scholarly journals Backward-forward linear-quadratic mean-field Stackelberg games

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Kehan Si ◽  
Zhen Wu
Keyword(s):  
Large Population ◽  
Consistency Condition ◽  
Market Price ◽  
Mean Field ◽  
Mapping Method ◽  
Open Loop ◽  
Stackelberg Games ◽  
Linear Quadratic ◽  
Population System ◽  
The Cost

AbstractThis paper studies a controlled backward-forward linear-quadratic-Gaussian (LQG) large population system in Stackelberg games. The leader agent is of backward state and follower agents are of forward state. The leader agent is dominating as its state enters those of follower agents. On the other hand, the state-average of all follower agents affects the cost functional of the leader agent. In reality, the leader and the followers may represent two typical types of participants involved in market price formation: the supplier and producers. This differs from standard MFG literature and is mainly due to the Stackelberg structure here. By variational analysis, the consistency condition system can be represented by some fully-coupled backward-forward stochastic differential equations (BFSDEs) with high dimensional block structure in an open-loop sense. Next, we discuss the well-posedness of such a BFSDE system by virtue of the contraction mapping method. Consequently, we obtain the decentralized strategies for the leader and follower agents which are proved to satisfy the ε-Nash equilibrium property.

scholarly journals Linear quadratic stochastic optimal control problems with operator coefficients: open-loop solutions

2019 ◽  
Vol 25 ◽  
pp. 17 ◽  
Author(s):  
Qingmeng Wei ◽  
Jiongmin Yong ◽  
Zhiyong Yu
Keyword(s):  
Optimal Control ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Optimal Control Problems ◽  
Mean Field ◽  
Open Loop ◽  
Control Problems ◽  
Linear Quadratic ◽  
Cost Functional ◽  
The Cost

An optimal control problem is considered for linear stochastic differential equations with quadratic cost functional. The coefficients of the state equation and the weights in the cost functional are bounded operators on the spaces of square integrable random variables. The main motivation of our study is linear quadratic (LQ, for short) optimal control problems for mean-field stochastic differential equations. Open-loop solvability of the problem is characterized as the solvability of a system of linear coupled forward-backward stochastic differential equations (FBSDE, for short) with operator coefficients, together with a convexity condition for the cost functional. Under proper conditions, the well-posedness of such an FBSDE, which leads to the existence of an open-loop optimal control, is established. Finally, as applications of our main results, a general mean-field LQ control problem and a concrete mean-variance portfolio selection problem in the open-loop case are solved.

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.

scholarly journals Linear-Quadratic Stackelberg Game for Mean-Field Backward Stochastic Differential System and Application

2019 ◽  
Vol 2019 ◽  
pp. 1-17 ◽  
Author(s):  
Kai Du ◽  
Zhen Wu
Keyword(s):  
Optimal Control ◽  
Differential System ◽  
Optimization Problems ◽  
Stackelberg Game ◽  
Mean Field ◽  
Open Loop ◽  
Stackelberg Equilibrium ◽  
Linear Quadratic ◽  
Stackelberg Differential Game ◽  
Lq Optimal Control

This paper is concerned with a new kind of Stackelberg differential game of mean-field backward stochastic differential equations (MF-BSDEs). By means of four Riccati equations (REs), the follower first solves a backward mean-field stochastic LQ optimal control problem and gets the corresponding open-loop optimal control with the feedback representation. Then the leader turns to solve an optimization problem for a 1×2 mean-field forward-backward stochastic differential system. In virtue of some high-dimensional and complicated REs, we obtain the open-loop Stackelberg equilibrium, and it admits a state feedback representation. Finally, as applications, a class of stochastic pension fund optimization problems which can be viewed as a special case of our formulation is studied and the open-loop Stackelberg strategy is obtained.

scholarly journals Dynamics and control for multi-agent networked systems: A finite-difference approach

2019 ◽  
Vol 29 (04) ◽  
pp. 755-790 ◽  
Author(s):  
Umberto Biccari ◽  
Dongnam Ko ◽  
Enrique Zuazua
Keyword(s):  
Parabolic Equations ◽  
Transport Model ◽  
Large Population ◽  
Mean Field ◽  
Mean Field Limit ◽  
Dynamics And Control ◽  
Multi Agent ◽  
Nonlocal Transport ◽  
The Cost ◽  
And Control

We analyze the dynamics of multi-agent collective behavior models and its control theoretical properties. We first derive a large population limit to parabolic diffusive equations. We also show that the nonlocal transport equations commonly derived as the mean-field limit, are subordinated to the first one. In other words, the solution of the nonlocal transport model can be obtained by a suitable averaging of the diffusive one. We then address the control problem in the linear setting, linking the multi-agent model with the spatial semi-discretization of parabolic equations. This allows us to use the existing techniques for parabolic control problems in the present setting and derive explicit estimates on the cost of controlling these systems as the number of agents tends to infinity. We obtain precise estimates on the time of control and the size of the controls needed to drive the system to consensus, depending on the size of the population considered. Our approach, inspired on the existing results for parabolic equations, possibly of fractional type, and in several space dimensions, shows that the formation of consensus may be understood in terms of the underlying diffusion process described by the heat semi-group. In this way, we are able to give precise estimates on the cost of controllability for these systems as the number of agents increases, both in what concerns the needed control time horizon and the size of the controls.

One-Player Dynamic Games

2017 ◽  
Author(s):  
João P. Hespanha
Keyword(s):  
Dynamic Programming ◽  
Discrete Time ◽  
Information Structure ◽  
Dynamic Games ◽  
Open Loop ◽  
Linear Quadratic ◽  
Time Dynamic ◽  
Discrete Time Dynamical System ◽  
Solution Methods ◽  
The Cost

This chapter focuses on one-player discrete time dynamic games, that is, the optimal control of a discrete time dynamical system. It first considers solution methods for one-player dynamic games, which are simple optimizations, before discussing discrete time cost-to-go. It shows that, regardless of the information structure (open loop, state feedback or other), it is not possible to obtain a cost lower than the cost-to-go. A computationally efficient recursive technique that can be used to compute the cost-to-go is dynamic programming. After providing an overview of discrete time dynamic programming, the chapter explores the complexity of computing the cost-to-go at all stages, the use of MATLAB to solve finite one-player games, and linear quadratic dynamic games. It concludes with a practice exercise and the corresponding solution, along with an additional exercise.

scholarly journals Linear quadratic mean field game with control input constraint

2018 ◽  
Vol 24 (2) ◽  
pp. 901-919 ◽  
Author(s):  
Ying Hu ◽  
Jianhui Huang ◽  
Xun Li
Keyword(s):  
Backward Stochastic Differential Equation ◽  
Consistency Condition ◽  
Mean Field ◽  
Control Process ◽  
Mean Field Games ◽  
Monotonicity Condition ◽  
Projection Operators ◽  
Control Input ◽  
Linear Quadratic ◽  
The Individual

In this paper, we study a class of linear-quadratic (LQ) mean-field games in which the individual control process is constrained in a closed convex subset Γ of full space ℝm. The decentralized strategies and consistency condition are represented by a class of mean-field forward-backward stochastic differential equation (MF-FBSDE) with projection operators on Γ. The wellposedness of consistency condition system is obtained using the monotonicity condition method. The related ϵ-Nash equilibrium property is also verified.

Linear-Quadratic Mean Field Stackelberg Games with State and Control Delays

2017 ◽  
Vol 55 (4) ◽  
pp. 2748-2781 ◽  
Author(s):  
A. Bensoussan ◽  
M. H. M. Chau ◽  
Y. Lai ◽  
S. C. P. Yam
Keyword(s):  
Mean Field ◽  
Stackelberg Games ◽  
Linear Quadratic ◽  
Quadratic Mean ◽  
And Control ◽  
Control Delays

Linear quadratic control problems of stochastic Volterra integral equations

2018 ◽  
Vol 24 (4) ◽  
pp. 1849-1879 ◽  
Author(s):  
Tianxiao Wang
Keyword(s):  
Integral Equations ◽  
Stochastic Systems ◽  
Open Loop ◽  
Volterra Integral Equations ◽  
Control Problems ◽  
Optimal Controls ◽  
Linear Quadratic Control ◽  
Linear Quadratic ◽  
The Cost ◽  
Quadratic Control Problems

This paper is concerned with linear quadratic control problems of stochastic differential equations (SDEs, in short) and stochastic Volterra integral equations (SVIEs, in short). Notice that for stochastic systems, the control weight in the cost functional is allowed to be indefinite. This feature is demonstrated here only by open-loop optimal controls but not limited to closed-loop optimal controls in the literature. As to linear quadratic problem of SDEs, some examples are given to point out the issues left by existing papers, and new characterizations of optimal controls are obtained in different manners. For the study of SVIEs with deterministic coefficients, a class of stochastic Fredholm−Volterra integral equations is introduced to replace conventional forward-backward SVIEs. Eventually, instead of using convex variation, we use spike variation to obtain some additional optimality conditions of linear quadratic problems for SVIEs.

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