# American Institute of Mathematical Sciences

September  2014, 13(5): 1719-1736. doi: 10.3934/cpaa.2014.13.1719

## Stochastic differential games with a varying number of players

 1 International Center for Decision and Risk Analysis, School of Management, P.O.Box 830688, SM 30, University of Texas at Dallas, Richardson, TX 75083-0688 2 Bonn University and Toulouse School of Economics, Germany, Germany

Received  September 2013 Revised  September 2013 Published  June 2014

We consider a non zero sum stochastic differential game with a maximum $n$ players, where the players control a diffusion in order to minimise a certain cost functional. During the game it is possible that present players may die or new players may appear. The death, respectively the birth time of a player is exponentially distributed with intensities that depend on the diffusion and the controls of the players who are alive. We show how the game is related to a system of partial differential equations with a special coupling in the zero order terms. We provide an existence result for solutions in appropriate spaces that allow to construct Nash optimal feedback controls. The paper is related to a previous result in a similar setting for two players leading to a parabolic system of Bellman equations [4]. Here, we study the elliptic case (infinite horizon) and present the generalisation to more than two players.
Citation: Alain Bensoussan, Jens Frehse, Christine Grün. Stochastic differential games with a varying number of players. Communications on Pure and Applied Analysis, 2014, 13 (5) : 1719-1736. doi: 10.3934/cpaa.2014.13.1719
##### References:
 [1] A. Bensoussan and J. Frehse, Nonlinear elliptic systems in stochastic game theory, J. Reine Angew. Math., 350 (1984), 23-67. [2] Alain Bensoussan and Jens Frehse, Diagonal elliptic Bellman systems to stochastic differential games with discount control and noncompact coupling, Rend. Mat. Appl., 29 (2009), 1-16. [3] A. Bensoussan and J. Frehse, Control and Nash games with mean field effect, Chin. Ann. Math. Ser. B, 34 (2013), 161-192. doi: 10.1007/s11401-013-0767-y. [4] A. Bensoussan, J. Frehse and C. Grün, On a system of PDEs associated to a game with a varying number of players,, 2013, (). [5] A. Bensoussan and A. Friedman, Nonzero-sum stochastic differential games with stopping times and free boundary problems, Trans. Amer. Math. Soc., 231 (1977), 275-327. [6] P. Bremaud, Point Processes and Queues: Martingale Dynamics, Springer Series in Statistics. Springer-Verlag, New York-Berlin, 1981. [7] R. Buckdahn, P. Cardaliaguet and C. Rainer, Nash equilibrium payoffs for nonzero-sum stochastic differential games, SIAM J. Control Optim., 43 (2004), 624-642. doi: 10.1137/S0363012902411556. [8] S. Campanato, Regolarizzazione negli spazi $L^{(2,\lambda )}$ delle soluzioni delle equazioni ellittiche del II ordine, BOOK Atti del Convegno su le Equazioni alle Derivate Parziali (Nervi, 1965), 33-35, Edizioni Cremonese, Rome, 1966. [9] J. Frehse, Bellman systems of stochastic differential games with three players, in Optimal Control and Partial Differential Equations. In Honour of Professor Alain Bensoussan's 60th Birthday. Proceedings of the Conference, Paris, France, December 4, 2000, Amsterdam, 2001, pp. 3-22. [10] A. Friedman, Stochastic differential games, J. Differential Equations, 11 (1972), 79-108. [11] S. Hamadene, J.-P. Lepeltier and S. Peng, BSDEs with continuous coefficients and stochastic differential games, Pitman Res. Notes Math. Ser., 364 (1997), 115-128. [12] S. Hamadène and J. Zhang, The continuous time nonzero-sum Dynkin game problem and application in game options, SIAM J. Control Optim., 48 (2009), 3659-3669. doi: 10.1137/080738933. [13] N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, Kodansha scientific books, North-Holland, 1989. [14] R. Isaacs, Differential Games. A Mathematical Theory with Applications to Warfare and Pursuit, Control and Optimization, John Wiley & Sons Inc., New York, 1965. [15] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Classics in Applied Mathematics, 31, SIAM, Philadelphia, PA, 2000. doi: 10.1137/1.9780898719451. [16] N. V. Krylov, Controlled Diffusion Processes, Springer-Verlag, Berlin, 2009. [17] O. A. Ladyzhenskaya and N. N. Uraltseva, Linear and Quasilinear Elliptic Equations, Translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis, Academic Press New York, 1968. [18] J. Leray and J. L. Lions, Quelques résultats de Visik sur les problèmes elliptiques non linèaires par les mèthodes de Minty-Browder, Bulletin de la Soc. Math. France, 93 (1965), 97-107. [19] Jr. Morrey and B. Charles, Multiple Integrals in the Calculus of Variations, Springer-Verlag, Berlin, 2008. doi: 10.1007/978-3-540-69952-1. [20] M. I. Višik, Quasilinear elliptic systems of equations containing subordinate terms, Dokl. Akad. Nauk SSSR, 144 (1962), 13-16. [21] K.-O. Widman, Hölder Continuity of Solutions of Elliptic Systems, Manuscripta Math., 5 (1971), 299-308.

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##### References:
 [1] A. Bensoussan and J. Frehse, Nonlinear elliptic systems in stochastic game theory, J. Reine Angew. Math., 350 (1984), 23-67. [2] Alain Bensoussan and Jens Frehse, Diagonal elliptic Bellman systems to stochastic differential games with discount control and noncompact coupling, Rend. Mat. Appl., 29 (2009), 1-16. [3] A. Bensoussan and J. Frehse, Control and Nash games with mean field effect, Chin. Ann. Math. Ser. B, 34 (2013), 161-192. doi: 10.1007/s11401-013-0767-y. [4] A. Bensoussan, J. Frehse and C. Grün, On a system of PDEs associated to a game with a varying number of players,, 2013, (). [5] A. Bensoussan and A. Friedman, Nonzero-sum stochastic differential games with stopping times and free boundary problems, Trans. Amer. Math. Soc., 231 (1977), 275-327. [6] P. Bremaud, Point Processes and Queues: Martingale Dynamics, Springer Series in Statistics. Springer-Verlag, New York-Berlin, 1981. [7] R. Buckdahn, P. Cardaliaguet and C. Rainer, Nash equilibrium payoffs for nonzero-sum stochastic differential games, SIAM J. Control Optim., 43 (2004), 624-642. doi: 10.1137/S0363012902411556. [8] S. Campanato, Regolarizzazione negli spazi $L^{(2,\lambda )}$ delle soluzioni delle equazioni ellittiche del II ordine, BOOK Atti del Convegno su le Equazioni alle Derivate Parziali (Nervi, 1965), 33-35, Edizioni Cremonese, Rome, 1966. [9] J. Frehse, Bellman systems of stochastic differential games with three players, in Optimal Control and Partial Differential Equations. In Honour of Professor Alain Bensoussan's 60th Birthday. Proceedings of the Conference, Paris, France, December 4, 2000, Amsterdam, 2001, pp. 3-22. [10] A. Friedman, Stochastic differential games, J. Differential Equations, 11 (1972), 79-108. [11] S. Hamadene, J.-P. Lepeltier and S. Peng, BSDEs with continuous coefficients and stochastic differential games, Pitman Res. Notes Math. Ser., 364 (1997), 115-128. [12] S. Hamadène and J. Zhang, The continuous time nonzero-sum Dynkin game problem and application in game options, SIAM J. Control Optim., 48 (2009), 3659-3669. doi: 10.1137/080738933. [13] N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, Kodansha scientific books, North-Holland, 1989. [14] R. Isaacs, Differential Games. A Mathematical Theory with Applications to Warfare and Pursuit, Control and Optimization, John Wiley & Sons Inc., New York, 1965. [15] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Classics in Applied Mathematics, 31, SIAM, Philadelphia, PA, 2000. doi: 10.1137/1.9780898719451. [16] N. V. Krylov, Controlled Diffusion Processes, Springer-Verlag, Berlin, 2009. [17] O. A. Ladyzhenskaya and N. N. Uraltseva, Linear and Quasilinear Elliptic Equations, Translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis, Academic Press New York, 1968. [18] J. Leray and J. L. Lions, Quelques résultats de Visik sur les problèmes elliptiques non linèaires par les mèthodes de Minty-Browder, Bulletin de la Soc. Math. France, 93 (1965), 97-107. [19] Jr. Morrey and B. Charles, Multiple Integrals in the Calculus of Variations, Springer-Verlag, Berlin, 2008. doi: 10.1007/978-3-540-69952-1. [20] M. I. Višik, Quasilinear elliptic systems of equations containing subordinate terms, Dokl. Akad. Nauk SSSR, 144 (1962), 13-16. [21] K.-O. Widman, Hölder Continuity of Solutions of Elliptic Systems, Manuscripta Math., 5 (1971), 299-308.
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