Monotone solutions for mean field games master equations: finite state space and optimal stopping

Bertucci, Charles

doi:10.5802/jep.167

Cited by 24 publications

(44 citation statements)

References 48 publications

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“…On the probability space pΩ, F, Pq, in addition to B, we assume that there is a sub-σ-algebra G independent of B and rich enough so that PpT d q " tLpξq : ξ : Ω Ñ T d is G-measurableu. For any 0 ď t ă T , we then denote by Π t the collection of F t -progressively-measurable R d -valued processes pπ s q tďsďT that are square-integrable (see (10)), where F t is the P-completion of the filtration gererated by G and pB s q tďsďT . We have…”

Section: Mean Field Control Problemmentioning

confidence: 99%

“…Connection with the MFG system. The analysis of the HJB equation ( 21) plays a key role in the derivation of existence and uniqueness of weak solutions to the master equation (11). Before we clarify the precise form of the master equation that we study next, we first recall the connection between the MFCP and the potential MFG introduced in Subsection 2.1.…”

Section: 3mentioning

confidence: 99%

“…Briefly, monotonicity is known to guarantee uniqueness of the equilibria to the corresponding MFG and also to enforce strong stability properties, which play a key role in the analysis of the regularity of the solutions to the master equation. Noticeably, several of the most recent works on the master equation are also written within the same monotone framework, see for instance [10,22,58] (see also [11], which is the counterpart of [10] but for MFGs on a finite state space). In all these papers, the objective is to define, in the monotone setting, a relevant notion of solution to the master equation that does not require the existence and the continuity of all the derivatives that appear therein.…”

Section: Introductionmentioning

confidence: 99%

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Weak solutions to the master equation of potential mean field games

Cecchin¹,

Delarue²

2022

Preprint

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The purpose of this work is to introduce a notion of weak solution to the master equation of a potential mean field game and to prove that existence and uniqueness hold under quite general assumptions. Remarkably, this is achieved without any monotonicity constraint on the coefficients. The key point is to interpret the master equation in a conservative sense and then to adapt to the infinite dimensional setting earlier arguments for hyperbolic systems deriving from a Hamilton-Jacobi-Bellman equation. Here, the master equation is indeed regarded as an infinite dimensional system set on the space of probability measures and is formally written as the derivative of the Hamilton-Jacobi-Bellman equation associated with the mean field control problem lying above the mean field game. To make the analysis easier, we assume that the coefficients are periodic, which allows to represent probability measures through their Fourier coefficients. Most of the analysis then consists in rewriting the master equation and the corresponding Hamilton-Jacobi-Bellman equation for the mean field control problem as partial differential equations set on the Fourier coefficients themselves. In the end, we establish existence and uniqueness of functions that are displacement semi-concave in the measure argument and that solve the Hamilton-Jacobi-Bellman equation in a suitable generalized sense and, subsequently, we get existence and uniqueness of functions that solve the master equation in an appropriate weak sense and that satisfy a weak one-sided Lipschitz inequality. As another new result, we also prove that the optimal trajectories of the associated mean field control problem are unique for almost every starting point, for a suitable probability measure on the space of probability measures.

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Section: Mean Field Control Problemmentioning

confidence: 99%

Section: 3mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Weak solutions to the master equation of potential mean field games

Cecchin¹,

Delarue²

2022

Preprint

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“…It is much more challenging to obtain a global classical solution, we refer to Buckdahn-Li-Peng-Rainer [14], Chassagneux-Crisan-Delarue [23], Cardaliaguet-Delarue-Lasry-Lions [19], Carmona-Delarue [22], Gangbo-Meszaros-Mou-Zhang [32] and, in the realm of potential MFGs, Bensoussan-Graber-Yam [8,9], Gangbo-Meszaros [31]. We also refer to Mou-Zhang [43], Bertucci [12], and Cardaliaguet-Souganidis [20] for global weak solutions which require much weaker regularity on the data, and Bayraktar-Cohen [3], Bertucci-Lasry-Lions [13], Cecchin-Delarue [25], Bertucci [11] for classical or weak solutions of finite state mean field game master equations. All the above global well-posedness results, with the exception [14] that considers linear master equations and thus no control or game is involved, require certain monotonicity condition, which we explain next.…”

Section: Introductionmentioning

confidence: 99%

“…One typical condition, extensively used in the literature [3,11,12,13,19,20,22,23,43], is the well-known Lasry-Lions monotonicity condition: for a function G :…”

Section: Introductionmentioning

confidence: 99%

Mean Field Game Master Equations with Anti-monotonicity Conditions

Mou¹,

Zhang²

2022

Preprint

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It is well known that the monotonicity condition, either in Lasry-Lions sense or in displacement sense, is crucial for the global well-posedness of mean field game master equations, as well as for the uniqueness of mean field equilibria and solutions to mean field game systems.In the literature, the monotonicity conditions are always taken in a fixed direction. In this paper we propose a new type of monotonicity condition in the opposite direction, which we call the anti-monotonicity condition, and establish the global well-posedness for mean field game master equations with nonseparable Hamiltonians. Our anti-monotonicity condition allows our data to violate both the Lasry-Lions monotonicity and the displacement monotonicity conditions.

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Master Equation for the Finite State Space Planning Problem

Bertucci

Lions²

2021

Arch Rational Mech Anal

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We present results of existence, regularity and uniqueness of solutions of the master equation associated with the mean field planning problem in the finite state space case, in the presence of a common noise. The results hold under monotonicity assumptions, which are used crucially in the different proofs of the paper. We also make a link with the trajectories induced by the solution of the master equation and start a discussion on the case of boundary conditions. Contents 1.2. Preliminary results 2. Planning problem master equation in R d 2.1. Statement of the problem 2.2. Properties of the Yosida approximation 2.3. The limit master equation 2.4. Links with the induced trajectories 3. A comment on the case of a restricted domain 3.1. Main differences with the previous case 3.2. The half space case with vanishing conditions

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Monotone solutions for mean field games master equations: finite state space and optimal stopping

Abstract: Cet article est mis à disposition selon les termes de la licence LICENCE INTERNATIONALE D'ATTRIBUTION CREATIVE COMMONS BY 4.0.

Cited by 24 publications

References 48 publications

Weak solutions to the master equation of potential mean field games

Weak solutions to the master equation of potential mean field games

Mean Field Game Master Equations with Anti-monotonicity Conditions

Master Equation for the Finite State Space Planning Problem

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