DELARUE , François

We here gather in a single note several original probabilistic works devoted to the analysis of the C^(1,1) regularity of the solution to the possibly degenerate complex Monge-Ampère equation. The whole analysis relies on a probabilistic writing of the solution as the value function of a stochastic optimal control problem. Such a representation...

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...

Cette note a pour but de donner un aperçu des travaux sur les équations aux dérivées partielles stochastiques singulières, qui ont valu la médaille Fields à Martin Hairer. Nous retraçons plus particulièrement le cheminement suivi par Martin Hairer pour aborder l'équation de Kardar-Parisi-Zhang et élaborer, à partir de là, la théorie plus...

International audience

We analyze a class of nonlinear partial dierential equations (PDEs) defined on the Euclidean space of dimension d times the Wasserstein space of d-dimensional probability measures with a finite second-order moment. We show that such equations admit a classical solutions for sufficiently small time intervals. Under additional constraints, we...

The goal of this paper is to demonstrate that common noise may serve as an exploration noise for learning the solution of a mean field game. This concept is here exemplified through a toy linear-quadratic model, for which a suitable form of common noise has already been proven to restore existence and uniqueness. We here go one step further and...

We here provide two sided bounds for the density of the solution of a system of n differential equations of dimension d, the first one being forced by a non-degenerate random noise and the (n-1) other ones being degenerate. The system formed by the n equations satisfies a suitable Hörmander condition: the second equation feels the noise plugged...

The paper is a continuation of the Kusuoka-Stroock programme of establishing smoothness properties of solutions of (possibly) degenerate partial differential equations by using probabilistic methods. We analyze here a class of semi-linear parabolic partial differential equations for which the linear part is a second order differential operator...

The purpose of this note is to provide an existence result for the solution of fully coupled Forward Backward Stochastic Differential Equations (FBSDEs) of the mean field type. These equations occur in the study of mean field games and the optimal control of dynamics of the McKean Vlasov type.

Motivated by earlier work on the use of fully-coupled Forward-Backward Stochastic Differential Equations (henceforth FBSDEs) in the analysis of mathematical models for the CO2 emissions markets, the present study is concerned with the analysis of these equations when the generator of the forward equation has a conservative degenerate structure...

The purpose of this paper is to provide a detailed probabilistic analysis of the optimal control of nonlinear stochastic dynamical systems of the McKean Vlasov type. Motivated by the recent interest in mean field games, we highlight the connection and the differences between the two sets of problems. We prove a new version of the stochastic...

Motivated by the recent advances in the theory of stochastic partial differential equations involving nonlinear functions of distributions, like the Kardar-Parisi-Zhang (KPZ) equation, we reconsider the unique solvability of one-dimensional stochastic differential equations, the drift of which is a distribution, by means of rough paths theory....

The zero-noise result for Peano phenomena of Bafico and Baldi (1982) is revisited. The original proof was based on explicit solutions to the elliptic equations for probabilities of exit times. The new proof given here is purely dynamical, based on a direct analysis of the SDE and the relative importance of noise and drift terms. The transition...

This two-volume book offers a comprehensive treatment of the probabilistic approach to mean field game models and their applications. The book is self-contained in nature and includes original material and applications with explicit examples throughout, including numerical solutions.Volume II tackles the analysis of mean field games in which...

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The goal of this paper is to provide a selection principle for potential mean field games on a finite state space and, in this respect, to show that equilibria that do not minimize the corresponding mean field control problem should be ruled out. Our strategy is a tailor-made version of the vanishing viscosity method for partial differential...

The purpose of this paper is to provide a complete probabilistic analysis of a large class of stochastic differential games for which the interaction between the players is of mean-field type in the sense that the private state on which a player bases his own strategy depends upon the empirical distribution of the private states of the other...

International audience

In this paper, we address an instance of uniquely solvable mean-field game with a common noise whose corresponding counterpart without common noise has several equilib-ria. We study the selection problem for this mean-field game without common noise via three approaches. A common approach is to select, amongst all the equilibria, those yielding...

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....

We study a sequence of symmetric $n$-player stochastic differential games driven by both idiosyncratic and common sources of noise, in which players interact with each other through their empirical distribution. The unique Nash equilibrium empirical measure of the $n$-player game is known to converge, as $n$ goes to infinity, to the unique...

An analysis of the error of the upwind scheme for transport equation with discontinuous coefficients is provided. We consider here a velocity field that is bounded and one-sided Lipschitz continuous. In this framework, solutions are defined in the sense of measures along the lines of Poupaud and Rascle's work. We study the convergence order of...

Motivated by a cap-and-trade model for the green house gas emissions regulation, we discuss and compare two methods of investigations for the asymptotic regime of stochastic differential games with a finite number of players as the number of players tends to the infinity. These two methods differ in the order in which optimization and passage...

In this paper we prove that the spatially homogeneous Landau equation for Maxwellian molecules can be represented through the product of two elementary processes. The first one is the Brownian motion on the group of rotations. The second one is, conditionally on the first one, a Gaussian process. Using this representation, we establish sharp...

We model the transmission of a message on the complete graph with n vertices and limited resources. The vertices of the graph represent servers that may broadcast the message at random. Each server has a random emission capital that decreases at each emission. Quantities of interest are the number of servers that receive the information before...