Geodesic
$Z$-equilibria in many-player stochastic differential games
Archivum mathematicum, Tome 29 (1993) no. 3-4, pp. 123-133 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

In this paper $N$-person nonzero-sum games are considered. The dynamics is described by Ito stochastic differential equations. The cost-functions are conditional expectations of functionals of Bolza type with respect to the initial situation. The notion of $Z$-equilibrium is introduced in many-player stochastic differential games. Some properties of $Z$-equilibria are analyzed. Sufficient conditions are established guaranteeing the $Z$-equilibrium for the strategies of the players. In a particular case of a linear-quadratic game the $Z$-equilibrium strategies are found in an explicit form.
In this paper $N$-person nonzero-sum games are considered. The dynamics is described by Ito stochastic differential equations. The cost-functions are conditional expectations of functionals of Bolza type with respect to the initial situation. The notion of $Z$-equilibrium is introduced in many-player stochastic differential games. Some properties of $Z$-equilibria are analyzed. Sufficient conditions are established guaranteeing the $Z$-equilibrium for the strategies of the players. In a particular case of a linear-quadratic game the $Z$-equilibrium strategies are found in an explicit form.
Classification : 90D15, 91A10, 91A60, 93E05
Keywords: nonzero-sum game; many-player game; stochastic differential equation; linear-quadratic game; Bolza functional; cost-function; strategy 
@article{ARM_1993_29_3-4_a0,
     author = {Gaidov, Svatoslav},
     title = {$Z$-equilibria in many-player stochastic differential games},
     journal = {Archivum mathematicum},
     pages = {123--133},
     year = {1993},
     volume = {29},
     number = {3-4},
     mrnumber = {1263113},
     zbl = {0816.93077},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ARM_1993_29_3-4_a0/}
}
TY  - JOUR
AU  - Gaidov, Svatoslav
TI  - $Z$-equilibria in many-player stochastic differential games
JO  - Archivum mathematicum
PY  - 1993
SP  - 123
EP  - 133
VL  - 29
IS  - 3-4
UR  - http://geodesic.mathdoc.fr/item/ARM_1993_29_3-4_a0/
LA  - en
ID  - ARM_1993_29_3-4_a0
ER  - 
%0 Journal Article
%A Gaidov, Svatoslav
%T $Z$-equilibria in many-player stochastic differential games
%J Archivum mathematicum
%D 1993
%P 123-133
%V 29
%N 3-4
%U http://geodesic.mathdoc.fr/item/ARM_1993_29_3-4_a0/
%G en
%F ARM_1993_29_3-4_a0
Gaidov, Svatoslav. $Z$-equilibria in many-player stochastic differential games. Archivum mathematicum, Tome 29 (1993) no. 3-4, pp. 123-133. http://geodesic.mathdoc.fr/item/ARM_1993_29_3-4_a0/

[1] Fleming, W. H., Rishel, R. W.: Deterministic and Stochastic Optimal Control. Springer-Verlag, Berlin – Heidelberg – New York, 1975. | MR

[2] Gaidov, S. D.: $Z$-Equilibrium in Stochastic Differential Games. (in Russian), In: Many Player Differential Games, Centre of Mathematics, Technical University, Rouse, Bulgaria (1984), 53-63.

[3] Gaidov, S. D.: Basic Optimal Strategies in Stochastic Differential Games. C. R. Acad. Bulgare Sci. 37 (1984), 457-460. | MR | Zbl

[4] Gaidov, S. D.: Pareto-Optimality in Stochastic Differential Games. Problems of Control and Information Theory 15 (1986), 439-450. | MR | Zbl

[5] Gaidov, S. D.: Nash Equilibrium in Stochastic Differential Games. Computers and Mathematics with Applications 12A (1986), 761-768. | MR | Zbl

[6] Gaidov, S. D.: Guaranteeing Strategies in Many Player Stochastic Differential Games. In: Mathematics and Educatin in Mathematics, Proceedings of 15th Spring Conference of UBM (1986), 379-383. | MR

[7] Roitenberg, Y. N.: Automatic Control. (in Russian), Nauka, Moscow, 1978. | MR

[8] Vaisbord, I. M., Zhukovskii, V. I.: Introduction into Many Player Differential Games. (in Russian), Sovetskoe Radio (1980), Moscow.