Nash Points for Nonzero-Sum Stochastic Differential Games with Separate Hamiltonians

We study a nonzero-sum stochastic differential game under the assumptions that the control sets are multidimensional convex compact, the game has separate dynamic and running costs and the multifunctions representing the optimal feedbacks have convex values. To prove the existence of Nash equilibria we reduce to study a system of uniformly parabolic equations strongly coupled by multivalued applications. We obtain the existence of Nash points in two different cases: (i) $\mathbb{R}$-valued process and general dynamic, (ii) multivalued process and affine dynamic.

[1]  Paola Mannucci,et al.  Nonzero-Sum Stochastic Differential Games with Discontinuous Feedback , 2004, SIAM J. Control. Optim..

[2]  Wen Shen,et al.  Semi-cooperative strategies for differential games , 2004, Int. J. Game Theory.

[3]  Thomas L. Vincent,et al.  Advances in Dynamic Game Theory , 2007 .

[4]  A. Friedman.  Stochastic Differential Equations and Applications , 1975 .

[5]  Avner Friedman,et al.  Stochastic differential games , 1972 .

[6]  Pierre Cardaliaguet,et al.  Existence and uniqueness of a Nash equilibrium feedback for a simple nonzero-sum differential game , 2003, Int. J. Game Theory.

[7]  A. Bensoussan,et al.  Nonlinear elliptic systems in stochastic game theory. , 1984 .

[8]  C. Berge,et al.  Espaces topologiques : fonctions multivoques , 1966 .

[9]  Alain Bensoussan,et al.  Regularity Results for Nonlinear Elliptic Systems and Applications , 2002 .

[10]  O. Ladyženskaja.  Linear and Quasilinear Equations of Parabolic Type , 1968 .

[11]  Geert Jan Olsder,et al.  On Open- and Closed-Loop Bang-Bang Control in Nonzero-Sum Differential Games , 2001, SIAM J. Control. Optim..

[12]  Pierre Cardaliaguet,et al.  On the Instability of the Feedback Equilibrium Payoff in a Nonzero-Sum Differential Game on the Line , 2007 .

[13]  W. Fleming,et al.  Controlled Markov processes and viscosity solutions , 1992 .

[14]  Catherine Rainer,et al.  Two Different Approaches to Nonzero-Sum Stochastic Differential Games , 2007 .

[15]  Catherine Rainer,et al.  Nash Equilibrium Payoffs for Nonzero-Sum Stochastic Differential Games , 2004, SIAM J. Control. Optim..

[16]  Wen Shen,et al.  Small BV Solutions of Hyperbolic Noncooperative Differential Games , 2004, SIAM J. Control. Optim..

[17]  M. K. Ghosh,et al.  Stochastic differential games: Occupation measure based approach , 1996 .

[18]  Alberto Bressan,et al.  Noncooperative Differential Games , 2011 .

[19]  A. Bensoussan.  Stochastic control by functional analysis methods , 1982 .

[20]  M. Landsberg,et al.  C. Berge, Espaces Topologiques — Fonetions Multivoques (Collection Universitaire de Mathématiques) XII + 272 S. m. 47 Abb. Paris 1959. Dunod Éditeur. Preis geb. 3,400 F , 1960 .

[21]  A. Bensoussan,et al.  Stochastic Games for N Players , 2000 .

[22]  A. Cellina.  A selection theorem , 1976 .

[23]  Harold J. Kushner.  Numerical Approximations for Nonzero-Sum Stochastic Differential Games , 2007, SIAM J. Control. Optim..

[24]  G. M. Lieberman.  SECOND ORDER PARABOLIC DIFFERENTIAL EQUATIONS , 1996 .

[25]  M. K. Ghosh,et al.  Zero-sum stochastic differential games with reflecting diffusions , 1997 .