دانلود کتاب Mean Field Games: Cetraro, Italy 2019 (Lecture Notes in Mathematics)

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Preface
Contents
1 An Introduction to Mean Field Game Theory
1.1 Introduction
1.2 Preliminaries
1.2.1 Optimal Control
1.2.1.1 Dynamic Programming and the Verification Theorem
1.2.1.2 Estimates on the SDE
1.2.2 The Space of Probability Measures
1.2.2.1 The Monge-Kantorovitch Distance
1.2.2.2 The d2 Distance
1.2.3 Mean Field Limits
1.2.3.1 The Glivenko-Cantelli Law of Large Numbers
1.2.3.2 The Well-Posedness of the McKean-Vlasov Equation
1.2.3.3 The Mean Field Limit
1.3 The Mean Field Game System
1.3.1 Heuristic Derivation of the MFG System
1.3.2 Second Order MFG System with Smoothing Couplings
1.3.3 Application to Games with Finitely Many Players
1.3.3.1 The N-Player Game
1.3.3.2 The MFG System and the N-Player Game
1.3.4 The Vanishing Viscosity Limit and the First Order System with Smoothing Couplings
1.3.5 Second Order MFG System with Local Couplings
1.3.5.1 Existence and Uniqueness of Solutions
1.3.6 The Long Time Ergodic Behavior and the Turnpike Property of Solutions
1.3.7 The Vanishing Viscosity Limit and the First Order System with Local Couplings
1.3.7.1 Existence and Uniqueness of Solutions
1.3.7.2 Variational Approach and Optimality Conditions
1.3.8 Further Comments, Related Topics and References
1.3.8.1 Boundary Conditions, Exit Time Problems, State Constraints, Planning Problem
1.3.8.2 Numerical Methods
1.3.8.3 MFG Systems with Several Populations
1.3.8.4 MFG of Control
1.3.8.5 MFG with Common Noise and with a Major Player
1.3.8.6 Miscellaneous
1.4 The Master Equation and the Convergence Problem
1.4.1 The Space of Probability Measures (Revisited)
1.4.1.1 The Monge-Kantorovich Distances
1.4.1.2 The Wasserstein Space of Probability Measures on Rd
1.4.2 Derivatives in the Space of Measures
1.4.2.1 The Flat Derivative
1.4.2.2 W-Differentiability
1.4.2.3 Link with the L-Derivative
1.4.2.4 Higher Order Derivatives
1.4.2.5 Comments
1.4.3 The Master Equation
1.4.3.1 Existence and Uniqueness of a Solution for the Master Equation
1.4.3.2 The Master Equation for MFG Problems on a Finite State Space
1.4.3.3 The MFG Problem with a Common Noise
1.4.3.4 Comments
1.4.4 Convergence of the Nash System
1.4.4.1 The Nash System
1.4.4.2 Finite Dimensional Projections of U
1.4.4.3 Convergence
1.4.4.4 Comments
Appendix: P.-L. Lions\' Courses on Mean Field Games at the Collège de France
Organization 2007–2008
Organization 2008–2009
Organization 2009–2010
Organization 2010–2011
Organization 2011–2012
Additional Notes
References
2 Lecture Notes on Variational Mean Field Games
2.1 Introduction and Modeling
2.1.1 A Coupled System of PDEs
2.1.2 Questions and Difficulties
2.2 Variational Formulation
2.2.1 Convex Duality
2.2.2 Lagrangian Formulation
2.2.3 Optimality Conditions on the Level of Single Agent Trajectories
2.3 Regularity via Duality
2.4 Regularity via OT, Time Discretization, and Flow Interchange
2.4.1 Tools from Optimal Transport and Wasserstein Spaces
2.4.2 Discretization in Time of Variational MFG and Optimality Conditions
2.5 Density-Constrained Mean Field Games
2.5.1 Optimality Conditions and Regularity of p
2.5.2 Optimality Conditions and Regularity of P
2.5.3 Approximation and Conclusions
References
3 Master Equation for Finite State Mean Field Games with Additive Common Noise
3.1 Introduction
3.1.1 Mean Field Games with a Common Noise
3.1.2 Master Equation
3.1.3 Finite State MFGs with a Common Noise
3.2 Formulation of the Finite State MFG with a Common Noise
3.2.1 Finite State MFG Without Common Noise
3.2.1.1 Optimal Control Problem
3.2.1.2 Definition of an MFG Equilibrium and Monotonicity Condition for Uniqueness
3.2.2 Common Noise
3.2.3 Assumption
3.2.3.1 Differentiability in μ
3.2.3.2 Detailed Regularity Assumptions
3.3 Stochastic MFG System
3.3.1 Stochastic HJB Equation
3.3.2 Formulation of the MFG System
3.3.3 Proof of the Solvability Result
3.4 Master Equation
3.4.1 Master Field
3.4.2 Representation of the Value Function Through the Master Field
3.4.3 Form of the Master Equation
3.4.3.1 Informal Derivation of the Equation
3.4.3.2 Solvability Result
3.5 Proof of the Smoothness of the Master Field
3.5.1 Linearized System
3.5.1.1 Form of the Linearized System
3.5.1.2 Stability Lemma
3.5.1.3 Existence and Uniqueness
3.5.2 Differentiability in p
3.5.3 Heat Kernel and Differentiability in x
References
4 Mean Field Games and Applications: Numerical Aspects
4.1 Introduction
4.2 Finite Difference Schemes
4.3 Variational Aspects of Some MFGs and Numerical Applications
4.3.1 An Optimal Control Problem Driven by a PDE
4.3.2 Discrete Version of the PDE Driven Optimal Control Problem
4.3.3 Recovery of the Finite Difference Scheme by Convex Duality
4.3.4 Alternating Direction Method of Multipliers
4.3.5 Chambolle and Pock\'s Algorithm
4.4 Multigrid Preconditioners
4.4.1 General Considerations on Multigrid Methods
4.4.2 Applications in the Context of Mean Field Games
4.4.3 Numerical Illustration
4.5 Algorithms for Solving the System of Non-Linear Equations
4.5.1 Combining Continuation Methods with Newton Iterations
4.5.2 A Recursive Algorithm Based on Elementary Solvers on Small Time Intervals
4.6 An Application to Pedestrian Flows
4.6.1 An Example with Two Populations, Congestion Effects and Various Boundary Conditions
4.6.1.1 The System of Partial Differential Equations
4.6.1.2 The Domain and the Boundary Conditions
4.6.2 Stationary Equilibria
4.6.3 A Stationary Equilibrium with ν=0.30
4.6.4 A Stationary Equilibrium with ν=0.16
4.6.5 Algorithm for Solving the System of Nonlinear Equations
4.7 Mean Field Type Control
4.7.1 Definition of the Problem
4.7.2 Numerical Simulations
4.8 MFGs in Macroeconomics
4.8.1 A Prototypical Model with Heterogeneous Agents: The Huggett Model
4.8.1.1 The Optimal Control Problem Solved by an Agent
4.8.1.2 The Ergodic Measure and the Coupling Condition
4.8.1.3 Summary
4.8.1.4 Theoretical Results
4.8.2 A Finite Difference Method for the Huggett Model
4.8.2.1 The numerical scheme
4.8.2.2 Numerical Simulations
4.8.3 The Model of Aiyagari
4.9 Conclusion
References