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

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  • Aïd, René
  • Campi, Luciano
  • Langrené, Nicolas
  • Pham, Huyên

Abstract

In this paper, we present a probabilistic numerical algorithm combining dynamic programming, Monte Carlo simulations and local basis regressions to solve non-stationary 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

  • Aïd, René & Campi, Luciano & Langrené, Nicolas & Pham, Huyên, 2014. "A probabilistic numerical method for optimal multiple switching problems in high dimension," LSE Research Online Documents on Economics 63011, London School of Economics and Political Science, LSE Library.
    • Handle: RePEc:ehl:lserod:63011
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    References listed on IDEAS

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    8. René Aïd & Luciano Campi & Adrien Nguyen Huu & Nizar Touzi, 2009. "A Structural Risk-Neutral Model Of Electricity Prices," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 12(07), pages 925-947.
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    Citations

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    Cited by:

    1. Kharroubi Idris & Langrené Nicolas & Pham Huyên, 2014. "A numerical algorithm for fully nonlinear HJB equations: An approach by control randomization," Monte Carlo Methods and Applications, De Gruyter, vol. 20(2), pages 145-165, June.
    2. Alasseur, C. & Féron, O., 2018. "Structural price model for coupled electricity markets," Energy Economics, Elsevier, vol. 75(C), pages 104-119.
    3. 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.
    4. Benedetti, Giuseppe & Campi, Luciano, 2016. "Utility indifference valuation for non-smooth payoffs with an application to power derivatives," LSE Research Online Documents on Economics 63016, London School of Economics and Political Science, LSE Library.
    5. 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.
    6. Magnus Perninge, 2018. "A limited-feedback approximation scheme for optimal switching problems with execution delays," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 87(3), pages 347-382, June.
    7. 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.
    8. Mihail Zervos & Carlos Oliveira & Kate Duckworth, 2018. "An investment model with switching costs and the option to abandon," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 88(3), pages 417-443, December.
    9. 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.
    10. 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.
    11. Rongju Zhang & Nicolas Langren'e & Yu Tian & Zili Zhu & Fima Klebaner & Kais Hamza, 2016. "Dynamic portfolio optimization with liquidity cost and market impact: a simulation-and-regression approach," Papers 1610.07694, arXiv.org, revised Jun 2019.
    12. 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, revised Apr 2024.
    13. 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.
    14. 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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    JEL classification:

    • C1 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General
    • R14 - Urban, Rural, Regional, Real Estate, and Transportation Economics - - General Regional Economics - - - Land Use Patterns
    • J01 - Labor and Demographic Economics - - General - - - Labor Economics: General

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