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A probabilistic numerical method for optimal multiple switching problems in high dimension

Author

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  • René Aïd

    (FiME Lab - Laboratoire de Finance des Marchés d'Energie - Université Paris Dauphine-PSL - PSL - Université Paris sciences et lettres - CREST - EDF R&D - EDF R&D - EDF - EDF, LEDa - Laboratoire d'Economie de Dauphine - IRD - Institut de Recherche pour le Développement - Université Paris Dauphine-PSL - PSL - Université Paris sciences et lettres - CNRS - Centre National de la Recherche Scientifique)

  • Luciano Campi

    (CEREMADE - CEntre de REcherches en MAthématiques de la DEcision - Université Paris Dauphine-PSL - PSL - Université Paris sciences et lettres - CNRS - Centre National de la Recherche Scientifique, LAGA - Laboratoire Analyse, Géométrie et Applications - UP8 - Université Paris 8 Vincennes-Saint-Denis - UP13 - Université Paris 13 - Institut Galilée - CNRS - Centre National de la Recherche Scientifique, CREST - Centre de Recherche en Économie et Statistique - ENSAI - Ecole Nationale de la Statistique et de l'Analyse de l'Information [Bruz] - X - École polytechnique - ENSAE Paris - École Nationale de la Statistique et de l'Administration Économique - CNRS - Centre National de la Recherche Scientifique)

  • Nicolas Langrené

    (LPMA - Laboratoire de Probabilités et Modèles Aléatoires - UPMC - Université Pierre et Marie Curie - Paris 6 - UPD7 - Université Paris Diderot - Paris 7 - CNRS - Centre National de la Recherche Scientifique, FiME Lab - Laboratoire de Finance des Marchés d'Energie - Université Paris Dauphine-PSL - PSL - Université Paris sciences et lettres - CREST - EDF R&D - EDF R&D - EDF - EDF)

  • Huyên Pham

    (LPMA - Laboratoire de Probabilités et Modèles Aléatoires - UPMC - Université Pierre et Marie Curie - Paris 6 - UPD7 - Université Paris Diderot - Paris 7 - CNRS - Centre National de la Recherche Scientifique, CREST - Centre de Recherche en Économie et Statistique - ENSAI - Ecole Nationale de la Statistique et de l'Analyse de l'Information [Bruz] - X - École polytechnique - ENSAE Paris - École Nationale de la Statistique et de l'Administration Économique - CNRS - Centre National de la Recherche Scientifique)

Abstract

In this paper, we present a probabilistic numerical algorithm combining dynamic programming, Monte Carlo simulations, and local basis regressions to solve nonstationary optimal multiple switching problems in infinite horizon. We provide the rate of convergence of the method in terms of the time step used to discretize the problem, of the regression basis used to approximate conditional expectations, and of the truncating time horizon. To make the method viable for problems in high dimension and long time horizon, we extend a memory reduction method to the general Euler scheme, so that, when performing the numerical resolution, the storage of the Monte Carlo simulation paths is not needed. Then, we apply this algorithm to a model of optimal investment in power plants in dimension eight, i.e., with two different technologies and six random factors.

Suggested Citation

  • René Aïd & Luciano Campi & Nicolas Langrené & Huyên Pham, 2014. "A probabilistic numerical method for optimal multiple switching problems in high dimension," Post-Print hal-02294328, HAL.
    • Handle: RePEc:hal:journl:hal-02294328
    DOI: 10.1137/120897298
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    Citations

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    Cited by:

    1. Alasseur, C. & Féron, O., 2018. "Structural price model for coupled electricity markets," Energy Economics, Elsevier, vol. 75(C), pages 104-119.
    2. Li Kai & Nyström Kaj & Olofsson Marcus, 2015. "Optimal switching problems under partial information," Monte Carlo Methods and Applications, De Gruyter, vol. 21(2), pages 91-120, June.
    3. Aïd, René & Basei, Matteo & Ferrari, Giorgio, 2023. "A Stationary Mean-Field Equilibrium Model of Irreversible Investment in a Two-Regime Economy," Center for Mathematical Economics Working Papers 679, Center for Mathematical Economics, Bielefeld University.
    4. Giorgia Callegaro & Luciano Campi & Valeria Giusto & Tiziano Vargiolu, 2017. "Utility indifference pricing and hedging for structured contracts in energy markets," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 85(2), pages 265-303, April.
    5. Rongju Zhang & Nicolas Langren'e & Yu Tian & Zili Zhu & Fima Klebaner & Kais Hamza, 2018. "Local Control Regression: Improving the Least Squares Monte Carlo Method for Portfolio Optimization," Papers 1803.11467, arXiv.org, revised Sep 2018.
    6. Magnus Perninge, 2020. "A finite horizon optimal switching problem with memory and application to controlled SDDEs," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 91(3), pages 465-500, June.
    7. Ivan Guo & Nicolas Langren'e & Jiahao Wu, 2023. "Simultaneous upper and lower bounds of American option prices with hedging via neural networks," Papers 2302.12439, arXiv.org.
    8. Cortazar, Gonzalo & Naranjo, Lorenzo & Sainz, Felipe, 2021. "Optimal decision policy for real options under general Markovian dynamics," European Journal of Operational Research, Elsevier, vol. 288(2), pages 634-647.
    9. Ren'e Aid & Matteo Basei & Giorgio Ferrari, 2023. "A Stationary Mean-Field Equilibrium Model of Irreversible Investment in a Two-Regime Economy," Papers 2305.00541, arXiv.org.

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