The Master Equation And The Convergence Problem In Mean Field Games

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The Master Equation and the Convergence Problem in Mean Field Games

Pierre Cardaliaguet,François Delarue,Jean-Michel Lasry,Pierre-Louis Lions
The Master Equation and the Convergence Problem in Mean Field Games Book PDF

Author : Pierre Cardaliaguet,François Delarue,Jean-Michel Lasry,Pierre-Louis Lions
Publisher : Princeton University Press
Page : 224 pages
File Size : 51,9 Mb
Release : 2019-08-13
Category : Mathematics
ISBN : 9780691190716

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The Master Equation and the Convergence Problem in Mean Field Games by Pierre Cardaliaguet,François Delarue,Jean-Michel Lasry,Pierre-Louis Lions Pdf

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.

The Master Equation and the Convergence Problem in Mean Field Games

Pierre Cardaliaguet,François Delarue,Jean-Michel Lasry,Pierre-Louis Lions
The Master Equation and the Convergence Problem in Mean Field Games Book PDF

Author : Pierre Cardaliaguet,François Delarue,Jean-Michel Lasry,Pierre-Louis Lions
Publisher : Princeton University Press
Page : 224 pages
File Size : 41,7 Mb
Release : 2019-08-13
Category : Mathematics
ISBN : 9780691193717

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The Master Equation and the Convergence Problem in Mean Field Games by Pierre Cardaliaguet,François Delarue,Jean-Michel Lasry,Pierre-Louis Lions Pdf

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.

Probabilistic Theory of Mean Field Games with Applications II

René Carmona,François Delarue
Probabilistic Theory of Mean Field Games with Applications II Book PDF

Author : René Carmona,François Delarue
Publisher : Springer
Page : 700 pages
File Size : 49,7 Mb
Release : 2018-03-08
Category : Mathematics
ISBN : 9783319564364

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Probabilistic Theory of Mean Field Games with Applications II by René Carmona,François Delarue Pdf

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.

Mean Field Games

Yves Achdou,Pierre Cardaliaguet,François Delarue,Alessio Porretta,Filippo Santambrogio
Mean Field Games Book PDF

Author : Yves Achdou,Pierre Cardaliaguet,François Delarue,Alessio Porretta,Filippo Santambrogio
Publisher : Springer Nature
Page : 316 pages
File Size : 45,9 Mb
Release : 2021-01-19
Category : Mathematics
ISBN : 9783030598372

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Mean Field Games by Yves Achdou,Pierre Cardaliaguet,François Delarue,Alessio Porretta,Filippo Santambrogio Pdf

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.

Mean Field Games

François Delarue
Mean Field Games Book PDF

Author : François Delarue
Publisher : American Mathematical Society
Page : 284 pages
File Size : 44,9 Mb
Release : 2021-12-14
Category : Mathematics
ISBN : 9781470455866

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Mean Field Games by François Delarue Pdf

This volume is based on lectures delivered at the 2020 AMS Short Course “Mean Field Games: Agent Based Models to Nash Equilibria,” held January 13–14, 2020, in Denver, Colorado. Mean field game theory offers a robust methodology for studying large systems of interacting rational agents. It has been extraordinarily successful and has continued to develop since its inception. The six chapters that make up this volume provide an overview of the subject, from the foundations of the theory to applications in economics and finance, including computational aspects. The reader will find a pedagogical introduction to the main ingredients, from the forward-backward mean field game system to the master equation. Also included are two detailed chapters on the connection between finite games and mean field games, with a pedestrian description of the different methods available to solve the convergence problem. The volume concludes with two contributions on applications of mean field games and on existing numerical methods, with an opening to machine learning techniques.

A Probabilistic Approach to Classical Solutions of the Master Equation for Large Population Equilibria

Jean-François Chassagneux,Dan Crisan,François Delarue
A Probabilistic Approach to Classical Solutions of the Master Equation for Large Population Equilibria Book PDF

Author : Jean-François Chassagneux,Dan Crisan,François Delarue
Publisher : American Mathematical Society
Page : 136 pages
File Size : 41,9 Mb
Release : 2022-11-10
Category : Mathematics
ISBN : 9781470453756

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A Probabilistic Approach to Classical Solutions of the Master Equation for Large Population Equilibria by Jean-François Chassagneux,Dan Crisan,François Delarue Pdf

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Probabilistic Theory of Mean Field Games with Applications I

René Carmona,François Delarue
Probabilistic Theory of Mean Field Games with Applications I Book PDF

Author : René Carmona,François Delarue
Publisher : Springer
Page : 714 pages
File Size : 46,5 Mb
Release : 2018-03-01
Category : Mathematics
ISBN : 9783319589206

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Probabilistic Theory of Mean Field Games with Applications I by René Carmona,François Delarue Pdf

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.

Stochastic Analysis, Filtering, and Stochastic Optimization

George Yin,Thaleia Zariphopoulou
Stochastic Analysis Filtering and Stochastic Optimization Book PDF

Author : George Yin,Thaleia Zariphopoulou
Publisher : Springer Nature
Page : 466 pages
File Size : 46,8 Mb
Release : 2022-04-22
Category : Mathematics
ISBN : 9783030985196

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Stochastic Analysis, Filtering, and Stochastic Optimization by George Yin,Thaleia Zariphopoulou Pdf

This volume is a collection of research works to honor the late Professor Mark H.A. Davis, whose pioneering work in the areas of Stochastic Processes, Filtering, and Stochastic Optimization spans more than five decades. Invited authors include his dissertation advisor, past collaborators, colleagues, mentees, and graduate students of Professor Davis, as well as scholars who have worked in the above areas. Their contributions may expand upon topics in piecewise deterministic processes, pathwise stochastic calculus, martingale methods in stochastic optimization, filtering, mean-field games, time-inconsistency, as well as impulse, singular, risk-sensitive and robust stochastic control.

Contemporary Research in Elliptic PDEs and Related Topics

Serena Dipierro
Contemporary Research in Elliptic PDEs and Related Topics Book PDF

Author : Serena Dipierro
Publisher : Springer
Page : 502 pages
File Size : 44,7 Mb
Release : 2019-07-12
Category : Mathematics
ISBN : 9783030189211

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Contemporary Research in Elliptic PDEs and Related Topics by Serena Dipierro Pdf

This volume collects contributions from the speakers at an INdAM Intensive period held at the University of Bari in 2017. The contributions cover several aspects of partial differential equations whose development in recent years has experienced major breakthroughs in terms of both theory and applications. The topics covered include nonlocal equations, elliptic equations and systems, fully nonlinear equations, nonlinear parabolic equations, overdetermined boundary value problems, maximum principles, geometric analysis, control theory, mean field games, and bio-mathematics. The authors are trailblazers in these topics and present their work in a way that is exhaustive and clearly accessible to PhD students and early career researcher. As such, the book offers an excellent introduction to a variety of fundamental topics of contemporary investigation and inspires novel and high-quality research.

Stochastic Dynamics Out of Equilibrium

Giambattista Giacomin,Stefano Olla,Ellen Saada,Herbert Spohn,Gabriel Stoltz
Stochastic Dynamics Out of Equilibrium Book PDF

Author : Giambattista Giacomin,Stefano Olla,Ellen Saada,Herbert Spohn,Gabriel Stoltz
Publisher : Springer
Page : 649 pages
File Size : 40,5 Mb
Release : 2019-06-30
Category : Mathematics
ISBN : 9783030150969

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Stochastic Dynamics Out of Equilibrium by Giambattista Giacomin,Stefano Olla,Ellen Saada,Herbert Spohn,Gabriel Stoltz Pdf

Stemming from the IHP trimester "Stochastic Dynamics Out of Equilibrium", this collection of contributions focuses on aspects of nonequilibrium dynamics and its ongoing developments. It is common practice in statistical mechanics to use models of large interacting assemblies governed by stochastic dynamics. In this context "equilibrium" is understood as stochastically (time) reversible dynamics with respect to a prescribed Gibbs measure. Nonequilibrium dynamics correspond on the other hand to irreversible evolutions, where fluxes appear in physical systems, and steady-state measures are unknown. The trimester, held at the Institut Henri Poincaré (IHP) in Paris from April to July 2017, comprised various events relating to three domains (i) transport in non-equilibrium statistical mechanics; (ii) the design of more efficient simulation methods; (iii) life sciences. It brought together physicists, mathematicians from many domains, computer scientists, as well as researchers working at the interface between biology, physics and mathematics. The present volume is indispensable reading for researchers and Ph.D. students working in such areas.

Traffic and Granular Flow '22

K. Ramachandra Rao (Associate professor in civil engineering),Armin Seyfried,Andreas Schadschneider
Traffic and Granular Flow 22 Book PDF

Author : K. Ramachandra Rao (Associate professor in civil engineering),Armin Seyfried,Andreas Schadschneider
Publisher : Springer Nature
Page : 518 pages
File Size : 53,6 Mb
Release : 2024
Category : Granular flow
ISBN : 9789819979769

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Traffic and Granular Flow '22 by K. Ramachandra Rao (Associate professor in civil engineering),Armin Seyfried,Andreas Schadschneider Pdf

This book gathers contributions on a variety of flowing collective systems. While primarily focusing on pedestrian dynamics, it also reflects the latest developments in areas such as vehicular traffic and granular flows and addresses related emerging topics such as self-propelled particles, data transport, swarm behaviour, intercellular transport, and individual interactions to complex systems. Combining fundamental research and practical applications in the various fields discussed, the book offers a valuable asset for researchers and professionals in areas such as civil and transportation engineering, mechanical engineering, electrical engineering, physics, computer science, and mathematics.

Many Agent Games in Socio-economic Systems: Corruption, Inspection, Coalition Building, Network Growth, Security

Vassili N. Kolokoltsov,Oleg A. Malafeyev
Many Agent Games in Socio economic Systems Corruption Inspection Coalition Building Network Growth Security Book PDF

Author : Vassili N. Kolokoltsov,Oleg A. Malafeyev
Publisher : Springer
Page : 196 pages
File Size : 41,8 Mb
Release : 2019-03-30
Category : Mathematics
ISBN : 9783030123710

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Many Agent Games in Socio-economic Systems: Corruption, Inspection, Coalition Building, Network Growth, Security by Vassili N. Kolokoltsov,Oleg A. Malafeyev Pdf

There has been an increase in attention toward systems involving large numbers of small players, giving rise to the theory of mean field games, mean field type control and nonlinear Markov games. Exhibiting various real world problems involving major and minor agents, this book presents a systematic continuous-space approximation approach for mean-field interacting agents models and mean-field games models. After describing Markov-chain methodology and a modeling of mean-field interacting systems, the text presents various structural conditions on the chain to yield respective socio-economic models, focusing on migration models via binary interactions. The specific applications are wide-ranging – including inspection and corruption, cyber-security, counterterrorism, coalition building and network growth, minority games, and investment policies and optimal allocation – making this book relevant to a wide audience of applied mathematicians interested in operations research, computer science, national security, economics, and finance.

Stochastic Optimal Control in Infinite Dimension

Giorgio Fabbri,Fausto Gozzi,Andrzej Święch
Stochastic Optimal Control in Infinite Dimension Book PDF

Author : Giorgio Fabbri,Fausto Gozzi,Andrzej Święch
Publisher : Springer
Page : 916 pages
File Size : 55,7 Mb
Release : 2017-06-22
Category : Mathematics
ISBN : 9783319530673

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Stochastic Optimal Control in Infinite Dimension by Giorgio Fabbri,Fausto Gozzi,Andrzej Święch Pdf

Providing an introduction to stochastic optimal control in infinite dimension, this book gives a complete account of the theory of second-order HJB equations in infinite-dimensional Hilbert spaces, focusing on its applicability to associated stochastic optimal control problems. It features a general introduction to optimal stochastic control, including basic results (e.g. the dynamic programming principle) with proofs, and provides examples of applications. A complete and up-to-date exposition of the existing theory of viscosity solutions and regular solutions of second-order HJB equations in Hilbert spaces is given, together with an extensive survey of other methods, with a full bibliography. In particular, Chapter 6, written by M. Fuhrman and G. Tessitore, surveys the theory of regular solutions of HJB equations arising in infinite-dimensional stochastic control, via BSDEs. The book is of interest to both pure and applied researchers working in the control theory of stochastic PDEs, and in PDEs in infinite dimension. Readers from other fields who want to learn the basic theory will also find it useful. The prerequisites are: standard functional analysis, the theory of semigroups of operators and its use in the study of PDEs, some knowledge of the dynamic programming approach to stochastic optimal control problems in finite dimension, and the basics of stochastic analysis and stochastic equations in infinite-dimensional spaces.

From Particle Systems to Partial Differential Equations

Cédric Bernardin,François Golse,Patrícia Gonçalves,Valeria Ricci,Ana Jacinta Soares
From Particle Systems to Partial Differential Equations Book PDF

Author : Cédric Bernardin,François Golse,Patrícia Gonçalves,Valeria Ricci,Ana Jacinta Soares
Publisher : Springer Nature
Page : 400 pages
File Size : 53,7 Mb
Release : 2021-05-30
Category : Mathematics
ISBN : 9783030697846

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From Particle Systems to Partial Differential Equations by Cédric Bernardin,François Golse,Patrícia Gonçalves,Valeria Ricci,Ana Jacinta Soares Pdf

This book includes the joint proceedings of the International Conference on Particle Systems and PDEs VI, VII and VIII. Particle Systems and PDEs VI was held in Nice, France, in November/December 2017, Particle Systems and PDEs VII was held in Palermo, Italy, in November 2018, and Particle Systems and PDEs VIII was held in Lisbon, Portugal, in December 2019. Most of the papers are dealing with mathematical problems motivated by different applications in physics, engineering, economics, chemistry and biology. They illustrate methods and topics in the study of particle systems and PDEs and their relation. The book is recommended to probabilists, analysts and to those mathematicians in general, whose work focuses on topics in mathematical physics, stochastic processes and differential equations, as well as to those physicists who work in statistical mechanics and kinetic theory.

Trends in Control Theory and Partial Differential Equations

Fatiha Alabau-Boussouira,Fabio Ancona,Alessio Porretta,Carlo Sinestrari
Trends in Control Theory and Partial Differential Equations Book PDF

Author : Fatiha Alabau-Boussouira,Fabio Ancona,Alessio Porretta,Carlo Sinestrari
Publisher : Springer
Page : 276 pages
File Size : 48,9 Mb
Release : 2019-07-04
Category : Mathematics
ISBN : 9783030179496

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Trends in Control Theory and Partial Differential Equations by Fatiha Alabau-Boussouira,Fabio Ancona,Alessio Porretta,Carlo Sinestrari Pdf

This book presents cutting-edge contributions in the areas of control theory and partial differential equations. Over the decades, control theory has had deep and fruitful interactions with the theory of partial differential equations (PDEs). Well-known examples are the study of the generalized solutions of Hamilton-Jacobi-Bellman equations arising in deterministic and stochastic optimal control and the development of modern analytical tools to study the controllability of infinite dimensional systems governed by PDEs. In the present volume, leading experts provide an up-to-date overview of the connections between these two vast fields of mathematics. Topics addressed include regularity of the value function associated to finite dimensional control systems, controllability and observability for PDEs, and asymptotic analysis of multiagent systems. The book will be of interest for both researchers and graduate students working in these areas.