Wellposedness of Second Order Master Equations for Mean Field Games with Nonsmooth Data

Mou, Chenchen; Zhang, Jianfeng

doi:10.48550/arxiv.1903.09907

Cited by 14 publications

(34 citation statements)

References 34 publications

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“…Alternatively, the monotone regime proved to be regularizing in the finite state space case [20,3,4]. More recently, several teams have addressed the issue of defining weak solutions of master equations in several context (which are not the monotone regime) : [10,9] propose ways of selecting a weak solution in finite state space, particularly in the potential case ; [22,13,14] introduce notions of weak solutions of the master equation which do not rely on monotonicity assumptions. Up to this point, no general framework has been proposed.…”

Section: Introductionmentioning

confidence: 99%

Monotone solutions for mean field games master equations : continuous state space and common noise

Bertucci

2021

Preprint

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We present the notion of monotone solution of mean field games master equations in the case of a continuous state space. We establish the existence, uniqueness and stability of such solutions under standard assumptions. This notion allows us to work with solutions which are merely continuous in the measure argument, in the case of first order master equations. We study several structures of common noises, in particular ones in which common jumps (or aggregate shocks) can happen randomly, and ones in which the correlation of randomness is carried by an additional parameter.

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Section: Introductionmentioning

confidence: 99%

Monotone solutions for mean field games master equations : continuous state space and common noise

Bertucci

2021

Preprint

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“…There has been great progress in recent years, beginning with the groundbreaking work [16] which heavily exploited the Lasry-Lions monotonicity condition. Well-posedness of the master equation has since been shown in several settings, including major player models [61], finite state space [8,11], lower regularity [59], degenerate idiosyncratic noise [18], and the recent paper [37] using the alternative weak (or displacement) monotonicity concept originating in [5].…”

Section: Introductionmentioning

confidence: 99%

Closed-loop convergence for mean field games with common noise

Lacker¹,

Flem²

2021

Preprint

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This paper studies the convergence problem for mean field games with common noise. We define a suitable notion of weak mean field equilibria, which we prove captures all subsequential limit points, as n → ∞, of closed-loop approximate equilibria from the corresponding n-player games. This extends to the common noise setting a recent result of the first author, while also simplifying a key step in the proof and allowing unbounded coefficients and non-i.i.d. initial conditions. Conversely, we show that every weak mean field equilibrium arises as the limit of some sequence of approximate equilibria for the n-player games, as long as the latter are formulated over a broader class of closed-loop strategies which may depend on an additional common signal.

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“…Lions also developed the Hilbertian approach [32] in order to handle equation of the form (0.1) or (0.2), which yields the existence of classical solutions under a structure condition on H and F ensuring the convexity of the solution with respect to the space variable. A partial list of references on the master equation is [2,3,5,6,7,8,9,11,17,27,28,33,34].…”

Section: Introductionmentioning

confidence: 99%

“…The recent papers Gangbo and Mészáros [27] and Gangbo, Mészáros, Mou and Zhang [28] overcome these difficulties by assuming a structure condition which ensures space convexity and, hence, the smoothness of the solution. First steps in the direction of dealing with nonsmooth solutions are the paper of Mou and Zhang [34], which discusses some notions of weak solution based on the behavior of the solution with respect to the solution of the mean field game system, as well as the aforementioned works [7,8].…”

Section: Introductionmentioning

confidence: 99%

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

Cardaliaguet¹,

Souganidis²

2021

Preprint

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We introduce a notion of weak solution of the master equation without idiosyncratic noise in Mean Field Game theory and establish its existence, uniqueness up to a constant and consistency with classical solutions when it is smooth. We work in a monotone setting and rely on Lions' Hilbert space approach. For the first-order master equation without idiosyncratic noise, we also give an equivalent definition in the space of measures and establish the well-posedness.

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Wellposedness of Second Order Master Equations for Mean Field Games with Nonsmooth Data

Cited by 14 publications

References 34 publications

Monotone solutions for mean field games master equations : continuous state space and common noise

Monotone solutions for mean field games master equations : continuous state space and common noise

Closed-loop convergence for mean field games with common noise

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

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