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Quenched Mass Transport of Particles Toward a Target

Author

Listed:
  • Bruno Bouchard

    (Université Paris-Dauphine, PSL University, CNRS)

  • Boualem Djehiche

    (KTH Royal Institute of Technology)

  • Idris Kharroubi

    (Sorbonne Université, Université de Paris)

Abstract

We consider the stochastic target problem of finding the collection of initial laws of a mean-field stochastic differential equation such that we can control its evolution to ensure that it reaches a prescribed set of terminal probability distributions, at a fixed time horizon. Here, laws are considered conditionally to the path of the Brownian motion that drives the system. This kind of problems is motivated by limiting behavior of interacting particles systems with applications in, for example, agricultural crop management. We establish a version of the geometric dynamic programming principle for the associated reachability sets and prove that the corresponding value function is a viscosity solution of a geometric partial differential equation. This provides a characterization of the initial masses that can be almost surely transported toward a given target, along the paths of a stochastic differential equation. Our results extend those of Soner and Touzi, Journal of the European Mathematical Society (2002) to our setting.

Suggested Citation

  • Bruno Bouchard & Boualem Djehiche & Idris Kharroubi, 2020. "Quenched Mass Transport of Particles Toward a Target," Journal of Optimization Theory and Applications, Springer, vol. 186(2), pages 345-374, August.
    • Handle: RePEc:spr:joptap:v:186:y:2020:i:2:d:10.1007_s10957-020-01704-y
    DOI: 10.1007/s10957-020-01704-y
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    References listed on IDEAS

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    1. Giorgio Fabbri & Fausto Gozzi & Andrzej Swiech, 2017. "Stochastic Optimal Control in Infinite Dimensions - Dynamic Programming and HJB Equations," Post-Print hal-01505767, HAL.
    2. Charalambos D. Aliprantis & Kim C. Border, 2006. "Infinite Dimensional Analysis," Springer Books, Springer, edition 0, number 978-3-540-29587-7, July.
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    Cited by:

    1. Bayraktar, Erhan & Yao, Song, 2024. "Stochastic control/stopping problem with expectation constraints," Stochastic Processes and their Applications, Elsevier, vol. 176(C).
    2. Camilo Hernández & Dylan Possamaï, 2024. "Time‐inconsistent contract theory," Mathematical Finance, Wiley Blackwell, vol. 34(3), pages 1022-1085, July.
    3. Maximilien Germain & Huy^en Pham & Xavier Warin, 2021. "A level-set approach to the control of state-constrained McKean-Vlasov equations: application to renewable energy storage and portfolio selection," Papers 2112.11059, arXiv.org, revised Nov 2022.
    4. Maximilien Germain & Huyên Pham & Xavier Warin, 2021. "A level-set approach to the control of state-constrained McKean-Vlasov equations: application to renewable energy storage and portfolio selection," Working Papers hal-03498263, HAL.
    5. Camilo Hern'andez & Dylan Possamai, 2023. "Time-inconsistent contract theory," Papers 2303.01601, arXiv.org.

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