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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.

Mathematics. --- Calculus of variations. --- Partial differential equations. --- Probabilities. --- Economic theory. --- Probability Theory and Stochastic Processes. --- Calculus of Variations and Optimal Control; Optimization. --- Partial Differential Equations. --- Economic Theory/Quantitative Economics/Mathematical Methods. --- Isoperimetrical problems --- Variations, Calculus of --- Maxima and minima --- Math --- Science --- Distribution (Probability theory. --- Mathematical optimization. --- Differential equations, partial. --- Economic theory --- Political economy --- Social sciences --- Economic man --- Partial differential equations --- Optimization (Mathematics) --- Optimization techniques --- Optimization theory --- Systems optimization --- Mathematical analysis --- Operations research --- Simulation methods --- System analysis --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probabilities --- Probability --- Statistical inference --- Combinations --- Mathematics --- Chance --- Least squares --- Mathematical statistics --- Risk

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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.

Quantitative methods (economics) --- Economic schools --- Partial differential equations --- Differential equations --- Numerical methods of optimisation --- Operational research. Game theory --- Probability theory --- differentiaalvergelijkingen --- waarschijnlijkheidstheorie --- stochastische analyse --- economie --- economisch denken --- wiskunde --- kansrekening --- optimalisatie

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Noire et inquiétante était la ville, la nuit. Dissipé par l'éclairage, les illuminations et les enseignes, les phantasmes nés de l'obscurité. Voilà la ville contemporaine transfigurée et prête à s'animer, se parer et travailler vingt-quatre heures sur vingt-quatre. La lumière s'impose comme une composante nouvelle et inattendue de la fabrique de la ville, grâce à des artistes tels que Laurent Fachard, Yann Kersalé, Mark Major, Roger Narboni et bien d'autres. Jouant sur les perceptions sensorielles et sur la virtualité, il lui revient non seulement d'embellir et de rassurer mais aussi de révéler des sites, marquer des repères, relier des fragments dispersés. L'analyse des cas comme Genève, Annemasse, Coventry, Lausanne, Lyon ou Saint-Nazaire démontre combien la lumière peut apporter de qualité aux cycles de la ville. La capacité de la lumière à transformer lieux et atmosphères comporte aussi des revers, risques d'une théâtralisation de la ville, d'une banalisation. Pourtant, par sa plasticité, sa capacité à encourager ou préfigurer des usages, elle suggère une ville autre qu'elle-même et contribue à créer un espace-temps différent. Deuxième ouvrage d'une trilogie, Penser la ville par la lumière fait suite à celui consacré au paysage et précède celui dédié à l'art contemporain. Ensemble ils témoignent du renouvellement des approches en matière de projet urbain, voire territorial.

Eclairage --- Eclairage public --- Lumière --- Paysage urbain --- Ville --- Municipal lighting --- Esthétique urbaine --- Social aspects --- Fachard, Laurent --- Kersalé, Yann --- Major, Mark --- Narboni, Roger --- 711.4 <44> --- -Cities and towns --- City lighting --- Metropolitan lighting --- Public lighting --- Lighting --- Municipal engineering --- Gemeentelijke planologie. Stadsplanning. Stedenbouw--Frankrijk --- -Gemeentelijke planologie. Stadsplanning. Stedenbouw--Frankrijk --- 711.4 <44> Gemeentelijke planologie. Stadsplanning. Stedenbouw--Frankrijk --- Municipal lighting. --- Cities and towns --- Environmental planning --- Social aspects. --- Aspect social --- Municipal lighting - Social aspects --- Lumière en architecture --- Éclairage architectural et décoratif --- Eclairage publique --- Villes --- Aspect symbolique --- Recherches --- Lumière en architecture --- Lumière --- Éclairage architectural et décoratif

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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 originating 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.

Convergence. --- Mean field theory. --- Many-body problem --- Statistical mechanics --- Functions --- A priori estimate. --- Approximation. --- Bellman equation. --- Boltzmann equation. --- Boundary value problem. --- C0. --- Chain rule. --- Compact space. --- Computation. --- Conditional probability distribution. --- Continuous function. --- Convergence problem. --- Convex set. --- Cooperative game. --- Corollary. --- Decision-making. --- Derivative. --- Deterministic system. --- Differentiable function. --- Directional derivative. --- Discrete time and continuous time. --- Discretization. --- Dynamic programming. --- Emergence. --- Empirical distribution function. --- Equation. --- Estimation. --- Euclidean space. --- Folk theorem (game theory). --- Folk theorem. --- Heat equation. --- Hermitian adjoint. --- Implementation. --- Initial condition. --- Integer. --- Large numbers. --- Linearization. --- Lipschitz continuity. --- Lp space. --- Macroeconomic model. --- Markov process. --- Martingale (probability theory). --- Master equation. --- Mathematical optimization. --- Maximum principle. --- Method of characteristics. --- Metric space. --- Monograph. --- Monotonic function. --- Nash equilibrium. --- Neumann boundary condition. --- Nonlinear system. --- Notation. --- Numerical analysis. --- Optimal control. --- Parameter. --- Partial differential equation. --- Periodic boundary conditions. --- Porous medium. --- Probability measure. --- Probability theory. --- Probability. --- Random function. --- Random variable. --- Randomization. --- Rate of convergence. --- Regime. --- Scientific notation. --- Semigroup. --- Simultaneous equations. --- Small number. --- Smoothness. --- Space form. --- State space. --- State variable. --- Stochastic calculus. --- Stochastic control. --- Stochastic process. --- Stochastic. --- Subset. --- Suggestion. --- Symmetric function. --- Technology. --- Theorem. --- Theory. --- Time consistency. --- Time derivative. --- Uniqueness. --- Variable (mathematics). --- Vector space. --- Viscosity solution. --- Wasserstein metric. --- Weak solution. --- Wiener process. --- Without loss of generality.