Mean field games and mean field type control theory Alain Bensoussan, Jens Frehse, Phillip Yam

Format:

Book

Author/Creator:

Bensoussan, Alain, author.

Frehse, J. (Jens), author.

Yam, Phillip (Sheung Chi Phillip), author.

Series:

SpringerBriefs in mathematics 2191-8198

SpringerBriefs in Mathematics 2191-8198

Language:

English

Subjects (All):

Mean field theory.

Control theory.

Game theory.

Distribution (Probability theory).

Game Theory.

Mathematics.

Systems Theory, Control.

Probability Theory and Stochastic Processes.

Partial Differential Equations.

distribution (statistics-related concept).

Medical Subjects:

Game Theory.

Local Subjects:

Mathematics.

Systems Theory, Control.

Probability Theory and Stochastic Processes.

Partial Differential Equations.

Physical Description:

1 online resource

Place of Publication:

New York Springer 2013

Language Note:

English

System Details:

PDF

text file

Summary:

Mean field games and Mean field type control introduce new problems in Control Theory. The terminology games may be confusing. In fact they are control problems, in the sense that one is interested in a single decision maker, whom we can call the representative agent. However, these problems are not standard, since both the evolution of the state and the objective functional is influenced but terms which are not directly related to the state or the control of the decision maker. They are however, indirectly related to him, in the sense that they model a very large community of agents similar to the representative agent. All the agents behave similarly and impact the representative agent. However, because of the large number an aggregation effect takes place. The interesting consequence is that the impact of the community can be modeled by a mean field term, but when this is done, the problem is reduced to a control problem

Contents:

Introduction

General Presentation of Mean Field Control Problems

Discussion of the Mean Field game

Discussion of the Mean Field Type Control

Approximation of Nash Games with a large number of players

Linear Quadratic Models

Stationary Problems- Different Populations

Nash differential games with Mean Field effect

Machine generated contents note: 1. Introduction

2. General Presentation of Mean Field Control Problems

2.1. Model and Assumptions

2.2. Definition of the Problems

3. Mean Field Games

3.1. HJB-FP Approach

3.2. Stochastic Maximum Principle

4. Mean Field Type Control Problems

4.1. HJB-FP Approach

4.2. Other Approaches

4.3. Stochastic Maximum Principle

4.4. Time Inconsistency Approach

5. Approximation of Nash Games with a Large Number of Players

5.1. Preliminaries

5.2. System of PDEs

5.3. Independent Trajectories

5.4. General Case

5.5. Nash Equilibrium Among Local Feedbacks

6. Linear Quadratic Models

6.1. Setting of the Model

6.2. Solution of the Mean Field Game Problem

6.3. Solution of the Mean Field Type Problem

6.4. Mean Variance Problem

6.5. Approximate N Player Differential Game

7. Stationary Problems

7.1. Preliminaries

7.2. Mean Field Game Set-Up

7.3. Additional Interpretations

7.4. Approximate N Player Nash Equilibrium

8. Different Populations

8.1. General Considerations

8.2. Multiclass Agents

8.3. Major Player

8.3.1. General Theory

8.3.2. Linear Quadratic Case

9. Nash Differential Games with Mean Field Effect

9.1. Description of the Problem

9.2. Mathematical Problem

9.3. Interpretation

9.4. Another Interpretation

9.5. Generalization

9.6. Approximate Nash Equilibrium for Large Communities

10. Analytic Techniques

10.1. General Set-Up

10.1.1. Assumptions

10.1.2. Weak Formulation

10.2. Priori Estimates for u

10.2.1. L[∞] Estimate for u

10.2.2. L2(W1,2)) Estimate for u

10.2.3. Cα Estimate for u

10.2.4. Lp(W2,P) Estimate for u

10.3. Priori Estimates for m

10.3.1. L2(W1,2) Estimate

10.3.2. L[∞](L[∞]) Estimates

10.3.3. Further Estimates

10.3.4. Statement of the Global A Priori Estimate Result

10.4. Existence Result

Notes:

Includes bibliographical references and index

Online resource; title from PDF title page (SpringerLink, viewed October 21, 2013)

Other Format:

Printed edition:

ISBN:

9781461485087

1461485088

146148507X

9781461485070

OCLC:

862941167

Access Restriction:

Restricted for use by site license