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Nash equilibrium payoffs for stochastic differential games with jumps and coupled nonlinear cost functionals. (English) Zbl 1333.60124
Summary: In this paper, we investigate Nash equilibrium payoffs for two-player nonzero-sum stochastic differential games with coupled nonlinear cost functionals. We obtain an existence theorem and a characterization theorem for Nash equilibrium payoffs. The novelty of this paper is to study Nash equilibrium payoffs for coupled cost functionals.

##### MSC:
 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 91A15 Stochastic games, stochastic differential games 91A23 Differential games (aspects of game theory) 91A05 2-person games 49N70 Differential games and control 49J55 Existence of optimal solutions to problems involving randomness
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##### References:
 [1] Barles, G.; Buckdahn, R.; Pardoux, E., Backward stochastic differential equations and integral-partial differential equations, Stoch. Stoch. Rep., 60, 57-83, (1997) · Zbl 0878.60036 [2] Barles, G.; Imbert, C., Second-order elliptic integro-differential equations: viscosity solutions’ theory revisited, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25, 567-585, (2008) · Zbl 1155.45004 [3] Bensoussan, A.; Frehse, J., Stochastic games for $$N$$ players, J. Optim. Theory Appl., 105, 543-565, (2000) · Zbl 0977.91006 [4] Biswas, I., On zero-sum stochastic differential games with jump-diffusion driven state: a viscosity solution framework, SIAM J. Control Optim., 50, 1823-1858, (2012) · Zbl 1253.91027 [5] Buckdahn, R.; Cardaliaguet, P.; Quincampoix, M., Some recent aspects of differential game theory, Dyn. Games Appl., 1, 74-114, (2011) · Zbl 1214.91013 [6] Buckdahn, R.; Cardaliaguet, P.; Rainer, C., Nash equilibrium payoffs for nonzero-sum stochastic differential games, SIAM J. Control Optim., 43, 624-642, (2004) · Zbl 1101.91010 [7] Buckdahn, R.; Hu, Y., Probabilistic interpretation of a coupled system of Hamilton-Jacobi-Bellman equations, J. Evol. Equ., 10, 529-549, (2010) · Zbl 1239.35036 [8] Buckdahn, R.; Hu, Y.; Li, J., Integral-partial gifferential equations of Isaacs type related to stochastic differential games with jumps, Stochastic Process. Appl., 121, 2715-2750, (2011) · Zbl 1243.91011 [9] Buckdahn, R.; Li, J., Stochastic differential games and viscosity solutions of Hamilton-Jacobi-Bellman-Isaacs equations, SIAM J. Control Optim., 47, 444-475, (2008) · Zbl 1157.93040 [10] Fleming, W. H.; Souganidis, P. E., On the existence of value functions of twoplayer, zero-sum stochastic differential games, Indiana Univ. Math. J., 38, 293-314, (1989) · Zbl 0686.90049 [11] Hamadène, S.; Lepeltier, J.; Peng, S., BSDEs with continuous coefficients and stochastic differential games, (El Karoui, Mazliak, Backward Stochastic Differential Equations, Pitman Research Notes in Math. Series, vol. 364, (1997)), 115-128 · Zbl 0892.60062 [12] Lasry, J.; Lions, P., Mean field games, Jpn. J. Math., 2, 229-260, (2007) · Zbl 1156.91321 [13] Lin, Q., A BSDE approach to Nash equilibrium payoffs for stochastic differential games with nonlinear cost functionals, Stochastic Process. Appl., 122, 357-385, (2012) · Zbl 1237.60047 [14] Lin, Q., Nash equilibrium payoffs for stochastic differential games with reflection, ESAIM Control Optim. Calc. Var., 19, 1189-1208, (2013) · Zbl 1283.49043 [15] Pardoux, E.; Pradeilles, F.; Rao, Z., Probabilistic interpretation of a system of semi-linear parabolic partial differential equations, Ann. Inst. H. Poincare Probab. Statist., 33, 467-490, (1997) · Zbl 0891.60054 [16] Peng, S., Backward stochastic differential equations—stochastic optimization theory and viscosity solutions of HJB equations, (Topics in Stochastic Analysis, (1997), Science Press Beijing), Yan, J., Peng, S., Fang, S., Wu, L.M. Ch.2 (Chinese vers.) [17] Royer, M., Backward stochastic differential equations with jumps and related non-linear expectations, Stochastic Process. Appl., 116, 1358-1376, (2006) · Zbl 1110.60062 [18] Tang, S.; Li, X., Necessary conditions for optimal control of stochastic systems with jumps, SIAM J. Control. Optim, 32, 1447-1475, (1994) · Zbl 0922.49021
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