# Tree approximation for discrete time stochastic processes: a process distance approach

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## Abstract

Approximating stochastic processes by scenario trees is important in decision analysis. In this paper we focus on improving the approximation quality of trees by smaller, tractable trees. In particular we propose and analyze an iterative algorithm to construct improved approximations: given a stochastic process in discrete time and starting with an arbitrary, approximating tree, the algorithm improves both, the probabilities on the tree and the related path-values of the smaller tree, leading to significantly improved approximations of the initial stochastic process. The quality of the approximation is measured by the process distance (nested distance), which was introduced recently. For the important case of quadratic process distances the algorithm finds locally best approximating trees in finitely many iterations by generalizing multistage k-means clustering.

## Keywords

Stochastic processes and trees Wasserstein and Kantorovich distance Tree approximation Optimal transport Facility location## Mathematics Subject Classification

90C15 60B05 90-08## Notes

### Acknowledgments

We thank the referees for their constructive criticism. We wish to thank two anonymous referees for their dedication to review the paper. Their valuable comments significantly improved the content and the presentation. Parts of this paper are addressed in the book *Multistage Stochastic Optimization* (Springer) by Pflug and Pichler, which also summarizes many more topics in multistage stochastic optimization and which had to be completed before final acceptance of this paper.

### Compliance with ethical standards

### Funding

This research was partially funded by the Austrian science fund FWF, project P 24125-N13 and by the Research Council of Norway, Grant 207690/E20.

## References

- Bally, V., Pagès, G., & Printems, J. (2005). A quantization tree method for pricing and hedging multidimensional American options.
*Mathematical Finance*,*15*(1), 119–168.CrossRefGoogle Scholar - Beiglböck, M., Goldstern, M., Maresch, G., & Schachermayer, W. (2009). Optimal and better transport plans.
*Journal of Functional Analysis*,*256*(6), 1907–1927.CrossRefGoogle Scholar - Beiglböck, M., Léonard, C., & Schachermayer, W. (2012). A general duality theorem for the Monge–Kantorovich transport problem.
*Studia Mathematica*,*209*, 151–167.CrossRefGoogle Scholar - Drezner, Z., & Hamacher, H. W. (2002).
*Facility location: Applications and theory*. New York, NY: Springer.CrossRefGoogle Scholar - Dudley, R. M. (1969). The speed of mean Glivenko–Cantelli convergence.
*The Annals of Mathematical Statistics*,*40*(1), 40–50.CrossRefGoogle Scholar - Dupačová, J., Gröwe-Kuska, N., & Römisch, W. (2003). Scenario reduction in stochastic programming.
*Mathematical Programming, Series A*,*95*(3), 493–511.CrossRefGoogle Scholar - Durrett, R. A. (2004).
*Probability: Theory and examples*(2nd ed.). Belmont, CA: Duxbury Press.Google Scholar - Graf, S., & Luschgy, H. (2000).
*Foundations of quantization for probability distributions*(vol. 1730), Lecture notes in mathematics. Berlin, Heidelberg: Springer.Google Scholar - Heitsch, H., & Römisch, W. (2003). Scenario reduction algorithms in stochastic programming.
*Computational Optimization and Applications*,*24*(2–3), 187–206.CrossRefGoogle Scholar - Heitsch, H., & Römisch, W. (2007). A note on scenario reduction for two-stage stochastic programs.
*Operations Research Letters*,*6*, 731–738.CrossRefGoogle Scholar - Heitsch, H., & Römisch, W. (2009a). Scenario tree modeling for multistage stochastic programs.
*Mathematical Programming Series A*,*118*, 371–406.Google Scholar - Heitsch, H., & Römisch, W. (2009b). Scenario tree reduction for multistage stochastic programs.
*Computational Management Science*,*2*, 117–133.Google Scholar - Heitsch, H., & Römisch, W. (2011). Stability and scenario trees for multistage stochastic programs. In G. Infanger (Ed.),
*Stochastic programming*, volume 150 of*international series in operations research & management science*, pp. 139–164. New York: Springer.Google Scholar - Heitsch, H., Römisch, W., & Strugarek, C. (2006). Stability of multistage stochastic programs.
*SIAM Journal on Optimization*,*17*(2), 511–525.CrossRefGoogle Scholar - Høyland, K., & Wallace, S. W. (2001). Generating scenario trees for multistage decision problems.
*Management Science*,*47*, 295–307.CrossRefGoogle Scholar - King, A. J., & Wallace, S. W. (2013).
*Modeling with stochastic programming*, volume XVI of*Springer Series in Operations Research and Financial Engineering*. Berlin: Springer.Google Scholar - Lloyd, S. P. (1982). Least square quantization in PCM.
*IEEE Transactions of Information Theory*,*28*(2), 129–137.CrossRefGoogle Scholar - Nocedal, J. (1980). Updating quasi-Newton matrices with limited storage.
*Mathematics of Computation*,*35*(151), 773–782.CrossRefGoogle Scholar - Pflug, G. C., & Römisch, W. (2007).
*Modeling, measuring and managing risk*. River Edge, NJ: World Scientific.CrossRefGoogle Scholar - Pflug, G. C. (2009). Version-independence and nested distribution in multistage stochastic optimization.
*SIAM Journal on Optimization*,*20*, 1406–1420.CrossRefGoogle Scholar - Pflug, G. C., & Pichler, A. (2012). A distance for multistage stochastic optimization models.
*SIAM Journal on Optimization*,*22*(1), 1–23.CrossRefGoogle Scholar - Pichler, A. (2013). Evaluations of risk measures for different probability measures.
*SIAM Journal on Optimization*,*23*(1), 530–551.CrossRefGoogle Scholar - Rachev, S. T. (1991).
*Probability metrics and the stability of stochastic models*. West Sussex: Wiley.Google Scholar - Rachev, S. T., & Rüschendorf, L. (1998).
*Mass transportation problems vol. I: Theory, vol. II: Applications*, volume XXV of*Probability and its applications*. New York: Springer.Google Scholar - Römisch, W. (2003). Stability of stochastic programming problems. In A. Ruszczyński & A. Shapiro (Eds.),
*Stochastic programming, handbooks in operations research and management science, volume 10, chapter 8*. Amsterdam: Elsevier.Google Scholar - Ruszczyński, A. (2006).
*Nonlinear optimization*. Princeton: Princeton University Press.Google Scholar - Schachermayer, W., & Teichmann, J. (2009). Characterization of optimal transport plans for the Monge–Kantorovich problem.
*Proceedings of the American Mathematical Society*,*137*(2), 519–529.CrossRefGoogle Scholar - Shapiro, A. (2010). Computational complexity of stochastic programming: Monte Carlo sampling approach. In
*Proceedings of the international congress of mathematicians*, pp. 2979–2995, Hyderabad, India.Google Scholar - Shapiro, A., & Nemirovski, A. (2005). On complexity of stochastic programming problems. In V. Jeyakumar & A. M. Rubinov (Eds.),
*Continuous optimization: Current trends and applications*(pp. 111–144). Berlin: Springer.CrossRefGoogle Scholar - Shiryaev, A. N. (1996).
*Probability*. New York: Springer.CrossRefGoogle Scholar - Vershik, A. M. (2006). Kantorovich metric: Initial history and little-known applications.
*Journal of Mathematical Sciences*,*133*(4), 1410–1417.CrossRefGoogle Scholar - Villani, C. (2003).
*Topics in optimal transportation*(vol. 58). Graduate Studies in Mathematics Providence, RI: American Mathematical Society.Google Scholar - Villani, C. (2009).
*Optimal transport, old and new*(vol. 338), Grundlehren der Mathematischen Wissenschaften. Berlin: Springer.Google Scholar - Williams, D. (1991).
*Probability with martingales*. Cambridge: Cambridge University Press.CrossRefGoogle Scholar