812708724
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). https://doi.org/10.1016/j.ejor.2011.10.056
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). https://doi.org/10.1287/opre.2013.1175
Pichler, A.: The natural Banach space for version independent risk measures. Insur. Math. Econ. 53(2), 405–415 (2003). https://doi.org/10.1016/j.insmatheco.2013.07.005
Pichler, A., Shapiro, A.: Minimal representations of insurance prices. Insur. Math. Econ. 62, 184–193 (2015). https://doi.org/10.1016/j.insmatheco.2015.03.011
Pichler, A., Shapiro, A.: Risk averse stochastic programming: time consistency and optimal stopping (2018). arXiv:1808.10807
Riedel, F.: Dynamic coherent risk measures. Stoch. Process. Appl. 112(2), 185–200 (2004). https://doi.org/10.1016/j.spa.2004.03.004
Rockafellar, R.T., Wets, R.J.-B.: Nonanticipativity and \({L}^1\)-martingales in stochastic optimization problems. Math. Program. Study 6, 170–187 (1976)
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). https://doi.org/10.1137/100811696
Ruszczyński, A.: Risk-averse dynamic programming for Markov decision processes. Math. Program. Ser. B 125, 235–261 (2010). https://doi.org/10.1007/s10107-010-0393-3
Ruszczyński, A., Shapiro, A.: Conditional risk mappings. Math. Oper. Res. 31(3), 544–561 (2006). https://doi.org/10.1287/moor.1060.0204
Shapiro, A.: On Kusuoka representation of law invariant risk measures. Math. Oper. Res. 38(1), 142–152 (2013). https://doi.org/10.1287/moor.1120.0563
Shapiro, A.: Rectangular sets of probability measures. Oper. Res. 64(2), 528–541 (2016). https://doi.org/10.1287/opre.2015.1466
Shapiro, A.: Interchangeability principle and dynamic equations in risk averse stochastic programming. Oper. Res. Lett. 45(4), 377–381 (2017). https://doi.org/10.1016/j.orl.2017.05.008
Shapiro, A., Dentcheva, D., Ruszczyński, A.: In: Lectures on Stochastic Programming. MOS-SIAM Series on Optimization. SIAM, second edition (2014). https://doi.org/10.1137/1.9780898718751
Shiryaev, A.N.: Probability. Springer, New York (1996). https://doi.org/10.1007/978-1-4757-2539-1
Villani, C.: Topics in Optimal Transportation, vol. 58 of Graduate Studies in Mathematics. American Mathematical Society, Providence (2003). ISBN 0-821-83312-X. https://doi.org/10.1090/gsm/058. http://books.google.com/books?id=GqRXYFxe0l0C
Xin, L., Shapiro, A.: Bounds for nested law invariant coherent risk measures. Oper. Res. Lett. 40, 431–435 (2012). https://doi.org/10.1016/j.orl.2012.09.002
Zhang, J.: Backward Stochastic Differential Equations. Springer, New York (2017). https://doi.org/10.1007/978-1-4939-7256-2
Acknowledgements
We would like to thank Prof. Shapiro for proposing to elaborate the continuity relations of nested risk measures with respect to the nested distance.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Special Issue Math. Prog. on “The interface between optimization and probability”.
Rights and permissions
About this article
Cite this article
Pichler, A., Schlotter, R. Martingale characterizations of risk-averse stochastic optimization problems. Math. Program. 181, 377–403 (2020). https://doi.org/10.1007/s10107-019-01391-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10107-019-01391-2