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Stochastics
An International Journal of Probability and Stochastic Processes
Volume 87, 2015 - Issue 1
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Articles

Existence of Nash equilibrium points for Markovian non-zero-sum stochastic differential games with unbounded coefficients

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Pages 85-111 | Received 02 Sep 2013, Accepted 14 Apr 2014, Published online: 08 Jul 2014

References

  • D.G.Aronson, Bounds for the fundamental solution of a parabolic equation, Bull. Am. Math. Soc.73(6) (1967), pp. 890–896.
  • A.Bensoussan and J.L.Lions, Application of Variational Inequalities in Stochastic Control, Studies in Mathematics and Its Applications, Vol. 12, North-Holland Publishing Co, Amsterdam, New York, 1982.
  • R.Buckdahn, P.Cardaliaguet, and C.Rainer, Nash equilibrium payoffs for nonzero-sum stochastic differential games, SIAM J. Control Optim.43(2) (2004), pp. 624–642.
  • R.Buckdahn and J.Li, Stochastic differential games and viscosity solutions for Hamilton–Jacobi–Bellman–Isaacs equations, SIAM J. Control Optim.47(1) (2008), pp. 444–475.
  • M.H.A.Davis and R.J.Elliott, Optimal play in a stochastic differential game, SIAM J. Control Optim.19(4) (1981), pp. 543–554.
  • N.El-Karoui and S.Hamadène, BSDEs and risk-sensitive control, zero-sum and nonzero-sum game problems of stochastic functional differential equations, Stoch. Process. Appl.107(1) (2003), pp. 145–169.
  • N.El Karoui, S.Peng, and M.C.Quenez, Backward stochastic differential equations in finance, Math. Finance7(1) (1997), pp. 1–71.
  • R.J.Elliott, The existence of value in stochastic differential games, SIAM J. Control Optim.14(1) (1976), pp. 85–94.
  • W.H.Fleming and P.E.Souganidis, On the existence of value functions of two-player, zero-sum stochastic differential games, Indiana Univ. Math. J.38(2) (1989), pp. 293–314.
  • A.Friedman, Differential Games, Wiley, New York, 1971.
  • A.Friedman, Stochastic differential games, J. Differ. Equ.11(1) (1972), pp. 79–108.
  • I.V.Girsanov, On transforming a certain class of stochastic processes by absolutely continuous substitution of measures, Theory Probab. Appl.5 (1960), pp. 285–301.
  • S.Hamadène, Backward–forward SDEs and stochastic differential games, Stoch. Process. Appl.77(1) (1998), pp. 1–15.
  • S.Hamadène, Nonzero sum linear-quadratic stochastic differential games and backward–forward equations, Stoch. Anal. Appl.17(1) (1999), pp. 117–130.
  • S.Hamadène and J.-P.Lepeltier, Points d'équilibre dans les jeux stochastiques de somme non nulle, Comptes rendus de l' Académie des sciences, tome 318, série 1, fev. 94, 1994, pp. 251–256.
  • S.Hamadène and J.-P.Lepeltier, Zero-sum stochastic differential games and backward equations, Syst. Control Lett.24(4) (1995), pp. 259–263.
  • S.Hamadène and J.-P.Lepeltier, Backward equations, stochastic control and zero-sum stochastic differential games, Stoch Int. J. Probab. Stoch. Process.54 (1995), pp. 221–231.
  • S.Hamadène, J.-P.Lepeltier, and S.Peng, BSDEs with Continuous Coefficients and Stochastic Differential Games, Pitman Research Notes in Mathematics Series, Taylor & Francis, Bosa Roca, 1997, pp. 115–128.
  • U.G.Haussmann, A Stochastic Maximum Principle for Optimal Control of Diffusions, John Wiley & Sons, New York, 1986.
  • R.Isaacs, Differential Games, Wiley, New York, 1965.
  • I.Karatzas and S.E.Shreve, Brownian Motion and Stochastic Calculus, 2nd ed., Springer Verlag, New York, 1991.
  • J.P.Lepeltier, Z.Wu, and Z.Yu, Nash equilibrium point for one kind of stochastic nonzero-sum game problem and BSDEs, Comptes Rendus Mathematique347(15) (2009), pp. 959–964.
  • Q.Lin, A BSDE approach to Nash equilibrium payoffs for stochastic differential games with nonlinear cost functionals, Stoch. Process. Appl.122(1) (2012), pp. 357–385.
  • P.Mannucci, Nonzero-sum stochastic differential games with discontinuous feedback, SIAM J. Control Optim.43(4) (2004), pp. 1222–1233.
  • T. Pham and J. Zhang, Two person zero-sum game in weak formulation and path dependent Bellman–Isaacs equation, preprint (2012). Available at arXiv:1209.6605v1.
  • P.Protter, Stochastic Integration and Differential Equations, 2nd ed., Springer-Verlag, Berlin, 2004.
  • C.Rainer, Two different approaches to nonzero-sum stochastic differential games, Appl. Math. Optim.56(1) (2007), pp. 131–144.
  • M. Sirbu, Stochastic Perrons method and elementary strategies for zero-sum dierential games, preprint (2013). Available at arXiv:1305.5083v1.

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