IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v191y2026ics0304414925002182.html

Optimal control of the nonlinear stochastic Fokker–Planck equation

Author

Listed:
  • Hambly, Ben
  • Jettkant, Philipp

Abstract

We consider a control problem for the nonlinear stochastic Fokker–Planck equation. This equation describes the evolution of the distribution of nonlocally interacting particles affected by a common source of noise. The system is directed by a controller that acts on the drift term with the goal of minimising a cost functional. We establish the well-posedness of the state equation, prove the existence of optimal controls, and formulate a stochastic maximum principle (SMP) that provides necessary and sufficient optimality conditions for the control problem. The adjoint process arising in the SMP is characterised by a nonlocal (semi)linear backward SPDE for which we study existence and uniqueness. We also rigorously connect the control problem for the nonlinear stochastic Fokker–Planck equation to the control of the corresponding McKean–Vlasov SDE that describes the motion of a representative particle. Our work extends existing results for the control of the Fokker–Planck equation to nonlinear and stochastic dynamics. In particular, the sufficient SMP, which we obtain by exploiting the special structure of the Fokker–Planck equation, seems to be novel even in the linear deterministic setting. We illustrate our results with an application to a model of government interventions in financial systems, supplemented by numerical illustrations.

Suggested Citation

  • Hambly, Ben & Jettkant, Philipp, 2026. "Optimal control of the nonlinear stochastic Fokker–Planck equation," Stochastic Processes and their Applications, Elsevier, vol. 191(C).
    • Handle: RePEc:eee:spapps:v:191:y:2026:i:c:s0304414925002182
    DOI: 10.1016/j.spa.2025.104774
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304414925002182
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spa.2025.104774?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to

    for a different version of it.

    References listed on IDEAS

    as
    1. Burzoni, Matteo & Campi, Luciano, 2023. "Mean field games with absorption and common noise with a model of bank run," Stochastic Processes and their Applications, Elsevier, vol. 164(C), pages 206-241.
    2. Ben Hambly & Andreas Søjmark, 2019. "An SPDE model for systemic risk with endogenous contagion," Finance and Stochastics, Springer, vol. 23(3), pages 535-594, July.
    3. Mao Fabrice Djete & Dylan Possamaï & Xiaolu Tan, 2022. "McKean–Vlasov Optimal Control: Limit Theory and Equivalence Between Different Formulations," Mathematics of Operations Research, INFORMS, vol. 47(4), pages 2891-2930, November.
    4. Tim Breitenbach & Alfio Borzì, 2020. "The Pontryagin maximum principle for solving Fokker–Planck optimal control problems," Computational Optimization and Applications, Springer, vol. 76(2), pages 499-533, June.
    5. Arthur Fleig & Roberto Guglielmi, 2017. "Optimal Control of the Fokker–Planck Equation with Space-Dependent Controls," Journal of Optimization Theory and Applications, Springer, vol. 174(2), pages 408-427, August.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Liu, Yanwei & Liang, Ziyong & Zhong, Wei & Xue, Yu & Wang, Yue & Tao, Naian & Lu, Yanbo, 2025. "Multi-objective predictive cruise control for electric heavy-duty trucks considering fleet battery swapping under cyber-physical system," Energy, Elsevier, vol. 321(C).
    2. He, Xihao, 2025. "On the limit theory of mean field optimal stopping with non-Markov dynamics and common noise," Stochastic Processes and their Applications, Elsevier, vol. 188(C).
    3. Christa Cuchiero & Stefan Rigger & Sara Svaluto-Ferro, 2020. "Propagation of minimality in the supercooled Stefan problem," Papers 2010.03580, arXiv.org, revised Jun 2022.
    4. Rudiger Frey & Theresa Traxler, 2024. "Playing with Fire? A Mean Field Game Analysis of Fire Sales and Systemic Risk under Regulatory Capital Constraints," Papers 2406.17528, arXiv.org.
    5. Burzoni, Matteo & Campi, Luciano, 2023. "Mean field games with absorption and common noise with a model of bank run," Stochastic Processes and their Applications, Elsevier, vol. 164(C), pages 206-241.
    6. Hambly, Ben & Petronilia, Aldaïr & Reisinger, Christoph & Rigger, Stefan & Søjmark, Andreas, 2025. "Contagious McKean–Vlasov problems with common noise: from smooth to singular feedback through hitting times," LSE Research Online Documents on Economics 128226, London School of Economics and Political Science, LSE Library.
    7. Zachary Feinstein & Andreas Søjmark, 2023. "Contagious McKean–Vlasov systems with heterogeneous impact and exposure," Finance and Stochastics, Springer, vol. 27(3), pages 663-711, July.
    8. Ludovic Tangpi & Shichun Wang, 2024. "Optimal bubble riding with price-dependent entry: a mean field game of controls with common noise," Mathematics and Financial Economics, Springer, volume 18, number 5, January.
    9. Tim Breitenbach & Alfio Borzì, 2020. "The Pontryagin maximum principle for solving Fokker–Planck optimal control problems," Computational Optimization and Applications, Springer, vol. 76(2), pages 499-533, June.
    10. Zachary Feinstein & Andreas Sojmark, 2020. "Dynamic Default Contagion in Heterogeneous Interbank Systems," Papers 2010.15254, arXiv.org, revised Jul 2021.
    11. Guanxing Fu & Paul Hager & Ulrich Horst, 2024. "A Mean-Field Game of Market Entry," Rationality and Competition Discussion Paper Series 517, CRC TRR 190 Rationality and Competition.
    12. Felix Hofer, 2025. "Price Levels in Heterogeneous-Agent Models," Papers 2510.26065, arXiv.org.
    13. Filippo de Feo & Fausto Gozzi & Andrzej 'Swik{e}ch & Lukas Wessels, 2025. "Stochastic Optimal Control of Interacting Particle Systems in Hilbert Spaces and Applications," Papers 2511.21646, arXiv.org.
    14. Mao Fabrice Djete & Gaoyue Guo & Nizar Touzi, 2023. "Mean field game of mutual holding with defaultable agents, and systemic risk," Papers 2303.07996, arXiv.org.
    15. Mathieu Lauri`ere & Ariel Neufeld & Kyunghyun Park, 2025. "Robust mean-field control under common noise uncertainty," Papers 2511.04515, arXiv.org.
    16. Zachary Feinstein & Andreas Sojmark, 2021. "Contagious McKean-Vlasov systems with heterogeneous impact and exposure," Papers 2104.06776, arXiv.org, revised Sep 2022.
    17. Toshiaki Yamanaka, 2025. "Coordinated Mean-Field Control for Systemic Risk," Papers 2512.04704, arXiv.org.
    18. Yucheng Guo & Qinxin Yan, 2025. "Particle Systems with Local Interactions via Hitting Times and Cascades on Graphs," Papers 2505.18448, arXiv.org, revised Mar 2026.
    19. Alain Bensoussan & Joohyun Kim & Bohan Li & Sheung Chi Phillip Yam, 2025. "Linear Quadratic Extended Mean Field Games and Control Problems," Journal of Optimization Theory and Applications, Springer, vol. 206(2), pages 1-47, August.
    20. Feinstein, Zachary & Sojmark, Andreas, 2023. "Contagious McKean–Vlasov systems with heterogeneous impact and exposure," LSE Research Online Documents on Economics 119457, London School of Economics and Political Science, LSE Library.

    More about this item

    Keywords

    ;
    ;
    ;
    ;

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:spapps:v:191:y:2026:i:c:s0304414925002182. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/505572/description#description .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.