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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

A. Bensoussan, J. Frehse and C. Grün, On a system of PDEs associated to a game with a varying number of players,, 2013, ().   Google Scholar

[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.  Google Scholar

[6]

P. Bremaud, Point Processes and Queues: Martingale Dynamics, Springer Series in Statistics. Springer-Verlag, New York-Berlin, 1981.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[10]

A. Friedman, Stochastic differential games, J. Differential Equations, 11 (1972), 79-108.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, Kodansha scientific books, North-Holland, 1989.  Google Scholar

[14]

R. Isaacs, Differential Games. A Mathematical Theory with Applications to Warfare and Pursuit, Control and Optimization, John Wiley & Sons Inc., New York, 1965.  Google Scholar

[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.  Google Scholar

[16]

N. V. Krylov, Controlled Diffusion Processes, Springer-Verlag, Berlin, 2009.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

M. I. Višik, Quasilinear elliptic systems of equations containing subordinate terms, Dokl. Akad. Nauk SSSR, 144 (1962), 13-16.  Google Scholar

[21]

K.-O. Widman, Hölder Continuity of Solutions of Elliptic Systems, Manuscripta Math., 5 (1971), 299-308.  Google Scholar

show all references

References:
[1]

A. Bensoussan and J. Frehse, Nonlinear elliptic systems in stochastic game theory, J. Reine Angew. Math., 350 (1984), 23-67.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

A. Bensoussan, J. Frehse and C. Grün, On a system of PDEs associated to a game with a varying number of players,, 2013, ().   Google Scholar

[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.  Google Scholar

[6]

P. Bremaud, Point Processes and Queues: Martingale Dynamics, Springer Series in Statistics. Springer-Verlag, New York-Berlin, 1981.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[10]

A. Friedman, Stochastic differential games, J. Differential Equations, 11 (1972), 79-108.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, Kodansha scientific books, North-Holland, 1989.  Google Scholar

[14]

R. Isaacs, Differential Games. A Mathematical Theory with Applications to Warfare and Pursuit, Control and Optimization, John Wiley & Sons Inc., New York, 1965.  Google Scholar

[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.  Google Scholar

[16]

N. V. Krylov, Controlled Diffusion Processes, Springer-Verlag, Berlin, 2009.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

M. I. Višik, Quasilinear elliptic systems of equations containing subordinate terms, Dokl. Akad. Nauk SSSR, 144 (1962), 13-16.  Google Scholar

[21]

K.-O. Widman, Hölder Continuity of Solutions of Elliptic Systems, Manuscripta Math., 5 (1971), 299-308.  Google Scholar

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