IDEAS home Printed from https://ideas.repec.org/a/inm/ormoor/v46y2021i2p471-502.html
   My bibliography  Save this article

A Probabilistic Approach to Extended Finite State Mean Field Games

Author

Listed:
  • René Carmona

    (Department of Operations Research and Financial Engineering, Princeton University, Princeton, New Jersey 08544)

  • Peiqi Wang

    (Bank of America, New York, New York 10281)

Abstract

We develop a probabilistic approach to continuous-time finite state mean field games. Based on an alternative description of continuous-time Markov chains by means of semimartingales and the weak formulation of stochastic optimal control, our approach not only allows us to tackle the mean field of states and the mean field of control at the same time, but also extends the strategy set of players from Markov strategies to closed-loop strategies. We show the existence and uniqueness of Nash equilibrium for the mean field game as well as how the equilibrium of a mean field game consists of an approximative Nash equilibrium for the game with a finite number of players under different assumptions of structure and regularity on the cost functions and transition rate between states.

Suggested Citation

  • René Carmona & Peiqi Wang, 2021. "A Probabilistic Approach to Extended Finite State Mean Field Games," Mathematics of Operations Research, INFORMS, vol. 46(2), pages 471-502, May.
    • Handle: RePEc:inm:ormoor:v:46:y:2021:i:2:p:471-502
    DOI: 10.1287/moor.2020.1071
    as

    Download full text from publisher

    File URL: http://dx.doi.org/10.1287/moor.2020.1071
    Download Restriction: no
    File URL: https://libkey.io/10.1287/moor.2020.1071?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
    ---><---

    References listed on IDEAS

    as
    1. Samuel N. Cohen & Robert J. Elliott, 2008. "Comparisons for backward stochastic differential equations on Markov chains and related no-arbitrage conditions," Papers 0810.0055, arXiv.org, revised Jan 2010.
    2. Pierre Cardaliaguet & Charles-Albert Lehalle, 2016. "Mean Field Game of Controls and An Application To Trade Crowding," Papers 1610.09904, arXiv.org, revised Sep 2017.
    3. Cecchin, Alekos & Pelino, Guglielmo, 2019. "Convergence, fluctuations and large deviations for finite state mean field games via the Master Equation," Stochastic Processes and their Applications, Elsevier, vol. 129(11), pages 4510-4555.
    4. Romuald Elie & Thibaut Mastrolia & Dylan Possamaï, 2019. "A Tale of a Principal and Many, Many Agents," Mathematics of Operations Research, INFORMS, vol. 44(2), pages 440-467, May.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Alexander Aurell & René Carmona & Gökçe Dayanıklı & Mathieu Laurière, 2022. "Finite State Graphon Games with Applications to Epidemics," Dynamic Games and Applications, Springer, vol. 12(1), pages 49-81, March.
    2. René Carmona, 2022. "The influence of economic research on financial mathematics: Evidence from the last 25 years," Finance and Stochastics, Springer, vol. 26(1), pages 85-101, January.

    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. Clémence Alasseur & Imen Ben Taher & Anis Matoussi, 2020. "An Extended Mean Field Game for Storage in Smart Grids," Journal of Optimization Theory and Applications, Springer, vol. 184(2), pages 644-670, February.
    2. Olivier Menoukeu Pamen, 2017. "Maximum Principles of Markov Regime-Switching Forward–Backward Stochastic Differential Equations with Jumps and Partial Information," Journal of Optimization Theory and Applications, Springer, vol. 175(2), pages 373-410, November.
    3. Ali Al-Aradi & Adolfo Correia & Danilo de Frietas Naiff & Gabriel Jardim & Yuri Saporito, 2019. "Extensions of the Deep Galerkin Method," Papers 1912.01455, arXiv.org, revised Apr 2022.
    4. Arvind V. Shrivats & Dena Firoozi & Sebastian Jaimungal, 2022. "A mean‐field game approach to equilibrium pricing in solar renewable energy certificate markets," Mathematical Finance, Wiley Blackwell, vol. 32(3), pages 779-824, July.
    5. Leippold, Markus & Schärer, Steven, 2017. "Discrete-time option pricing with stochastic liquidity," Journal of Banking & Finance, Elsevier, vol. 75(C), pages 1-16.
    6. Akihiro Kaneko, 2023. "Multi-stage Euler-Maruyama methods for backward stochastic differential equations driven by continuous-time Markov chains," Papers 2311.08826, arXiv.org, revised Nov 2023.
    7. Nendel, Max, 2019. "On Nonlinear Expectations and Markov Chains under Model Uncertainty," Center for Mathematical Economics Working Papers 628, Center for Mathematical Economics, Bielefeld University.
    8. Marcel Nutz & Yuchong Zhang, 2021. "Mean Field Contest with Singularity," Papers 2103.04219, arXiv.org.
    9. Steven Campbell & Yichao Chen & Arvind Shrivats & Sebastian Jaimungal, 2021. "Deep Learning for Principal-Agent Mean Field Games," Papers 2110.01127, arXiv.org.
    10. René Carmona & Peiqi Wang, 2021. "Finite-State Contract Theory with a Principal and a Field of Agents," Management Science, INFORMS, vol. 67(8), pages 4725-4741, August.
    11. Al-Aradi, Ali & Correia, Adolfo & Jardim, Gabriel & de Freitas Naiff, Danilo & Saporito, Yuri, 2022. "Extensions of the deep Galerkin method," Applied Mathematics and Computation, Elsevier, vol. 430(C).
    12. Bastien Baldacci & Philippe Bergault, 2021. "Optimal incentives in a limit order book: a SPDE control approach," Papers 2112.00375, arXiv.org, revised Oct 2022.
    13. Dena Firoozi & Arvind V Shrivats & Sebastian Jaimungal, 2021. "Principal agent mean field games in REC markets," Papers 2112.11963, arXiv.org, revised Jun 2022.
    14. E. Savku & G.-W Weber, 2022. "Stochastic differential games for optimal investment problems in a Markov regime-switching jump-diffusion market," Annals of Operations Research, Springer, vol. 312(2), pages 1171-1196, May.
    15. Dilip Madan & Martijn Pistorius & Mitja Stadje, 2013. "On dynamic spectral risk measures, a limit theorem and optimal portfolio allocation," Papers 1301.3531, arXiv.org, revised Apr 2017.
    16. Christoph Belak & Daniel Hoffmann & Frank T. Seifried, 2020. "Continuous-Time Mean Field Games with Finite StateSpace and Common Noise," Working Paper Series 2020-05, University of Trier, Research Group Quantitative Finance and Risk Analysis.
    17. Olivier Menoukeu-Pamen & Romuald Hervé Momeya, 2017. "A maximum principle for Markov regime-switching forward–backward stochastic differential games and applications," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 85(3), pages 349-388, June.
    18. René Carmona & Gökçe Dayanıklı & Mathieu Laurière, 2022. "Mean Field Models to Regulate Carbon Emissions in Electricity Production," Dynamic Games and Applications, Springer, vol. 12(3), pages 897-928, September.
    19. René Carmona, 2022. "The influence of economic research on financial mathematics: Evidence from the last 25 years," Finance and Stochastics, Springer, vol. 26(1), pages 85-101, January.
    20. Camilo Hern'andez & Dylan Possamai, 2023. "Time-inconsistent contract theory," Papers 2303.01601, arXiv.org.

    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:inm:ormoor:v:46:y:2021:i:2:p:471-502. 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: Chris Asher (email available below). General contact details of provider: https://edirc.repec.org/data/inforea.html .

    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.