American Institute of Mathematical Sciences

Advanced
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 and Management Optimization, 2006, 2 (1) : 95-117. doi: 10.3934/jimo.2006.2.95
[1]

Alain Bensoussan, Shaokuan Chen, Suresh P. Sethi. Linear quadratic differential games with mixed leadership: The open-loop solution. Numerical Algebra, Control and Optimization, 2013, 3 (1) : 95-108. doi: 10.3934/naco.2013.3.95
[2]

Kehan Si, Zhenda Xu, Ka Fai Cedric Yiu, Xun Li. Open-loop solvability for mean-field stochastic linear quadratic optimal control problems of Markov regime-switching system. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021074
[3]

Tyrone E. Duncan. Some linear-quadratic stochastic differential games for equations in Hilbert spaces with fractional Brownian motions. Discrete and Continuous Dynamical Systems, 2015, 35 (11) : 5435-5445. doi: 10.3934/dcds.2015.35.5435
[4]

Dean A. Carlson. Finding open-loop Nash equilibrium for variational games. Conference Publications, 2005, 2005 (Special) : 153-163. doi: 10.3934/proc.2005.2005.153
[5]

Hanxiao Wang, Jingrui Sun, Jiongmin Yong. Weak closed-loop solvability of stochastic linear-quadratic optimal control problems. Discrete and Continuous Dynamical Systems, 2019, 39 (5) : 2785-2805. doi: 10.3934/dcds.2019117
[6]

Jingrui Sun, Hanxiao Wang. Mean-field stochastic linear-quadratic optimal control problems: Weak closed-loop solvability. Mathematical Control and Related Fields, 2021, 11 (1) : 47-71. doi: 10.3934/mcrf.2020026
[7]

Jun Moon. Linear-quadratic mean-field type stackelberg differential games for stochastic jump-diffusion systems. Mathematical Control and Related Fields, 2022, 12 (2) : 371-404. doi: 10.3934/mcrf.2021026
[8]

Ishak Alia. Open-loop equilibriums for a general class of time-inconsistent stochastic optimal control problems. Mathematical Control and Related Fields, 2021  doi: 10.3934/mcrf.2021053
[9]

Martino Bardi. Explicit solutions of some linear-quadratic mean field games. Networks and Heterogeneous Media, 2012, 7 (2) : 243-261. doi: 10.3934/nhm.2012.7.243
[10]

Michael Schönlein. Computation of open-loop inputs for uniformly ensemble controllable systems. Mathematical Control and Related Fields, 2021  doi: 10.3934/mcrf.2021046
[11]

Valery Y. Glizer, Oleg Kelis. Singular infinite horizon zero-sum linear-quadratic differential game: Saddle-point equilibrium sequence. Numerical Algebra, Control and Optimization, 2017, 7 (1) : 1-20. doi: 10.3934/naco.2017001
[12]

Mohamed Aliane, Mohand Bentobache, Nacima Moussouni, Philippe Marthon. Direct method to solve linear-quadratic optimal control problems. Numerical Algebra, Control and Optimization, 2021, 11 (4) : 645-663. doi: 10.3934/naco.2021002
[13]

Jianhui Huang, Shujun Wang, Zhen Wu. Backward-forward linear-quadratic mean-field games with major and minor agents. Probability, Uncertainty and Quantitative Risk, 2016, 1 (0) : 8-. doi: 10.1186/s41546-016-0009-9
[14]

René Carmona, Kenza Hamidouche, Mathieu Laurière, Zongjun Tan. Linear-quadratic zero-sum mean-field type games: Optimality conditions and policy optimization. Journal of Dynamics and Games, 2021, 8 (4) : 403-443. doi: 10.3934/jdg.2021023
[15]

Jiannan Zhang, Ping Chen, Zhuo Jin, Shuanming Li. Open-loop equilibrium strategy for mean-variance portfolio selection: A log-return model. Journal of Industrial and Management Optimization, 2021, 17 (2) : 765-777. doi: 10.3934/jimo.2019133
[16]

Robert S. Anderssen, Martin Kružík. Modelling of wheat-flour dough mixing as an open-loop hysteretic process. Discrete and Continuous Dynamical Systems - B, 2013, 18 (2) : 283-293. doi: 10.3934/dcdsb.2013.18.283
[17]

Liming Zhang, Rongming Wang, Jiaqin Wei. Open-loop equilibrium mean-variance reinsurance, new business and investment strategies with constraints. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021140
[18]

Jianhui Huang, Xun Li, Jiongmin Yong. A linear-quadratic optimal control problem for mean-field stochastic differential equations in infinite horizon. Mathematical Control and Related Fields, 2015, 5 (1) : 97-139. doi: 10.3934/mcrf.2015.5.97
[19]

Yadong Shu, Bo Li. Linear-quadratic optimal control for discrete-time stochastic descriptor systems. Journal of Industrial and Management Optimization, 2022, 18 (3) : 1583-1602. doi: 10.3934/jimo.2021034
[20]

Shigeaki Koike, Hiroaki Morimoto, Shigeru Sakaguchi. A linear-quadratic control problem with discretionary stopping. Discrete and Continuous Dynamical Systems - B, 2007, 8 (2) : 261-277. doi: 10.3934/dcdsb.2007.8.261

2020 Impact Factor: 1.801

Tools

Metrics

  • PDF downloads (159)
  • HTML views (0)
  • Cited by (16)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy