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Risk-Sensitive Nonzero-Sum Stochastic Differential Game with Unbounded Coefficients

Abstract

This article is related to risk-sensitive nonzero-sum stochastic differential games in the Markovian framework. This game takes into account the attitudes of the players towards risks, and the utilities are of exponential forms. We show the existence of a Nash equilibrium point for the game when the drift is no longer bounded and only satisfies a linear growth condition. The main tool is the notion of backward stochastic differential equation, which in our case, is multidimensional with continuous generator involving both a quadratic term and a stochastic linear growth component with respect to the volatility process.

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References

  1. Aronson DG (1967) Bounds for the fundamental solution of a parabolic equation. Bull Am Math Soc 73(6):890–896

    MathSciNet  Article  Google Scholar 

  2. Başar T (1999) Nash equilibria of risk-sensitive nonlinear stochastic differential games. J Optim Theory Appl 100(3):479–498

    MathSciNet  Article  Google Scholar 

  3. Doléan-Dade C, Dellacherie C, Meyer PA (1970) Diffusions à coefficients continus, d’après Stroock et Varadhan. Séminaire de Probabilités (Strasbourg) 4:240–282

    Google Scholar 

  4. El-Karoui N, Hamadène S (2003) BSDEs and risk-sensitive control, zero-sum and nonzero-sum game problems of stochastic functional differential equations. Stoch Process Appl 107(1):145–169

    MathSciNet  Article  Google Scholar 

  5. El-Karoui N, Peng S, Quenez MC (1997) Backward stochastic differential equations in finance. Math Finance 7(1):1–71

    MathSciNet  Article  Google Scholar 

  6. Fleming WH (2006) Risk sensitive stochastic control and differential games. Commun Inf Syst 6(3):161–177

    MathSciNet  MATH  Google Scholar 

  7. Fleming WH, McEneaney WM (1992) Risk sensitive optimal control and differential games. Springer, Berlin

    Book  Google Scholar 

  8. Girsanov IV (1960) On transforming a certain class of stochastic processes by absolutely continuous substitution of measures. Theory Probab Appl 5:285–301

    MathSciNet  Article  Google Scholar 

  9. Hamadène S (1998) Backward-forward SDE’s and stochastic differential games. Stoch Process Appl 77(1):1–15

    MathSciNet  Article  Google Scholar 

  10. Hamadène S, Lepeltier J-P, Peng S (1997) BSDEs with continuous coefficients and stochastic differential games. In: El Karoui N, Mazliak L (eds) Pitman Res Notes Math Ser, vol 364. Longman, Harlow, UK, pp 115–128

  11. Hamadène S, Mu R (2015) Existence of Nash equilibrium points for Markovian non-zero-sum stochastic differential games with unbounded coefficients. Stoch Int J Probab Stoch Process 87(1):85–111

    MathSciNet  Article  Google Scholar 

  12. Haussmann UG (1986) A stochastic maximum principle for optimal control of diffusions. Wiley, Hoboken

    MATH  Google Scholar 

  13. Hu Y, Tang S (2016) Multi-dimensional backward stochastic differential equations of diagonally quadratic generators. Stoch Process Appl 126(4):1066–1086

    MathSciNet  Article  Google Scholar 

  14. James Matthew R (1992) Asymptotic analysis of nonlinear stochastic risk-sensitive control and differential games. Math Control Signals Syst 5(4):401–417

    MathSciNet  Article  Google Scholar 

  15. Karatzas I, Shreve SE (1991) Brownian motion and stochastic calculus, 2nd edn. Springer, Berlin

    MATH  Google Scholar 

  16. Kobylanski M (2000) Backward stochastic differential equations and partial differential equations with quadratic growth. Ann Probab 28(2):558–602

    MathSciNet  Article  Google Scholar 

  17. Moon J, Duncan TE, Basar T (2019) Risk-sensitive zero-sum differential games. IEEE Trans Autom Control 64(4):1503–1518

    MathSciNet  Article  Google Scholar 

  18. Pardoux E, Peng S (1990) Adapted solution of a backward stochastic differential equation. Syst Control Lett 14(1):55–61

    MathSciNet  Article  Google Scholar 

  19. Pardoux E, Peng S (1992) Backward stochastic differential equations and quasilinear parabolic partial differential equations. Stochastic partial differential equations and their applications. Springer, Berlin, pp 200–217

    MATH  Google Scholar 

  20. Peng S (2011) Backward stochastic differential equation, nonlinear expectation and their applications. In: Proceedings of the international congress of mathematicians, pp 393–432

  21. Tembine H, Zhu Q, Başar T (2011) Risk-sensitive mean-field stochastic differential games. In: Proceedings of 18th IFAC World Congress

  22. Xing H, Žitković G (2018) A class of globally solvable Markovian quadratic BSDE systems and applications. Ann Probab 46(1):491–550

    MathSciNet  Article  Google Scholar 

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Correspondence to Rui Mu.

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The second author is supported in part by National Natural Science Foundation for Young Scientists of China (Grant No. 11701404) and Natural Science Foundation for Young Scientists of Jiangsu Province of China (Grant No. BK20160300)

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Hamadène, S., Mu, R. Risk-Sensitive Nonzero-Sum Stochastic Differential Game with Unbounded Coefficients. Dyn Games Appl 11, 84–108 (2021). https://doi.org/10.1007/s13235-020-00353-0

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  • DOI: https://doi.org/10.1007/s13235-020-00353-0

Keywords

  • Risk-sensitive
  • Nonzero-sum stochastic differential games
  • Nash equilibrium point
  • Backward stochastic differential equations