A general characterization of the mean field limit for stochastic differential games

Lacker, Daniel

doi:10.1007/s00440-015-0641-9

Cited by 173 publications

(203 citation statements)

References 43 publications

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“…The representation formula of the ergodic value function is shown to be quite useful in proving Theorem 5.2. Let us also mention that the convergence results for Nash equilibria we present are somewhat stronger than those obtained in [18,32]. In fact, we show that the maximum distance between the invariant measures in the Nash equilibrium tuple tends to 0 as the number of players increases to infinity (see Theorem 5.2 (b)).…”

Section: Introductionmentioning

confidence: 51%

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On solutions of mean field games with ergodic cost

Arapostathis

Biswas

Carroll

2017

Journal de Mathématiques Pures et Appliquées

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Abstract. A general class of mean field games are considered where the governing dynamics are controlled diffusions in R d . The optimization criterion is the long time average of a running cost function. Under various sets of hypotheses, we establish the existence of mean field game solutions. We also study the long time behavior of the mean field game solutions associated with the finite horizon problem, and under the assumption of geometric ergodicity for the dynamics, we show that these converge to the ergodic mean field game solution as the horizon tends to infinity. Lastly, we study the associated N -player games, show existence of Nash equilibria, and establish the convergence of the solutions associated to Nash equilibria of the game to a mean field game solution as N → ∞.

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Section: Introductionmentioning

confidence: 51%

“…On the other hand, it is also known that one can construct ε-Nash equilibria for N -player games from mean field game solutions. See for example [14,15,28,31,32]. Mean field games have seen a wide variety of applications, and have been studied extensively during the last decade using both analytic and probabilistic techniques.…”

Section: Introductionmentioning

confidence: 99%

On solutions of mean field games with ergodic cost

Arapostathis

Biswas

Carroll

2017

Journal de Mathématiques Pures et Appliquées

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“…By (12), (13) and (14), and by Proposition 7, we deduce that there exists a sequence of random variables pδ N q N ě1 constructed on pΞ, G, Pq such that Proceeding similarly with the terms driven by f in (11) …”

Section: Theorem 1 Assume That F and G Are Bounded And Lipschitz Conmentioning

confidence: 84%

“…Regarding the convergence problem, the picture is then as follows. When the finite player equilibria are taken over open loop controls, compactness arguments, without any need for asymptotic uniqueness, may be used, see for instance [10,13]; however, this strategy fails when equilibria are computed over controls in closed loop form. In the latter case, the only strategy that has been known so far for tackling the convergence problem requires uniqueness, see [3].…”

Section: Introductionmentioning

confidence: 99%

Mean field games: A toy model on an Erdös-Renyi graph.

Delarue

2017

ESAIM: Procs

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Abstract. The purpose of this short article is to address a simple example of a game with a large number of players in mean field interaction when the graph connection between them is not complete but is of the Erdös-Renyi type. We study the quenched convergence of the equilibria towards the solution of a mean field game. To do so, we follow recent works on the convergence problem for mean field games and we heavily use the fact that the master equation of the asymptotic game has a strong solution.

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“…We formulate the problem as a continuum-player game-this abstraction allows us to obtain computationally tractable results (cf. Aumann, 1964;Carmona, 2013;Schmeidler, 1973, for the concept of a continuum-player game, and Cardaliaguet, 2010;Carmona & Delarue, 2013;Lacker, 2016;Lasry & Lions, 2007, for the subclass of mean field games). The paper is organized as follows.…”

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confidence: 99%

Liquidity effects of trading frequency

Gayduk

Nadtochiy

2017

Mathematical Finance

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In this article, we present a discrete time modeling framework, in which the shape and dynamics of a Limit Order Book (LOB) arise endogenously from an equilibrium between multiple market participants (agents). We use the proposed modeling framework to analyze the effects of trading frequency on market liquidity in a very general setting. In particular, we demonstrate the dual effect of high trading frequency. On the one hand, the higher frequency increases market efficiency, if the agents choose to provide liquidity in equilibrium. On the other hand, it also makes markets more fragile, in the sense that the agents choose to provide liquidity in equilibrium only if they are market-neutral (i.e., their beliefs satisfy certain martingale property). Even a very small deviation from market-neutrality may cause the agents to stop providing liquidity, if the trading frequency is sufficiently high, which represents an endogenous liquidity crisis (aka flash crash) in the market. This framework enables us to provide more insight into how such a liquidity crisis unfolds, connecting it to the so-called adverse selection effect.

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A general characterization of the mean field limit for stochastic differential games

Cited by 173 publications

References 43 publications

On solutions of mean field games with ergodic cost

On solutions of mean field games with ergodic cost

Mean field games: A toy model on an Erdös-Renyi graph.

Liquidity effects of trading frequency

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