# 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.

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