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Risk-sensitive mean field stochastic differential games

Author

Listed:
  • Hamidou Tembine

    (E3S - Supélec Sciences des Systèmes - Ecole Supérieure d'Electricité - SUPELEC (FRANCE))

  • Quanyan Zhu

    (CSL - Coordinated Science Laboratory - University of Illinois System)

  • Tamer Basar

    (CSL - Coordinated Science Laboratory - University of Illinois System)

Abstract

In this paper, we study a class of risk-sensitive mean-field stochastic di fferential games. Under regularity assumptions, we use results from standard risk-sensitive di fferential game theory to show that the mean- field value of the exponentiated cost functional coincides with the value function of a Hamilton-Jacobi-Bellman-Fleming (HJBF) equation with an additional quadratic term. We provide an explicit solution of the mean- field best response when the instantaneous cost functions are log-quadratic and the state dynamics are affine in the control. An equivalent mean- field risk-neutral problem is formulated and the corresponding mean-fi eld equilibria are characterized in terms of backward-forward macroscopic McKean-Vlasov equations, Fokker- Planck-Kolmogorov equations and HJBF equations.

Suggested Citation

  • Hamidou Tembine & Quanyan Zhu & Tamer Basar, 2011. "Risk-sensitive mean field stochastic differential games," Post-Print hal-00643547, HAL.
    • Handle: RePEc:hal:journl:hal-00643547
    DOI: 10.3182/20110828-6-IT-1002.02247
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    Citations

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    Cited by:

    1. A. Bensoussan & K. Sung & S. Yam, 2013. "Linear–Quadratic Time-Inconsistent Mean Field Games," Dynamic Games and Applications, Springer, vol. 3(4), pages 537-552, December.
    2. Said Hamadène & Rui Mu, 2021. "Risk-Sensitive Nonzero-Sum Stochastic Differential Game with Unbounded Coefficients," Dynamic Games and Applications, Springer, vol. 11(1), pages 84-108, March.
    3. Dario Bauso & Quanyan Zhu & Tamer Başar, 2016. "Decomposition and Mean-Field Approach to Mixed Integer Optimal Compensation Problems," Journal of Optimization Theory and Applications, Springer, vol. 169(2), pages 606-630, May.
    4. Minyi Huang, 2013. "A Mean Field Capital Accumulation Game with HARA Utility," Dynamic Games and Applications, Springer, vol. 3(4), pages 446-472, December.
    5. Boualem Djehiche & Minyi Huang, 2016. "A Characterization of Sub-game Perfect Equilibria for SDEs of Mean-Field Type," Dynamic Games and Applications, Springer, vol. 6(1), pages 55-81, March.
    6. Dario Bauso & Hamidou Tembine & Tamer Başar, 2016. "Robust Mean Field Games," Dynamic Games and Applications, Springer, vol. 6(3), pages 277-303, September.
    7. Fabio Bagagiolo & Dario Bauso, 2014. "Mean-Field Games and Dynamic Demand Management in Power Grids," Dynamic Games and Applications, Springer, vol. 4(2), pages 155-176, June.
    8. Jun Moon & Tamer Başar, 2019. "Risk-Sensitive Mean Field Games via the Stochastic Maximum Principle," Dynamic Games and Applications, Springer, vol. 9(4), pages 1100-1125, December.
    9. A. Bensoussan & K. C. J. Sung & S. C. P. Yam & S. P. Yung, 2016. "Linear-Quadratic Mean Field Games," Journal of Optimization Theory and Applications, Springer, vol. 169(2), pages 496-529, May.
    10. Dario Bauso & Raffaele Pesenti & Marco Tolotti, 2016. "Opinion Dynamics and Stubbornness Via Multi-Population Mean-Field Games," Journal of Optimization Theory and Applications, Springer, vol. 170(1), pages 266-293, July.

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