Probabilistic theory of mean field games with applications. Mean field games with common noise and master equations / II, ([2018])

• Record Type: Book
• Other Titles: Mean field games with common noise and master equations
• Further Information: Note: René Carmona, François Delarue.
• Authors: Carmona, R (René); Delarue, F (François)
• Contents: Intro; Foreword; Preface of Volume II; Organization of the Book: Volume II Organigram; Contents; Contents of Volume I; Part I MFGs with a Common Noise; 1 Optimization in a Random Environment; 1.1 FBSDEs in a Random Environment; 1.1.1 Immersion and Compatibility; 1.1.2 Compatible Probabilistic Set-Up; More on the Compatibility Condition; 1.1.3 Kunita-Watanabe Decomposition and Definition of a Solution; New Formulation of the Forward-Backward System; Definition of a Solution; 1.2 Strong Versus Weak Solution of an FBSDE; 1.2.1 Notions of Uniqueness; 1.2.2 Canonical Spaces. Transfer of Solutions to the Canonical Space1.2.3 Yamada-Watanabe Theorem for FBSDEs; Measurability of the Law of the Output with Respect to the Law of the Input; Main Statement; Compatible Product Probabilistic Set-Up; 1.3 Initial Information, Small Time Solvability, and Decoupling Field; 1.3.1 Disintegration of FBSDEs; Initial Information; Disintegration of FBSDEs; 1.3.2 Small Time Solvability and Decoupling Field; Small Time Solvability; Decoupling Field; 1.3.3 Induction Procedure; General Mechanism; Stability Property; 1.4 Optimization with Random Coefficients; 1.4.1 Optimization Problem. Description of the DynamicsCost Functionals; Standing Assumptions; 1.4.2 Value Function and Stochastic HJB Equation; 1.4.3 Revisiting the Connection Between the HJB Equations and BSDEs; Main Statement; An Intermediate Result; End of the Proof of Theorem 1.57; 1.4.4 Revisiting the Pontryagin Stochastic Maximum Principle; Necessary Condition; Sufficiency; 1.5 Notes & Complements; 2 MFGs with a Common Noise: Strong and Weak Solutions; 2.1 Conditional Propagation of Chaos; 2.1.1 N-Player Games with a Common Noise; 2.1.2 Set-Up for a Conditional McKean-Vlasov Theory. Exchangeable Sequences of Random VariablesNonlinear Stochastic Fokker-Planck Equation; 2.1.3 Formulation of the Limit Problem; Conditional Distributions Given F0; Solving the Conditional McKean-Vlasov SDE; More About the Flow of Conditional Marginal Distributions Îơ; 2.1.4 Conditional Propagation of Chaos; 2.2 Strong Solutions to MFGs with Common Noise; 2.2.1 Solution Strategy for Mean Field Games; 2.2.2 Revisiting the Probabilistic Set-Up; Specification of the Initial Condition; Compatibility Condition in the Optimal Control Problem; Summary. 2.2.3 FBSDE Formulation of the First Step in the Search for a Solution2.2.4 Strong MFG Matching Problem: Solutions and Strong Solvability; Equilibrium on an Arbitrary Space; Strong Solvability and Strong Solutions; 2.3 Weak Solutions for MFGs with Common Noise; 2.3.1 Weak MFG Matching Problem; Canonical Space and Distribution of an Equilibrium; 2.3.2 Yamada-Watanabe Theorem for MFG Equilibria; 2.3.3 Infinite Dimensional Stochastic FBSDEs; 2.4 Notes & Complements; 3 Solving MFGs with a Common Noise; 3.1 Introduction; 3.1.1 Road Map to Weak Solutions; 3.1.2 Statement of the Problem.
• Publisher Details: Cham, Switzerland : Springer
• Publication Date: 2018
• Extent: 1 online resource
• Subjects: 530.14/4; Mathematics; Mean field theory; Game theory; SCIENCE -- Energy; SCIENCE -- Mechanics -- General; SCIENCE -- Physics -- General; Game theory; Mean field theory; Mathematics -- Calculus; Mathematics -- Differential Equations; Business & Economics -- Economics -- Theory; Calculus of variations; Differential calculus & equations; Economic theory & philosophy; Distribution (Probability theory); Mathematical optimization; Differential equations, partial; Economic theory; Mathematics -- Probability & Statistics -- General; Probability & statistics; Electronic books
• Languages: English
• ISBNs: 9783319564364; 3319564366
• Related ISBNs: 9783319564357; 3319564358
• Notes: Note: Includes bibliographical references and indexes.; Note: Online resource; title from PDF title page (SpringerLink, viewed March 13, 2018).
• Access Rights: Legal Deposit; Only available on premises controlled by the deposit library and to one user at any one time; The Legal Deposit Libraries (Non-Print Works) Regulations (UK).
• Access Usage: Restricted: Printing from this resource is governed by The Legal Deposit Libraries (Non-Print Works) Regulations (UK) and UK copyright law currently in force.