Weak solutions of the master equation for Mean Field Games with no idiosyncratic noise

Cardaliaguet, Pierre; Souganidis, Panagiotis E.

doi:10.48550/arxiv.2109.14911

Cited by 3 publications

(5 citation statements)

References 29 publications

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“…Various notions of weak solutions to master equations are proposed by Mou and Zhang [26] and Bertucci [2,3]. Through a combination of the Hilbertian interpretation and Bertucci's notion of monotone solutions, Cardaliaguet and Souganidis build a unique, global, weak solution to the master equation with common noise but without idiosyncratic noise [9]; we note that this approach does not seem to adapt to problems with idiosyncratic noise.…”

Section: 2mentioning

confidence: 97%

Mean field games with common noise and degenerate idiosyncratic noise

Cardaliaguet¹,

Seeger²,

Souganidis³

2022

Preprint

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We study the forward-backward system of stochastic partial differential equations describing a mean field game for a large population of small players subject to both idiosyncratic and common noise. The unique feature of the problem is that the idiosyncratic noise coefficient may be degenerate, so that the system does not admit smooth solutions in general. We develop a new notion of weak solutions for backward stochastic Hamilton-Jacobi-Bellman equations, and use this to build probabilistically weak solutions of the mean field game system.

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Section: 2mentioning

confidence: 97%

Mean field games with common noise and degenerate idiosyncratic noise

Cardaliaguet¹,

Seeger²,

Souganidis³

2022

Preprint

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“…and thus (18) becomes µ * ,ξ t = k(t, ν * ,ξ t , ϕ * ,ξ t ). We note that (19) implies k(t, •, •) is uniformly Lipschitz continuous in x and y, uniformly in [0, T ]. Taking the conditional expectation of (15) on…”

Section: Nce Systemmentioning

confidence: 99%

“…There are also some studies on the global well-posedness of the weak solutions to the master equation, see e.g. Mou-Zhang [54], Bertucci [9,10], Cardaliaguet-Souganidis [19], Cecchin-Delarue [24].…”

Section: Introductionmentioning

confidence: 99%

Linear-Quadratic Mean Field Games of Controls with Non-Monotone Data

Li¹,

Mou²,

Zhang³

et al. 2022

Preprint

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In this paper, we study a class of linear-quadratic (LQ) mean field games of controls with common noises and their corresponding N -player games. The theory of mean field game of controls considers a class of mean field games where the interaction is via the joint law of both the state and control. By the stochastic maximum principle, we first analyze the limiting behavior of the representative player and obtain his/her optimal control in a feedback form with the given distributional flow of the population and its control. The mean field equilibrium is determined by the Nash certainty equivalence (NCE) system. Thanks to the common noise, we do not require any monotonicity conditions for the solvability of the NCE system. We also study the master equation arising from the LQ mean field game of controls, which is a finite-dimensional second-order parabolic equation. It can be shown that the master equation admits a unique classical solution over an arbitrary time horizon without any monotonicity conditions. Beyond that, we can solve the N -player game directly by further assuming the non-degeneracy of the idiosyncratic noises. As byproducts, we prove the quantitative convergence results from the N -player game to the mean field game and the propagation of chaos property for the related optimal trajectories.

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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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Weak solutions of the master equation for Mean Field Games with no idiosyncratic noise

Cited by 3 publications

References 29 publications

Mean field games with common noise and degenerate idiosyncratic noise

Mean field games with common noise and degenerate idiosyncratic noise

Linear-Quadratic Mean Field Games of Controls with Non-Monotone Data

Mean Field Game Master Equations with Anti-monotonicity Conditions

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