Jeux à champ moyen. I – Le cas stationnaire
TL;DR: Lasry et al. as mentioned in this paper introduce an approche generale for modeliser des jeux avec un tres grand nombre of joueurs, and consider des equilibres de Nash a N joues for des problemes stochastiques en temps long and deduisons rigoureusement les equations de type « champ moyen » quand N tend vers l'infini.
About: This article is published in Comptes Rendus Mathematique.The article was published on 2006-11-01. It has received 802 citations till now.
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TL;DR: In this paper, the authors present three examples of the mean-field approach to modelling in economics and finance (or other related subjects) and show that these nonlinear problems are essentially well-posed problems with unique solutions.
Abstract: We survey here some recent studies concerning what we call mean-field models by analogy with Statistical Mechanics and Physics. More precisely, we present three examples of our mean-field approach to modelling in Economics and Finance (or other related subjects...). Roughly speaking, we are concerned with situations that involve a very large number of “rational players” with a limited information (or visibility) on the “game”. Each player chooses his optimal strategy in view of the global (or macroscopic) informations that are available to him and that result from the actions of all players. In the three examples we mention here, we derive a mean-field problem which consists in nonlinear differential equations. These equations are of a new type and our main goal here is to study them and establish their links with various fields of Analysis. We show in particular that these nonlinear problems are essentially well-posed problems i.e., have unique solutions. In addition, we give various limiting cases, examples and possible extensions. And we mention many open problems.
2,385 citations
TL;DR: Lasry et al. as mentioned in this paper considered the case of Nash equilibria for stochastic control type problems in finite horizon and presented general existence and uniqueness results for the partial differential equations systems that they introduced.
776 citations
01 Jan 2011
TL;DR: The Course Bachelier 2009 as discussed by the authors was inspired from a course inspired by the work of Jean-Michel Lasry, and the course was based upon the articles of the three authors and upon unpublished materials they developed.
Abstract: This text is inspired from a “Cours Bachelier” held in January 2009 and taught by Jean-Michel Lasry. This course was based upon the articles of the three authors and upon unpublished materials they developed. Proofs were not presented during the conferences and are now available. So are some issues that were only rapidly tackled during class.
479 citations
01 Dec 2010
TL;DR: The paper shows that under certain mild conditions, there exists a unique Nash equilibrium that almost satisfies the control objective to minimize electricity generation costs by establishing a PEV charging schedule that fills the overnight demand valley.
Abstract: The paper develops a novel decentralized charging control strategy for large populations of plug-in electric vehicles (PEVs). We consider the situation where PEV agents are rational and weakly coupled via their operation costs. At an established Nash equilibrium, each of the PEV agents reacts optimally with respect to the average charging strategy of all the PEV agents. Each of the average charging strategies can be approximated by an infinite population limit which is the solution of a fixed point problem. The control objective is to minimize electricity generation costs by establishing a PEV charging schedule that fills the overnight demand valley. The paper shows that under certain mild conditions, there exists a unique Nash equilibrium that almost satisfies that goal. Moreover, the paper establishes a sufficient condition under which the system converges to the unique Nash equilibrium. The theoretical results are illustrated through various numerical examples.
479 citations
TL;DR: Numerical methods for the approximation of the stationary and evolutive versions of stochastic differential game models are proposed here and existence and uniqueness properties as well as bounds for the solutions of the discrete schemes are investigated.
Abstract: Mean field type models describing the limiting behavior, as the number of players tends to $+\infty$, of stochastic differential game problems, have been recently introduced by J-M Lasry and P-L Lions [C R Math Acad Sci Paris, 343 (2006), pp 619-625; C R Math Acad Sci Paris, 343 (2006), pp 679-684; Jpn J Math, 2 (2007), pp 229-260] Numerical methods for the approximation of the stationary and evolutive versions of such models are proposed here In particular, existence and uniqueness properties as well as bounds for the solutions of the discrete schemes are investigated Numerical experiments are carried out
347 citations
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TL;DR: In this paper, it is shown that the core of a market coincides with the set of its equilibrium allocations, i.e., allocations which constitute a competitive equilibrium when combined with an appropriate price structure.
Abstract: It is suggested that the most natural mathematical model for a market with "perfect competition" is one in which there is a continuum of traders (like the continuum of points on a line). It is shown that the core of such a market coincides with the set of its "equilibrium allocations", i.e., allocations which constitute a competitive equilibrium when combined with an appropriate price structure.
1,124 citations
TL;DR: On etudie des systemes d'equations aux derivees partielles non lineaires provenant de la theorie des jeux differentiels stochastiques au sens de Nash as discussed by the authors.
Abstract: On etudie des systemes d'equations aux derivees partielles non lineaires provenant de la theorie des jeux differentiels stochastiques au sens de Nash
67 citations
TL;DR: In this paper, the authors studied the convergence of stochastic games with cost functions to the ergodic Bellman equation and showed that the average cost converges with respect to the discount ρ → 0.
Abstract: Stochastic games with cost functionals J ( i ) ρ, x ( v ) = E ∫ ∞ 0 e – ρ t l i ( y, v ) d t , i = 1, 2 with controls v = v 1 , v 2 and state y ( t ) with y (0) = x are considered. Each player wants to minimize his (her) cost functional. E denotes the expected value and the state variables y are coupled with the controls v via a stochastic differential equation with initial value x . The corresponding Bellman system, which is used for the calculation of feedback controls v = v ( y ) and the solvability of the game, leads to a class of diagonal second-order nonlinear elliptic systems, which also occur in other branches of analysis. Their behaviour concerning existence and regularity of solutions is, despite many positive results, not yet well understood, even in the case where the l i , are simple quadratic functions. The objective of this paper is to give new insight to these questions for fixed ρ > 0, and, primarily, to analyse the limiting behaviour as the discount ρ → 0. We find that the modified solutions of the stochastic games converge, for subsequences, to the solution of the so-called ergodic Bellman equation and that the average cost converges. A former restriction of the space dimension has been removed. A reasonable class of quadratic integrands may be treated. More specifically, we consider the Bellman systems of equations – ∆ z + λ = H ( x, Dz ), where the space variable x belongs to a periodic cube (for the sake of simplifying the presentation). They are shown to have smooth solutions. If u ρ is the solution of – ∆ u ρ + ρ u ρ = H ( x, Du ρ ) then the convergence of u ρ — ῡ ρ to z , as ρ tends to 0, is established. The conditions on H are such that some quadratic growth in Du is allowed.
26 citations
TL;DR: In this paper, a theory nouvelle on the formation of the volatilite is proposed, which prend en compte l'influence de la couverture d'options sur la dynamique du prix du sous-jacent.
22 citations