Master Equation and the Convergence Problem in Mean Field Games (eBook)

Pierre Cardaliaguet, Francois Delarue, Jean-Michel Lasry, Pierre-Louis Lions (Autoren)

eBook Download: PDF

2019
224 Seiten
Princeton University Press (Verlag)
978-0-691-19371-7 (ISBN)

Infos

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.

Autor

Pierre Cardaliaguet is professor of mathematics at Paris Dauphine University. François Delarue is professor of mathematics at the University of Nice Sophia Antipolis. Jean-Michel Lasry is associate researcher of mathematics at Paris Dauphine University. Pierre-Louis Lions is professor of partial differential equations and their applications at the Collège de France.

Produkt-Details

• Erscheint lt. Verlag: 13.8.2019
• Reihe/Serie: Annals of Mathematics Studies
• Reihe/Serie: Annals of Mathematics Studies
• Verlagsort: Princeton
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Mathematik / Informatik ► Mathematik ► Angewandte Mathematik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Finanz- / Wirtschaftsmathematik
• Schlagworte: Approximation • A priori estimate • 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 • Folk theorem (game theory) • 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 • probability measure • Probability Theory • random function • Randomization • Random Variable • Rate of Convergence • Regime • scientific notation • Semigroup • Simultaneous Equations • small number • Smoothness • space form • State Space • state variable • stochastic • Stochastic Calculus • stochastic control • Stochastic process • 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
• ISBN-10: 0-691-19371-1 / 0691193711
• ISBN-13: 978-0-691-19371-7 / 9780691193717