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January  2006, 2(1): 95-117. doi: 10.3934/jimo.2006.2.95

Two-person zero-sum linear quadratic stochastic differential games by a Hilbert space method

Libin Mou 1, and  Jiongmin Yong 2,

1. 

Department of Mathematics, Bradley University, Peoria, IL 616125, United States
2. 

Department of Mathematics, University of Central Florida, Orlando, FL 32816, United States

Received  September 2005 Revised  December 2005 Published  January 2006

An open-loop two-person zero-sum linear quadratic (LQ for short) stochastic differential game is considered. The controls for both players are allowed to appear in both the drift and diffusion of the state equation, the weighting matrices in the payoff/cost functional are not assumed to be definite/non-singular, and the cross-terms between two controls are allowed to appear. A forward-backward stochastic differential equation (FBSDE, for short) and a generalized differential Riccati equation are introduced, whose solvability leads to the existence of the open-loop saddle points for the corresponding two-person zero-sum LQ stochastic differential game, under some additional mild conditions. The main idea is a thorough study of general two-person zero-sum LQ games in Hilbert spaces.
Keywords: Hilbert method., open-loop controls, Stochastic games, linear-quadratic, saddle points.
Mathematics Subject Classification: 91A23, 91A15, 91A05, 49N70, 49N1.
Citation: Libin Mou, Jiongmin Yong. Two-person zero-sum linear quadratic stochastic differential games by a Hilbert space method. Journal of Industrial & Management Optimization, 2006, 2 (1) : 95-117. doi: 10.3934/jimo.2006.2.95
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