Probabilistic Theory of Mean Field Games with Applications I: Mean Field FBSDEs, Control, and Games

Description

This two-volume book offers a comprehensive treatment of the probabilistic approach to mean field game models and their applications. The book is self-contained in nature and includes original material and applications with explicit examples throughout, including numerical solutions. Volume I of the book is entirely devoted to the theory of mean field games without a common noise. The first half of the volume provides a self-contained introduction to mean field games, starting from concrete illustrations of games with a finite number of players, and ending with ready-for-use solvability results. Readers are provided with the tools necessary for the solution of forward-backward stochastic differential equations of the McKean-Vlasov type at the core of the probabilistic approach. The second half of this volume focuses on the main principles of analysis on the Wasserstein space. It includes Lions' approach to the Wasserstein differential calculus, and the applications of its results to the analysis of stochastic mean field control problems. Together, both Volume I and Volume II will greatly benefit mathematical graduate students and researchers interested in mean field games. The authors provide a detailed road map through the book allowing different access points for different readers and building up the level of technical detail. The accessible approach and overview will allow interested researchers in the applied sciences to obtain a clear overview of the state of the art in mean field games. Front Matter ....Pages i-xxv Front Matter ....Pages 1-1 Learning by Examples: What Is a Mean Field Game? (René Carmona, François Delarue)....Pages 3-65 Probabilistic Approach to Stochastic Differential Games (René Carmona, François Delarue)....Pages 67-127 Stochastic Differential Mean Field Games (René Carmona, François Delarue)....Pages 129-213 FBSDEs and the Solution of MFGs Without Common Noise (René Carmona, François Delarue)....Pages 215-345 Front Matter ....Pages 347-347 Spaces of Measures and Related Differential Calculus (René Carmona, François Delarue)....Pages 349-512 Optimal Control of SDEs of McKean-Vlasov Type (René Carmona, François Delarue)....Pages 513-617 Extensions for Volume I (René Carmona, François Delarue)....Pages 619-680 Back Matter ....Pages 681-713