Comptes Rendus
no. 15-16
Probability Theory
Nash equilibrium point for one kind of stochastic nonzero-sum game problem and BSDEs
[Equilibre de Nash pour un jeu de somme non nulle et équations rétrogrades]

Présenté par : Marc Yor

Jean-Pierre Lepeltier1 ; Zhen Wu2 ; Zhiyong Yu3, 4
1 Département de Mathématiques, Université du Maine, avenue O. Messiaen, 72085 Le Mans cedex 9, France
2 School of Mathematics and System Science, Shandong University, Jinan 250100, China
3 School of Economics, Shandong University, Jinan 250100, China
4 Département de Mathématiques, Université d'Évry Val d'Essonne, 91025 Évry cedex, France
Comptes Rendus. Mathématique, Volume 347 (2009) no. 15-16, pp. 959-964.

Dans cette Note nous nous intéressons à un problème particulier de jeu différentiel stochastique de somme non nulle à N joueurs. En utilisant la théorie des équations différentielles stochastiques rétrogrades et le calcul de Malliavin nous donnons la forme explicite d'un équilibre de Nash.

In this Note, we deal with one kind of stochastic nonzero-sum differential game problem for N players. Using the theory of backward stochastic differential equations and Malliavin calculus, we give the explicit form of a Nash equilibrium point.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2009.04.033
Jean-Pierre Lepeltier 1 ; Zhen Wu 2 ; Zhiyong Yu 3, 4

1 Département de Mathématiques, Université du Maine, avenue O. Messiaen, 72085 Le Mans cedex 9, France
2 School of Mathematics and System Science, Shandong University, Jinan 250100, China
3 School of Economics, Shandong University, Jinan 250100, China
4 Département de Mathématiques, Université d'Évry Val d'Essonne, 91025 Évry cedex, France 
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Jean-Pierre Lepeltier; Zhen Wu; Zhiyong Yu. Nash equilibrium point for one kind of stochastic nonzero-sum game problem and BSDEs. Comptes Rendus. Mathématique, Volume 347 (2009) no. 15-16, pp. 959-964. doi : 10.1016/j.crma.2009.04.033. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.04.033/

[1] J.P. Aubin Mathematical Methods of Game and Economic Theory, Studies in Mathematics and Its Applications, North-Holland, Amsterdam, 1976

[2] A. Bensoussan; J. Frehse Stochastic games for N players, J. Optim. Theory Appl., Volume 105 (2000), pp. 543-565

[3] S. Hamadène; J.P. Lepeltier Zero-sum stochastic differential games and backward equations, Systems Control Lett., Volume 24 (1995), pp. 259-263

[4] S. Hamadène; J.P. Lepeltier; S. Peng BSDEs with continuous coefficients and stochastic differential games (El. Karoui; L. Mazliak, eds.), Pitman Research Notes in Mathematics Series, vol. 364, Langman, Harlow, 1997, pp. 115-128

[5] N. El Karoui; S. Peng; M.C. Quenez Backward stochastic differential equations in finance, Math. Finance, Volume 7 (1997) no. 1, pp. 1-71

[6] M. Kobylanski Backward stochastic differential equations and partial differential equations with quadratic growth, Ann. Probab., Volume 28 (2000), pp. 558-602

[7] J.P. Lepeltier; J. San Martin Existence for BSDE with superlinear-quadratic coefficient, Stochastics Stochastics Rep., Volume 63 (1997), pp. 227-240

[8] J. Nash Equilibrium points in n-person games, Proc. Natl. Acad. Sci. USA, Volume 36 (1950), pp. 48-49

[9] D. Nualart The Malliavin Calculus and Related Topics, Probability and Its Applications, Springer-Verlag, New York and Berlin, 1995

Cité par Sources :

☆ This work is supported by the Natural Science Foundation of China (10671112), the National Basic Research Program of China (973 Program, No. 2007CB814901 and No. 2007CB814904), the Natural Science Foundation of Shandong Province (JQ200801 and 2008BS01024) and the Doctoral Fund of Education Ministry of China.

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