scholarly journals Linear Quadratic Stochastic Differential Games: Open-Loop and Closed-Loop Saddle Points

2014 ◽  
Vol 52 (6) ◽  
pp. 4082-4121 ◽  
Author(s):  
Jingrui Sun ◽  
Jiongmin Yong
Keyword(s):  
Differential Games ◽  
Closed Loop ◽  
Saddle Points ◽  
Stochastic Differential Games ◽  
Open Loop ◽  
Linear Quadratic

scholarly journals Mean-field linear-quadratic stochastic differential games in an infinite horizon

2021 ◽  
Author(s):  
Xun Li ◽  
Jingtao Shi ◽  
Jiongmin Yong
Keyword(s):  
Nash Equilibrium ◽  
Differential Games ◽  
Closed Loop ◽  
Infinite Horizon ◽  
Mean Field ◽  
Riccati Equations ◽  
Stochastic Differential Games ◽  
Open Loop ◽  
Linear Quadratic ◽  
Algebraic Riccati Equations

This paper is concerned with two-person mean-field linear-quadratic non-zero sum stochastic differential games in an infinite horizon. Both open-loop and closed-loop Nash equilibria are introduced. Existence of an open-loop Nash equilibrium is characterized by the solvability of a system of mean-field forward-backward stochastic differential equations in an infinite horizon and the convexity of the cost functionals, and the closed-loop representation of an open-loop Nash equilibrium is given through the solution to a system of two coupled non-symmetric algebraic Riccati equations. The existence of a closed-loop Nash equilibrium is  characterized by the solvability of a system of two coupled symmetric algebraic Riccati equations. Two-person mean-field linear-quadratic zero-sum stochastic differential games in an infinite time horizon are also considered. Both the existence of open-loop and closed-loop saddle points are characterized by the solvability of a system of two coupled generalized algebraic Riccati equations with static stabilizing solutions. Mean-field linear-quadratic stochastic optimal control problems in an infinite horizon are discussed as well, for which it is proved that the open-loop solvability and closed-loop solvability are equivalent.

scholarly journals Mean-field backward–forward stochastic differential equations and nonzero sum stochastic differential games

2020 ◽  
pp. 2150036
Author(s):  
Yinggu Chen ◽  
Boualem Djehiche ◽  
Said Hamadène
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Differential Games ◽  
Mean Field ◽  
Stochastic Differential Games ◽  
Open Loop ◽  
Random Coefficients ◽  
Linear Quadratic ◽  
Uniqueness Results ◽  
Weak Monotonicity

We study a general class of fully coupled backward–forward stochastic differential equations of mean-field type (MF-BFSDE). We derive existence and uniqueness results for such a system under weak monotonicity assumptions and without the non-degeneracy condition on the forward equation. This is achieved by suggesting an implicit approximation scheme that is shown to converge to the solution of the system of MF-BFSDE. We apply these results to derive an explicit form of open-loop Nash equilibrium strategies for nonzero sum mean-field linear-quadratic stochastic differential games with random coefficients. These strategies are valid for any time horizon of the game.

On closed-loop equilibrium strategies for mean-field stochastic linear quadratic problems

2020 ◽  
Vol 26 ◽  
pp. 41
Author(s):  
Tianxiao Wang
Keyword(s):  
Optimal Control ◽  
Closed Loop ◽  
Optimal Control Problems ◽  
Mean Field ◽  
Open Loop ◽  
Time Inconsistency ◽  
Control Problems ◽  
Linear Quadratic ◽  
Linear Quadratic Optimal Control ◽  
Equilibrium Strategies

This article is concerned with linear quadratic optimal control problems of mean-field stochastic differential equations (MF-SDE) with deterministic coefficients. To treat the time inconsistency of the optimal control problems, linear closed-loop equilibrium strategies are introduced and characterized by variational approach. Our developed methodology drops the delicate convergence procedures in Yong [Trans. Amer. Math. Soc. 369 (2017) 5467–5523]. When the MF-SDE reduces to SDE, our Riccati system coincides with the analogue in Yong [Trans. Amer. Math. Soc. 369 (2017) 5467–5523]. However, these two systems are in general different from each other due to the conditional mean-field terms in the MF-SDE. Eventually, the comparisons with pre-committed optimal strategies, open-loop equilibrium strategies are given in details.

One-Player Differential Games

2017 ◽  
Author(s):  
João P. Hespanha
Keyword(s):  
Dynamical System ◽  
Differential Games ◽  
Continuous Time ◽  
Information Structure ◽  
Dynamic Games ◽  
State Feedback ◽  
Open Loop ◽  
Linear Quadratic ◽  
Time Dynamic ◽  
Termination Time

This chapter focuses on one-player continuous time dynamic games, that is, the optimal control of a continuous time dynamical system. It begins by considering a one-player continuous time differential game in which the (only) player wants to minimize either using an open-loop policy or a state-feedback policy. It then discusses continuous time cost-to-go, with the following conclusion: regardless of the information structure considered (open loop, state feedback, or other), it is not possible to obtain a cost lower than cost-to-go. It also explores continuous time dynamic programming, linear quadratic dynamic games, and differential games with variable termination time before concluding with a practice exercise and the corresponding solution.

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