Mean field games with common noise

By René A. Carmona, François Delarue, Daniel Lacker. 2014

291 citations recorded by the source corpus.

A theory of existence and uniqueness is developed for general stochastic differential mean field games with common noise. The concepts of strong and weak solutions are introduced in analogy with the theory of stochastic differential equations, and existence of weak solutions for mean field games is shown to hold under very general assumptions. Examples and counter-examples are provided to enlighten the underpinnings of the existence theory. Finally, an analog of the famous result of Yamada and Watanabe is derived, and it is used to prove existence and uniqueness of a strong solution under additional assumptions.

Subfields: Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d0a311da2ef6b85f1bbed146cb4eff666840da72

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Citations in the corpus

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].”
Mean Field Games of Timing and Models for Bank Runs (2016, unknown): “This notion of compatibility arises naturally because of the central role played by weak convergence arguments in our analysis; essentially the same issue appears in the papers [12, 28], which deal with more traditional mean field game models.”
Linear Quadratic Mean Field Type Control and Mean Field Games with Common Noise, with Application to Production of an Exhaustible Resource (2016, background): “The first largely comprehensive account of this problem for mean field games with common noise was given by Lacker in [44].”
Superposition and mimicking theorems for conditional McKean–Vlasov equations (2020, background): “General results on existence, uniqueness, and limit theory require working with a notion of weak equilibrium which is well-suited for weak convergence arguments [18, 53, 17].”
McKean–Vlasov optimal control: The dynamic programming principle (2019, methodology): “Motivated by the notion of weak solution of classical SDEs, and similarly to the ideas used by El Karoui and Tan [37], and Carmona, Delarue, and Lacker [28] in a mean–field games context, our first task is to provide an appropriate ‘relaxation‘ of the problem.”
Mean–field moral hazard for optimal energy demand response management (2019, background): “As pointed out by Carmona, Delarue, and Lacker [14], it is commonplace for control problems in weak formulation, and actually already for weak solutions to SDEs, that the underlying driving noise is not rich enough to obtain a solution which is only adapted to it.”
Mean field and n‐agent games for optimal investment under relative performance criteria (2017, background): “We refer the reader to [16] for a detailed discussion of mean field games with common noise, and to [12, 39] for general results on limits of n-agent games.”
On non-uniqueness and uniqueness of solutions in finite-horizon Mean Field Games (2017, unknown): “The ﬁrst and best known is the case of H convex and F, G increasing with respect to m in the L 1 sense, which means that imitating the other agents is costly, see [10, 39] for the PDE proof and [2, 14] for probabilistic proofs, which also cover MFGs with a common noise.”
On the convergence of closed-loop Nash equilibria to the mean field game limit (2018, methodology): “Our notion of weak MFE turns out to be equivalent in a certain sense to the notion of weak MFG solution introduced in [17, 43].”
Splitting methods and short time existence for the master equations in mean field games (2020, background): “Besides the PDE approach we just mentioned, MFG with common noise can also be handled through a probabilistic formulation: see the pioneering result [12], as well as [2, 21] and the monograph [11].”