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

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  • Raimund Kovacevic
  • Alois Pichler

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. Copyright Springer Science+Business Media New York 2015

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  • Raimund Kovacevic & Alois Pichler, 2015. "Tree approximation for discrete time stochastic processes: a process distance approach," Annals of Operations Research, Springer, vol. 235(1), pages 395-421, December.
    • Handle: RePEc:spr:annopr:v:235:y:2015:i:1:p:395-421:10.1007/s10479-015-1994-2
    DOI: 10.1007/s10479-015-1994-2
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    References listed on IDEAS

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    1. Kjetil Høyland & Stein W. Wallace, 2001. "Generating Scenario Trees for Multistage Decision Problems," Management Science, INFORMS, vol. 47(2), pages 295-307, February.
    2. Vlad Bally & Gilles Pagès & Jacques Printems, 2005. "A Quantization Tree Method For Pricing And Hedging Multidimensional American Options," Mathematical Finance, Wiley Blackwell, vol. 15(1), pages 119-168, January.
    3. Holger Heitsch & Werner Römisch, 2009. "Scenario tree reduction for multistage stochastic programs," Computational Management Science, Springer, vol. 6(2), pages 117-133, May.
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    Cited by:

    1. Martin Glanzer & Georg Ch. Pflug, 2020. "Multiscale stochastic optimization: modeling aspects and scenario generation," Computational Optimization and Applications, Springer, vol. 75(1), pages 1-34, January.
    2. Raimund M. Kovacevic, 2019. "Valuation and pricing of electricity delivery contracts: the producer’s view," Annals of Operations Research, Springer, vol. 275(2), pages 421-460, April.
    3. Weiguo Zhang & Xiaolei He, 2022. "A New Scenario Reduction Method Based on Higher-Order Moments," INFORMS Journal on Computing, INFORMS, vol. 34(4), pages 1903-1918, July.
    4. Delgado, Felipe & Trincado, Ricardo & Pagnoncelli, Bernardo K., 2019. "A multistage stochastic programming model for the network air cargo allocation under capacity uncertainty," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 131(C), pages 292-307.
    5. Alois Pichler & Michael Weinhardt, 2022. "The nested Sinkhorn divergence to learn the nested distance," Computational Management Science, Springer, vol. 19(2), pages 269-293, June.
    6. Giovanni Pantuso & Trine K. Boomsma, 2020. "On the number of stages in multistage stochastic programs," Annals of Operations Research, Springer, vol. 292(2), pages 581-603, September.
    7. Markéta Horejšová & Sebastiano Vitali & Miloš Kopa & Vittorio Moriggia, 2020. "Evaluation of scenario reduction algorithms with nested distance," Computational Management Science, Springer, vol. 17(2), pages 241-275, June.
    8. Hannes Schwarz & Valentin Bertsch & Wolf Fichtner, 2018. "Two-stage stochastic, large-scale optimization of a decentralized energy system: a case study focusing on solar PV, heat pumps and storage in a residential quarter," OR Spectrum: Quantitative Approaches in Management, Springer;Gesellschaft für Operations Research e.V., vol. 40(1), pages 265-310, January.
    9. Nichols, J.M. & Spendelow, J.A. & Nichols, J.D., 2017. "Using Optimal Transport Theory to Estimate Transition Probabilities in Metapopulation Dynamics," Ecological Modelling, Elsevier, vol. 359(C), pages 311-319.
    10. W. Ackooij & X. Warin, 2020. "On conditional cuts for stochastic dual dynamic programming," EURO Journal on Computational Optimization, Springer;EURO - The Association of European Operational Research Societies, vol. 8(2), pages 173-199, June.
    11. Yi Yang & Sixin Wang & Wei Xu & Kunlun Wei, 2018. "Reliability evaluation of wireless multimedia sensor networks based on instantaneous availability," International Journal of Distributed Sensor Networks, , vol. 14(11), pages 15501477188, November.

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