The Master equation in mean field theory

By Alain Bensoussan, Jens Frehse, Phillip S. C. Yam. 2014

164 citations recorded by the source corpus.

In his lectures at College de France, P.L. Lions introduced the concept of Master equation, see [5] for Mean Field Games. It is introduced in a heuristic fashion, from the system of partial differential equations, associated to a Nash equilibrium for a large, but finite, number of players. The method, also explained in[2], consists in a formal analogy of terms. The interest of this equation is that it contains interesting particular cases, which can be studied directly, in particular the system of HJB-FP (Hamilton-Jacobi-Bellman, Fokker-Planck) equations obtained as the limit of the finite Nash equilibrium game, when the trajectories are independent, see [4]. Usually, in mean field theory, one can bypass the large Nash equilibrium, by introducing the concept of representative agent, whose action is influenced by a distribution of similar agents, and obtains directly the system of HJB-FP equations of interest, see for instance [1]. Apparently, there is no such approach for the Master equation. We show here that it is possible. We first do it for the Mean Field type control problem, for which we interpret completely the Master equation. For the Mean Field Games itself, we solve a related problem, and obtain again the Master equation.

Subfields: Economics and Econometrics, Finance, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/855ed41846c7170a7fc709394a09f218386c40e3

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Citations in the corpus

Mean field and n‐agent games for optimal investment under relative performance criteria (2017, background): “Equation (30) essentially provides the solution of the so-called master equation; see [4] or [14] for an introduction to the master equation in MFG theory.”
Linear-Quadratic Mean Field Games (2014, methodology): “investigated MFGs using a master equation approach in [38].”
Wellposedness of Second Order Master Equations for Mean Field Games with Nonsmooth Data (2019, background): “, Bensoussan-Frehse-Yam [4, 5] and Carmona-Delarue [18] where master equations are derived formally following different approaches.”
Bellman equation and viscosity solutions for mean-field stochastic control problem (2015, unknown): “…partial diﬀerential equation (PDE) of second order in the Wassertein space of probability measures extends previous results in the literature [5, 12, 28]: it reduces in particular to the Bellman equation in the space of density functions derived by Bensoussan, Frehse and Yam [6] when the…”
Superposition and mimicking theorems for conditional McKean–Vlasov equations (2020, background): “1], and also [57, 36, 16, 10, 19, 55, 1, 8, 18, 53, 2, 38, 46]), as well as the dynamics of a particle in the large-scale limit of a particle system with mean field interactions and common noise (see, e.”
A Probabilistic Approach to Classical Solutions of the Master Equation for Large Population Equilibria (2014, background): “Recently, in the independent works [1, 8, 17, 18] and with di erent approaches, several authors revisited, mostly heuristically, the master equation in the case when the dynamics of the players are stochastic.”
Master Bellman equation in the Wasserstein space: Uniqueness of viscosity solutions (2021, methodology): “Such partial differential equations are also known as Master Bellman equations or Bellman equations in the Wasserstein space, see for instance [6, 14, 46].”
Existence of a solution to an equation arising from the theory of Mean Field Games (2015, methodology): “ŚWIE֒CH After the manuscript was completed we learned about the papers [10, 13] which deal with formal derivation of the Master Equations in both deterministic and stochastic cases and their analysis.”
McKean–Vlasov optimal control: The dynamic programming principle (2019, background): “This was first achieved by Lauriére and Pironneau [56], and later by Bensoussan, Frehse, and Yam [10; 11; 12], who assumed the existence at all times of a density for the marginal distribution of the state process, and reformulated the problem as a deterministic density control problem, with a famil”
From the master equation to mean field game limit theory: a central limit theorem (2018, methodology): “For more details on the connection between the master equation and more common PDE or probabilistic formulations of mean field games, see [4, 5, 7] or [6, Section 1.”
Smoothing properties of McKean–Vlasov SDEs (2017, background): “Papers [3, 4, 22] present further details of the relevance of the class of nonlinear partial differential equations (1.”
From the master equation to mean field game limit theory: Large deviations and concentration of measure (2018, background): “For more details on the connection between the master equation and more common PDE or probabilistic formulations of mean field games, see [4, 5, 11] or [10, Section 1.”
A general characterization of the mean field limit for stochastic differential games (2014, background): “The literature on mean field games with common noise is quite scarce so far, but some general analysis is provided in the recent papers [1,5,9,11], and some specific models were studied in [12,19].”
Finite state mean field games with Wright–Fisher common noise (2019, methodology): “Although we provide a longer review on MFG later in the text, we feel useful to quote, at this early stage of the introduction, [33, 34] and [9, Chapter 7] as references on the master equation for finite state mean field games, and [7, 5, 10, 8, 16] as references for continuous state mean field game”