François Delarue

This book describes the latest advances in the theory of mean field games, which are optimal control problems with a continuum of players, each of them interacting with the whole statistical distribution of a population. While it originated in economics, this theory now has applications in areas as diverse as mathematical finance, crowd phenomena, epidemiology, and cybersecurity. Because mean field games concern the interactions of infinitely many players in an optimal control framework, one expects them to appear as the limit for Nash equilibria of differential games with finitely many players as the number of players tends to infinity. This book rigorously establishes this convergence, which has been an open problem until now. The limit of the system associated with differential games with finitely many players is described by the so-called master equation, a nonlocal transport equation in the space of measures. After defining a suitable notion of differentiability in the space of measures, the authors provide a complete self-contained analysis of the master equation. Their analysis includes the case of common noise problems in which all the players are affected by a common Brownian motion. They then go on to explain how to use the master equation to prove the mean field limit. This groundbreaking book presents two important new results in mean field games that contribute to a unified theoretical framework for this exciting and fast-developing area of mathematics.

This book describes the latest advances in the theory of mean field games, which are optimal control problems with a continuum of players, each of them interacting with the whole statistical distribution of a population. While it originated in economics, this theory now has applications in areas as diverse as mathematical finance, crowd phenomena, epidemiology, and cybersecurity. Because mean field games concern the interactions of infinitely many players in an optimal control framework, one expects them to appear as the limit for Nash equilibria of differential games with finitely many players as the number of players tends to infinity. This book rigorously establishes this convergence, which has been an open problem until now. The limit of the system associated with differential games with finitely many players is described by the so-called master equation, a nonlocal transport equation in the space of measures. After defining a suitable notion of differentiability in the space of measures, the authors provide a complete self-contained analysis of the master equation. Their analysis includes the case of common noise problems in which all the players are affected by a common Brownian motion. They then go on to explain how to use the master equation to prove the mean field limit. This groundbreaking book presents two important new results in mean field games that contribute to a unified theoretical framework for this exciting and fast-developing area of mathematics.

De la Septennalite et de ses consequences, par F. Delarue, ... Date de l'edition originale: 1824Ce livre est la reproduction fidele d une uvre publiee avant 1920 et fait partie d une collection de livres reimprimes a la demande editee par Hachette Livre, dans le cadre d un partenariat avec la Bibliotheque nationale de France, offrant l opportunite d acceder a des ouvrages anciens et souvent rares issus des fonds patrimoniaux de la BnF. Les uvres faisant partie de cette collection ont ete numerisees par la BnF et sont presentes sur Gallica, sa bibliotheque numerique. En entreprenant de redonner vie a ces ouvrages au travers d une collection de livres reimprimes a la demande, nous leur donnons la possibilite de rencontrer un public elargi et participons a la transmission de connaissances et de savoirs parfois difficilement accessibles. Nous avons cherche a concilier la reproduction fidele d un livre ancien a partir de sa version numerisee avec le souci d un confort de lecture optimal. Nous esperons que les ouvrages de cette nouvelle collection vous apporteront entiere satisfaction. Pour plus d informations, rendez-vous sur www.hachettebnf.fr

Le Guide de la sante, ou Tableaux complets des maladies secretes... par F. Delarue. 3e editionDate de l'edition originale: 1833Ce livre est la reproduction fidele d'une oeuvre publiee avant 1920 et fait partie d'une collection de livres reimprimes a la demande editee par Hachette Livre, dans le cadre d'un partenariat avec la Bibliotheque nationale de France, offrant l'opportunite d'acceder a des ouvrages anciens et souvent rares issus des fonds patrimoniaux de la BnF. Les oeuvres faisant partie de cette collection ont ete numerisees par la BnF et sont presentes sur Gallica, sa bibliotheque numerique. En entreprenant de redonner vie a ces ouvrages au travers d'une collection de livres reimprimes a la demande, nous leur donnons la possibilite de rencontrer un public elargi et participons a la transmission de connaissances et de savoirs parfois difficilement accessibles. Nous avons cherche a concilier la reproduction fidele d'un livre ancien a partir de sa version numerisee avec le souci d'un confort de lecture optimal. Nous esperons que les ouvrages de cette nouvelle collection vous apporteront entiere satisfaction. Pour plus d'informations, rendez-vous sur www.hachettebnf.fr

Catalogue de fonds et d'assortiment de Francois Delarue, editeur d'estampes... Date de l'edition originale: 1858" Collection (Art). Francois Delarue. Paris. 1858]"Ce livre est la reproduction fidele d'une oeuvre publiee avant 1920 et fait partie d'une collection de livres reimprimes a la demande editee par Hachette Livre, dans le cadre d'un partenariat avec la Bibliotheque nationale de France, offrant l'opportunite d'acceder a des ouvrages anciens et souvent rares issus des fonds patrimoniaux de la BnF. Les oeuvres faisant partie de cette collection ont ete numerisees par la BnF et sont presentes sur Gallica, sa bibliotheque numerique. En entreprenant de redonner vie a ces ouvrages au travers d'une collection de livres reimprimes a la demande, nous leur donnons la possibilite de rencontrer un public elargi et participons a la transmission de connaissances et de savoirs parfois difficilement accessibles. Nous avons cherche a concilier la reproduction fidele d'un livre ancien a partir de sa version numerisee avec le souci d'un confort de lecture optimal. Nous esperons que les ouvrages de cette nouvelle collection vous apporteront entiere satisfaction. Pour plus d'informations, rendez-vous sur www.hachettebnf.fr

Featuring extensive, peer-reviewed research in pure and applied mathematics, this series publishes long papers and coordinated groups of related studies in book format. Overseen by the AMS Transactions Editorial Committee, the collection delves into advanced mathematical theories and methodologies.

This volume provides an introduction to the theory of Mean Field Games, suggested by J.-M. Lasry and P.-L. Lions in 2006 as a mean-field model for Nash equilibria in the strategic interaction of a large number of agents. Besides giving an accessible presentation of the main features of mean-field game theory, the volume offers an overview of recent developments which explore several important directions: from partial differential equations to stochastic analysis, from the calculus of variations to modeling and aspects related to numerical methods. Arising from the CIME Summer School "Mean Field Games" held in Cetraro in 2019, this book collects together lecture notes prepared by Y. Achdou (with M. Laurière), P. Cardaliaguet, F. Delarue, A. Porretta and F. Santambrogio. These notes will be valuable for researchers and advanced graduate students who wish to approach this theory and explore its connections with several different fields in mathematics.

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 resultsto 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.

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 II tackles the analysis of mean field games in which the players are affected by a common source of noise. The first part of the volume introduces and studies the concepts of weak and strong equilibria, and establishes general solvability results. The second part is devoted to the study of the master equation, a partial differential equation satisfied by the value function of the game over the space of probability measures. Existence of viscosity and classical solutions are proven and used to study asymptotics of games with finitely many players. 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.

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 II tackles the analysis of mean field games in which the players are affected by a common source of noise. The first part of the volume introduces and studies the concepts of weak and strong equilibria, and establishes general solvability results. The second part is devoted to the study of the master equation, a partial differential equation satisfied by the value function of the game over the space of probability measures. Existence of viscosity and classical solutions are proven and used to study asymptotics of games with finitely many players. 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.

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 resultsto 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.