Regularity Theory For Mean Field Game Systems

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Regularity Theory for Mean-Field Game Systems

Diogo A. Gomes,Edgard A. Pimentel,Vardan Voskanyan
Regularity Theory for Mean Field Game Systems Book PDF

Author : Diogo A. Gomes,Edgard A. Pimentel,Vardan Voskanyan
Publisher : Springer
Page : 156 pages
File Size : 45,9 Mb
Release : 2016-09-14
Category : Mathematics
ISBN : 9783319389349

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Regularity Theory for Mean-Field Game Systems by Diogo A. Gomes,Edgard A. Pimentel,Vardan Voskanyan Pdf

Beginning with a concise introduction to the theory of mean-field games (MFGs), this book presents the key elements of the regularity theory for MFGs. It then introduces a series of techniques for well-posedness in the context of mean-field problems, including stationary and time-dependent MFGs, subquadratic and superquadratic MFG formulations, and distinct classes of mean-field couplings. It also explores stationary and time-dependent MFGs through a series of a-priori estimates for solutions of the Hamilton-Jacobi and Fokker-Planck equation. It shows sophisticated a-priori systems derived using a range of analytical techniques, and builds on previous results to explain classical solutions. The final chapter discusses the potential applications, models and natural extensions of MFGs. As MFGs connect common problems in pure mathematics, engineering, economics and data management, this book is a valuable resource for researchers and graduate students in these fields.

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 : 43,5 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.

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 : 48,6 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.

Advances in Dynamic and Mean Field Games

Joseph Apaloo,Bruno Viscolani
Advances in Dynamic and Mean Field Games Book PDF

Author : Joseph Apaloo,Bruno Viscolani
Publisher : Birkhäuser
Page : 363 pages
File Size : 46,6 Mb
Release : 2018-01-03
Category : Mathematics
ISBN : 9783319706191

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Advances in Dynamic and Mean Field Games by Joseph Apaloo,Bruno Viscolani Pdf

This contributed volume considers recent advances in dynamic games and their applications, based on presentations given at the 17th Symposium of the International Society of Dynamic Games, held July 12-15, 2016, in Urbino, Italy. Written by experts in their respective disciplines, these papers cover various aspects of dynamic game theory including mean-field games, stochastic and pursuit-evasion games, and computational methods for dynamic games. Topics covered include Pedestrian flow in crowded environments Models for climate change negotiations Nash Equilibria for dynamic games involving Volterra integral equations Differential games in healthcare markets Linear-quadratic Gaussian dynamic games Aircraft control in wind shear conditions Advances in Dynamic and Mean-Field Games presents state-of-the-art research in a wide spectrum of areas. As such, it serves as a testament to the continued vitality and growth of the field of dynamic games and their applications. It will be of interest to an interdisciplinary audience of researchers, practitioners, and graduate students.

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 : 49,5 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.

PDE Models for Multi-Agent Phenomena

Pierre Cardaliaguet,Alessio Porretta,Francesco Salvarani
PDE Models for Multi Agent Phenomena Book PDF

Author : Pierre Cardaliaguet,Alessio Porretta,Francesco Salvarani
Publisher : Springer
Page : 218 pages
File Size : 47,9 Mb
Release : 2018-12-22
Category : Mathematics
ISBN : 9783030019471

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PDE Models for Multi-Agent Phenomena by Pierre Cardaliaguet,Alessio Porretta,Francesco Salvarani Pdf

This volume covers selected topics addressed and discussed during the workshop “PDE models for multi-agent phenomena,” which was held in Rome, Italy, from November 28th to December 2nd, 2016. The content mainly focuses on kinetic equations and mean field games, which provide a solid framework for the description of multi-agent phenomena. The book includes original contributions on the theoretical and numerical study of the MFG system: the uniqueness issue and finite difference methods for the MFG system, MFG with state constraints, and application of MFG to market competition. The book also presents new contributions on the analysis and numerical approximation of the Fokker-Planck-Kolmogorov equations, the isotropic Landau model, the dynamical approach to the quantization problem and the asymptotic methods for fully nonlinear elliptic equations. Chiefly intended for researchers interested in the mathematical modeling of collective phenomena, the book provides an essential overview of recent advances in the field and outlines future research directions.

Mean Field Games

François Delarue
Mean Field Games Book PDF

Author : François Delarue
Publisher : American Mathematical Society
Page : 284 pages
File Size : 43,6 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.

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 : 43,5 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.

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 : 47,8 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.

Mean Field Games and Mean Field Type Control Theory

Alain Bensoussan,Jens Frehse,Phillip Yam
Mean Field Games and Mean Field Type Control Theory Book PDF

Author : Alain Bensoussan,Jens Frehse,Phillip Yam
Publisher : Springer Science & Business Media
Page : 132 pages
File Size : 45,6 Mb
Release : 2013-10-16
Category : Science
ISBN : 9781461485087

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Mean Field Games and Mean Field Type Control Theory by Alain Bensoussan,Jens Frehse,Phillip Yam Pdf

​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. ​

New Trends in Analysis and Geometry

Mohamed A. Khamsi
New Trends in Analysis and Geometry Book PDF

Author : Mohamed A. Khamsi
Publisher : Cambridge Scholars Publishing
Page : 401 pages
File Size : 40,5 Mb
Release : 2020-01-24
Category : Mathematics
ISBN : 9781527546127

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New Trends in Analysis and Geometry by Mohamed A. Khamsi Pdf

This unique mathematical volume brings together geometers, analysts, differential equations specialists and graph-theorists to provide a glimpse on recent mathematical trends whose commonalities have hitherto remained, for the most part, unnoticed. The applied mathematician will be pleasantly surprised with the interpretation of a voting system in terms of the fixed points of a mapping given in the book, as much as the classical analyst will be enthusiastic to find detailed discussions on the generalization of the notion of metric space, in which the metric takes values on an abstract monoid. Classical themes on fixed point theory are adapted to the diverse setting of graph theory, thus uncovering a set of tools whose power and versatility will be appreciated by mathematicians working on either area. The volume also includes recent results on variable exponent spaces which reveal much-needed connections with partial differential equations, while the incipient field of variational inequalities on manifolds, also explored here, will be of interest to researchers from a variety of fields.

Paris-Princeton Lectures on Mathematical Finance 2010

Areski Cousin,Stéphane Crépey,Olivier Guéant,David Hobson,Monique Jeanblanc,Jean-Michel Lasry,Jean-Paul Laurent,Pierre-Louis Lions,Peter Tankov
Paris Princeton Lectures on Mathematical Finance 2010 Book PDF

Author : Areski Cousin,Stéphane Crépey,Olivier Guéant,David Hobson,Monique Jeanblanc,Jean-Michel Lasry,Jean-Paul Laurent,Pierre-Louis Lions,Peter Tankov
Publisher : Springer
Page : 366 pages
File Size : 53,9 Mb
Release : 2010-10-06
Category : Mathematics
ISBN : 9783642146602

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Paris-Princeton Lectures on Mathematical Finance 2010 by Areski Cousin,Stéphane Crépey,Olivier Guéant,David Hobson,Monique Jeanblanc,Jean-Michel Lasry,Jean-Paul Laurent,Pierre-Louis Lions,Peter Tankov Pdf

The Paris-Princeton Lectures in Financial Mathematics, of which this is the fourth volume, publish cutting-edge research in self-contained, expository articles from outstanding specialists - established or on the rise! The aim is to produce a series of articles that can serve as an introductory reference source for research in the field. The articles are the result of frequent exchanges between the finance and financial mathematics groups in Paris and Princeton. The present volume sets standards with five articles by: 1. Areski Cousin, Monique Jeanblanc and Jean-Paul Laurent, 2. Stéphane Crépey, 3. Olivier Guéant, Jean-Michel Lasry and Pierre-Louis Lions, 4. David Hobson and 5. Peter Tankov.

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 : 40,8 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.

Mean-Field-Type Games for Engineers

Julian Barreiro-Gomez,Hamidou Tembine
Mean Field Type Games for Engineers Book PDF

Author : Julian Barreiro-Gomez,Hamidou Tembine
Publisher : CRC Press
Page : 526 pages
File Size : 48,7 Mb
Release : 2021-11-18
Category : Technology & Engineering
ISBN : 9781000473537

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Mean-Field-Type Games for Engineers by Julian Barreiro-Gomez,Hamidou Tembine Pdf

The contents of this book comprise an appropriate background to start working and doing research on mean-field-type control and game theory. To make the exposition and explanation even easier, we first study the deterministic optimal control and differential linear-quadratic games. Then, we progressively add complexity step-by-step and little-by-little to the problem settings until we finally study and analyze mean-field-type control and game problems incorporating several stochastic processes, e.g., Brownian motions, Poisson jumps, and random coefficients. We go beyond the Nash equilibrium, which provides a solution for non- cooperative games, by analyzing other game-theoretical concepts such as the Berge, Stackelberg, adversarial/robust, and co-opetitive equilibria. For the mean-field-type game analysis, we provide several numerical examples using a Matlab-based user-friendly toolbox that is available for the free use to the readers of this book. We present several engineering applications in both continuous and discrete time. Among these applications we find the following: water distribution systems, micro-grid energy storage, stirred tank reactor, mechanism design for evolutionary dynamics, multi-level building evacuation problem, and the COVID-19 propagation control. Julian Barreiro-Gomez Hamidou Tembine With such a demand from engineering audiences, this book is very timely and provides a thorough study of mean-field-type game theory. The strenuous protagonist of this book is to bridge between the theoretical findings and engineering solutions. The book introduces the basics first, and then mathematical frameworks are elaborately explained. The engineering application examples are shown in detail, and the popular learning approaches are also investigated. Those advantageous characteristics will make this book a comprehensive handbook of many engineering fields for many years, and I will buy one when it gets published. Zhu Han

Fokker-Planck-Kolmogorov Equations

Vladimir I. Bogachev,Nicolai V. Krylov,Michael Röckner,Stanislav V. Shaposhnikov
Fokker Planck Kolmogorov Equations Book PDF

Author : Vladimir I. Bogachev,Nicolai V. Krylov,Michael Röckner,Stanislav V. Shaposhnikov
Publisher : American Mathematical Soc.
Page : 482 pages
File Size : 42,8 Mb
Release : 2015-12-17
Category : Fokker-Planck equation
ISBN : 9781470425586

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Fokker-Planck-Kolmogorov Equations by Vladimir I. Bogachev,Nicolai V. Krylov,Michael Röckner,Stanislav V. Shaposhnikov Pdf

This book gives an exposition of the principal concepts and results related to second order elliptic and parabolic equations for measures, the main examples of which are Fokker-Planck-Kolmogorov equations for stationary and transition probabilities of diffusion processes. Existence and uniqueness of solutions are studied along with existence and Sobolev regularity of their densities and upper and lower bounds for the latter. The target readership includes mathematicians and physicists whose research is related to diffusion processes as well as elliptic and parabolic equations.