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,, \emph{J. Reine Angew. Math.}, 350 (1984), 23. Google Scholar

[2]

Alain Bensoussan and Jens Frehse, Diagonal elliptic Bellman systems to stochastic differential games with discount control and noncompact coupling,, \emph{Rend. Mat. Appl.}, 29 (2009), 1. Google Scholar

[3]

A. Bensoussan and J. Frehse, Control and Nash games with mean field effect,, \emph{Chin. Ann. Math. Ser. B}, 34 (2013), 161. 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,, \emph{Trans. Amer. Math. Soc.}, 231 (1977), 275. Google Scholar

[6]

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

[7]

R. Buckdahn, P. Cardaliaguet and C. Rainer, Nash equilibrium payoffs for nonzero-sum stochastic differential games,, \emph{SIAM J. Control Optim.}, 43 (2004), 624. doi: 10.1137/S0363012902411556. Google Scholar

[8]

S. Campanato, Regolarizzazione negli spazi $L^{(2,\lambda )}$ delle soluzioni delle equazioni ellittiche del II ordine,, \emph{BOOK Atti del Convegno su le Equazioni alle Derivate Parziali (Nervi, (1965), 33. Google Scholar

[9]

J. Frehse, Bellman systems of stochastic differential games with three players,, in \emph{Optimal Control and Partial Differential Equations. In Honour of Professor Alain Bensoussan's 60th Birthday. Proceedings of the Conference, (2000), 3. Google Scholar

[10]

A. Friedman, Stochastic differential games,, \emph{J. Differential Equations}, 11 (1972), 79. Google Scholar

[11]

S. Hamadene, J.-P. Lepeltier and S. Peng, BSDEs with continuous coefficients and stochastic differential games,, \emph{Pitman Res. Notes Math. Ser.}, 364 (1997), 115. Google Scholar

[12]

S. Hamadène and J. Zhang, The continuous time nonzero-sum Dynkin game problem and application in game options,, \emph{SIAM J. Control Optim.}, 48 (2009), 3659. doi: 10.1137/080738933. Google Scholar

[13]

N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes,, Kodansha scientific books, (1989). Google Scholar

[14]

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

[15]

D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications,, Classics in Applied Mathematics, (2000). doi: 10.1137/1.9780898719451. Google Scholar

[16]

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

[17]

O. A. Ladyzhenskaya and N. N. Uraltseva, Linear and Quasilinear Elliptic Equations,, Translated from the Russian by Scripta Technica, (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,, \emph{Bulletin de la Soc. Math. France}, 93 (1965), 97. Google Scholar

[19]

Jr. Morrey and B. Charles, Multiple Integrals in the Calculus of Variations,, Springer-Verlag, (2008). doi: 10.1007/978-3-540-69952-1. Google Scholar

[20]

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

[21]

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

show all references

References:
[1]

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

[2]

Alain Bensoussan and Jens Frehse, Diagonal elliptic Bellman systems to stochastic differential games with discount control and noncompact coupling,, \emph{Rend. Mat. Appl.}, 29 (2009), 1. Google Scholar

[3]

A. Bensoussan and J. Frehse, Control and Nash games with mean field effect,, \emph{Chin. Ann. Math. Ser. B}, 34 (2013), 161. 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,, \emph{Trans. Amer. Math. Soc.}, 231 (1977), 275. Google Scholar

[6]

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

[7]

R. Buckdahn, P. Cardaliaguet and C. Rainer, Nash equilibrium payoffs for nonzero-sum stochastic differential games,, \emph{SIAM J. Control Optim.}, 43 (2004), 624. doi: 10.1137/S0363012902411556. Google Scholar

[8]

S. Campanato, Regolarizzazione negli spazi $L^{(2,\lambda )}$ delle soluzioni delle equazioni ellittiche del II ordine,, \emph{BOOK Atti del Convegno su le Equazioni alle Derivate Parziali (Nervi, (1965), 33. Google Scholar

[9]

J. Frehse, Bellman systems of stochastic differential games with three players,, in \emph{Optimal Control and Partial Differential Equations. In Honour of Professor Alain Bensoussan's 60th Birthday. Proceedings of the Conference, (2000), 3. Google Scholar

[10]

A. Friedman, Stochastic differential games,, \emph{J. Differential Equations}, 11 (1972), 79. Google Scholar

[11]

S. Hamadene, J.-P. Lepeltier and S. Peng, BSDEs with continuous coefficients and stochastic differential games,, \emph{Pitman Res. Notes Math. Ser.}, 364 (1997), 115. Google Scholar

[12]

S. Hamadène and J. Zhang, The continuous time nonzero-sum Dynkin game problem and application in game options,, \emph{SIAM J. Control Optim.}, 48 (2009), 3659. doi: 10.1137/080738933. Google Scholar

[13]

N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes,, Kodansha scientific books, (1989). Google Scholar

[14]

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

[15]

D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications,, Classics in Applied Mathematics, (2000). doi: 10.1137/1.9780898719451. Google Scholar

[16]

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

[17]

O. A. Ladyzhenskaya and N. N. Uraltseva, Linear and Quasilinear Elliptic Equations,, Translated from the Russian by Scripta Technica, (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,, \emph{Bulletin de la Soc. Math. France}, 93 (1965), 97. Google Scholar

[19]

Jr. Morrey and B. Charles, Multiple Integrals in the Calculus of Variations,, Springer-Verlag, (2008). doi: 10.1007/978-3-540-69952-1. Google Scholar

[20]

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

[21]

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

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