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Probabilistic Theory of Mean Field Games with Applications I pp 67–127Cite as

Probabilistic Approach to Stochastic Differential Games

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Part of the book series: Probability Theory and Stochastic Modelling ((PTSM,volume 83))

Abstract

This chapter offers a crash course on the theory of nonzero sum stochastic differential games. Its goal is to introduce the jargon and the notation of this class of game models. As the main focus of the text is the probabilistic approach to the solution of stochastic games, we review the strategy based on the stochastic Pontryagin maximum principle and show how BSDEs and FBSDEs can be brought to bear in the search for Nash equilibria. We emphasize the differences between open loop and closed loop or Markov equilibria and we illustrate the results of the chapter with detailed analyses of some of the models introduced in Chapter 1

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References

  1. T.T.K. An and B. Oksendal. Maximum principle for stochastic differential games with partial information. Journal of Optimization Theory and Applications, 139:463–483, 2008.

    Article  MathSciNet  MATH  Google Scholar 

  2. T.T.K. An and B. Oksendal. A maximum principle for stochastic differential games with g-expectations and partial information. Stochastics, 84:137–155, 2012.

    Article  MathSciNet  MATH  Google Scholar 

  3. A. Bensoussan and J. Frehse. Nonlinear elliptic systems in stochastic game theory. Journal für die reine und angewandte Mathematik, 350:23–67, 1984.

    MathSciNet  MATH  Google Scholar 

  4. A. Bensoussan and J. Frehse. Stochastic games for N players. Journal of Optimization Theory and Applications, 105:543–565, 2000.

    Article  MathSciNet  MATH  Google Scholar 

  5. A. Bensoussan and J. Frehse. Smooth solutions of systems of quasilinear parabolic equations. ESAIM: Control, Optimisation and Calculus of Variations, 8:169–193, 2010.

    Article  MathSciNet  MATH  Google Scholar 

  6. R. Carmona. Lectures on BSDEs, Stochastic Control and Stochastic Differential Games. SIAM, 2015.

    Google Scholar 

  7. R. Carmona, J.P. Fouque, M. Moussavi, and L.H. Sun. Systemic risk and stochastic games with delay. Technical report, 2016. https://arxiv.org/abs/1607.06373

    Google Scholar 

  8. R. Carmona, J.P. Fouque, and L.H. Sun. Mean field games and systemic risk: a toy model. Communications in Mathematical Sciences, 13:911–933, 2015.

    Article  MathSciNet  MATH  Google Scholar 

  9. F. Delarue and G. Guatteri. Weak existence and uniqueness for FBSDEs. Stochastic Processes and their Applications, 116:1712–1742, 2006.

    Article  MathSciNet  MATH  Google Scholar 

  10. A. Friedman. Partial differential equations of parabolic type. Prentice-Hall, Englewood Cliffs, N.J., first edition, 1964.

    MATH  Google Scholar 

  11. A. Friedman. Stochastic differential games. Journal of Differential Equations, 11:79–108, 1972.

    Article  MathSciNet  MATH  Google Scholar 

  12. S. Hamadène. Backward-forward SDE’s and stochastic differential games. Stochastic Processes and their Applications, 77:1–15, 1998.

    Article  MathSciNet  MATH  Google Scholar 

  13. S. Hamadène. Nonzero-sum linear quadratic stochastic differential games and backward forward equations. Stochastic Analysis and Applications, 17:117–130, 1999.

    Article  MathSciNet  MATH  Google Scholar 

  14. N. Krylov. Controlled Diffusion Processes. Stochastic Modelling and Applied Probability. Springer-Verlag Berlin Heidelberg, 1980.

    Book  MATH  Google Scholar 

  15. O.A. Ladyzenskaja, V.A. Solonnikov, and N. N. Ural’ceva. Linear and Quasi-linear Equations of Parabolic Type. Translations of Mathematical Monographs. American Mathematical Society, 1968.

    Google Scholar 

  16. G M Lieberman. Second Order Parabolic Differential Equations. World Scientific, 1996.

    Book  MATH  Google Scholar 

  17. A. Y. Veretennikov. Strong solutions and explicit formulas for solutions of stochastic integral equations. Matematicheskii Sbornik, 111:434–452, 1980.

    MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. ORFE Department, Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ, USA

    René Carmona

  2. Institut Universitaire de France & Laboratoire J.A. Dieudonné, Université Nice Sophia Antipolis, Nice, France

    François Delarue

Authors
  1. René Carmona
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  2. François Delarue
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Carmona, R., Delarue, F. (2018). Probabilistic Approach to Stochastic Differential Games. In: Probabilistic Theory of Mean Field Games with Applications I. Probability Theory and Stochastic Modelling, vol 83. Springer, Cham. https://doi.org/10.1007/978-3-319-58920-6_2

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