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Linear-Quadratic Mean Field Games

  • A. Bensoussan
  • K. C. J. Sung
  • S. C. P. Yam
  • S. P. Yung
Article

Abstract

We provide a comprehensive study of a general class of linear-quadratic mean field games. We adopt the adjoint equation approach to investigate the unique existence of their equilibrium strategies. Due to the linearity of the adjoint equations, the optimal mean field term satisfies a forward–backward ordinary differential equation. For the one-dimensional case, we establish the unique existence of the equilibrium strategy. For a dimension greater than one, by applying the Banach fixed point theorem under a suitable norm, a sufficient condition for the unique existence of the equilibrium strategy is provided, which is independent of the coefficients of controls in the underlying dynamics and is always satisfied whenever the coefficients of the mean field term are vanished, and hence, our theories include the classical linear-quadratic stochastic control problems as special cases. As a by-product, we also establish a neat and instructive sufficient condition, which is apparently absent in the literature and only depends on coefficients, for the unique existence of the solution for a class of nonsymmetric Riccati equations. Numerical examples of nonexistence of the equilibrium strategy will also be illustrated. Finally, a similar approach has been adopted to study the linear-quadratic mean field type stochastic control problems and their comparisons with mean field games.

Keywords

Mean field games Mean field type stochastic control problems Adjoint equations Linear quadratic 

Notes

Acknowledgments

We are grateful to many seminar and conference participants such as those in the workshop of Sino-French Summer Institute 2011 and 15th International Congress on Insurance Mathematics and Economics 2011 for their valuable comments and suggestions on the preliminary version of the present work. The first author acknowledges the financial support by National Science Foundation DMS-1303775 and The Hong Kong RGC GRF 500113. The third author—Phillip Yam—acknowledges the financial support from The Hong Kong RGC GRF 502909, The Hong Kong RGC GRF 500111, The Hong Kong RGC GRF 404012 with the project title: Advanced Topics In Multivariate Risk Management In Finance And Insurance, The Chinese University of Hong Kong Direct Grant 2010/2011 Project ID: 2060422, and The Chinese University of Hong Kong Direct Grant 2011/2012 Project ID: 2060444. Phillip Yam also expresses his sincere gratitude to the hospitality of both Hausdorff Center for Mathematics of the University of Bonn and Mathematisches Forschungsinstitut Oberwolfach (MFO) in the German Black Forest during the preparation of the present work. The last author thanks the support from the Hung Hing Ying Physical Science Research Fund of code 30129.203730745.018746.22500.406.01.

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • A. Bensoussan
    • 1
    • 2
  • K. C. J. Sung
    • 3
  • S. C. P. Yam
    • 4
  • S. P. Yung
    • 5
  1. 1.International Center for Decision and Risk Analysis, Jindal School of ManagementThe University of Texas at DallasRichardsonUSA
  2. 2.Department of Systems Engineering and Engineering Management, College of Science and EngineeringCity University of Hong KongHong KongChina
  3. 3.Department of Statistics and Actuarial ScienceThe University of Hong KongHong KongChina
  4. 4.Department of StatisticsThe Chinese University of Hong KongHong KongChina
  5. 5.Department of MathematicsThe University of Hong KongHong KongChina

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