LASRY Jean Michel

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This paper presents a novel approach to characterize the dynamics of the limit spectrum of large random matrices. This approach is based upon the notion we call "spectral dominance". In particular, we show that the limit spectral measure can be determined as the derivative of the unique viscosity solution of a partial integro-differential equation. This also allows to make general and "short" proofs for the convergence problem. We treat the cases of Dyson Brownian motions, Wishart processes and present a general class of models for which this characterization holds.

We propose a (toy) MFG model for the evolution of residents and firms densities, coupled both by labour market equilibrium conditions and competition for land use (congestion). This results in a system of two Hamilton-Jacobi-Bellman and two Fokker-Planck equations with a new form of coupling related to optimal transport. This MFG has a convex potential which enables us to find weak solutions by a variational approach. In the case of quadratic Hamiltonians, the problem can be reformulated in Lagrangian terms and solved numerically by an IPFP/Sinkhorn-like scheme as in [4]. We present numerical results based on this approach, these simulations exhibit different behaviours with either agglomeration or segregation dominating depending on the initial conditions and parameters.

This is a work in progress. The aim is to propose a plausible mechanism for the short term dynamics of the oil market based on the interaction of economic agents. This is a theoretical research which by no means aim at describing all the aspects of the oil market. In particular, we use the tools and terminology of game theory, but we do not claim that this game actually exists in the real world. In parallel, we are currently studying and calibrating a long term model for the oil industry, which addresses the interactions of a monopolists with a competitive fringe of small producers. It is the object of another paper that will be available soon. The present premiminary version does not contain all the economic arguments and all the connections with our long term model. It mostly addresses the description of the model, the equations and numerical simulations focused on the oil industry short term dynamics. A more complete version will be available soon.

We present results of existence, regularity and uniqueness of solutions of the master equation associated with the mean field planning problem in the finite state space case, in the presence of a common noise. The results hold under monotonicity assumptions, which are used crucially in the different proofs of the paper. We also make a link with the trajectories induced by the solution of the master equation and start a discussion on the case of boundary conditions.

This note is concerned with a modeling question arising from the mean field games theory. We show how to model mean field games involving a major player which has a strategic advantage, while only allowing closed loop markovian strategies for all the players. We illustrate this property through three examples.

We study in this paper three aspects of Mean Field Games. The first one is the case when the dynamics of each player depend on the strategies of the other players. The second one concerns the modeling of " noise " in discrete space models and the formulation of the Master Equation in this case. Finally, we show how Mean Field Games reduce to agent based models when the intertemporal preference rate goes to infinity, i.e. when the anticipation of the players vanishes.

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This thesis deals with the study of new medium field game models. We first study optimal stopping and impulse control models in the absence of common noise. We construct for these models a notion of adapted solution for which we prove existence and uniqueness results under natural assumptions. Then, we focus on several properties of mean-field games. We study the limit of these models to pure evolution models when the players' anticipation tends to 0. We show the uniqueness of equilibria for strongly coupled systems (coupled by strategies) under certain assumptions. We then prove some regularity results on a master equation that models a mean field game with common noise in a discrete state space. We then present a generalization of the standard Uzawa algorithm and apply it to the numerical solution of some mean-field game models, in particular optimal stopping or impulse control. Finally, we present a concrete case of mean-field game that comes from problems involving a large number of connected devices in telecommunications.

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We study in this paper three aspects of Mean Field Games. The first one is the case when the dynamics of each player depend on the strategies of the other players. The second one concerns the modeling of " noise " in discrete space models and the formulation of the Master Equation in this case. Finally, we show how Mean Field Games reduce to agent based models when the intertemporal preference rate goes to infinity, i.e. when the anticipation of the players vanishes.

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Mean field game theory was introduced in 2006 by Jean-Michel Lasry and Pierre-Louis Lions. It allows the study of game theory in certain configurations where the number of players is too large to hope for a practical resolution. We study the theory of mean-field games on graphs based on the work of Olivier Guéant which we will extend to more general Hilbertian forms. We will also study the links between K-means and mean-field games, which will in principle allow us to propose new algorithms for K-means using numerical resolution techniques specific to mean-field games. Finally, we will study a mean-field game, namely the "meeting start time" problem by extending it to situations where agents can choose between two meetings. We will study analytically and numerically the existence and multiplicity of solutions of this problem.

A parcimonious long term model is proposed for a mining industry. Knowing the dynamics of the global reserve, the strategy of each production unit consists of an optimal control problem with two controls, first the flux invested into prospection and the building of new extraction facilities, second the production rate. In turn, the dynamics of the global reserve depends on the individual strategies of the producers, so the models leads to an equilibrium, which is described by low dimensional systems of partial differential equations. The dimen-sionality depends on the number of technologies that a mining producer can choose. In some cases, the systems may be reduced to a Hamilton-Jacobi equation which is degenerate at the boundary and whose right hand side may blow up at the boundary. A mathematical analysis is supplied. Then numerical simulations for models with one or two technologies are described. In particular, a numerical calibration of the model in order to fit the historical data is carried out.

The ad-trading desks of media-buying agencies are increasingly relying on complex algorithms for purchasing advertising inventory. In particular, Real-Time Bidding (RTB) algorithms respond to many auctions -- usually Vickrey auctions -- throughout the day for buying ad-inventory with the aim of maximizing one or several key performance indicators (KPI). The optimization problems faced by companies building bidding strategies are new and interesting for the community of applied mathematicians. In this article, we introduce a stochastic optimal control model that addresses the question of the optimal bidding strategy in various realistic contexts: the maximization of the inventory bought with a given amount of cash in the framework of audience strategies, the maximization of the number of conversions/acquisitions with a given amount of cash, etc. In our model, the sequence of auctions is modeled by a Poisson process and the \textit{price to beat} for each auction is modeled by a random variable following almost any probability distribution. We show that the optimal bids are characterized by a Hamilton-Jacobi-Bellman equation, and that almost-closed form solutions can be found by using a fluid limit. Numerical examples are also carried out.

This paper deals with a stochastic order-driven market model with waiting costs, for order books with heterogenous traders. Offer and demand of liquidity drives price formation and traders anticipate future evolutions of the order book. The natural framework we use is mean field game theory, a class of stochastic differential games with a continuum of anonymous players. Several sources of heterogeneity are considered including the mean size of orders. Thus we are able to consider the coexistence of Institutional Investors and High Frequency Traders (HFT). We provide both analytical solutions and numerical experiments. Implications on classical quantities are explored: order book size, prices, and effective bid/ask spread. According to the model, in markets with Institutional Investors only we show the existence of inefficient liquidity imbalances in equilibrium, with two symmetrical situations corresponding to what we call liquidity calls for liquidity. During these situations the transaction price significantly moves away from the fair price. However this macro phenomenon disappears in markets with both Institutional Investors and HFT, although a more precise study shows that the benefits of the new situation go to HFT only, leaving Institutional Investors even with higher trading costs.

In spite of the growing consideration for optimal execution issues in the financial mathematics literature, numerical approximations of optimal trading curves are almost never discussed. In this article, we present a numerical method to approximate the optimal strategy of a trader willing to unwind a large portfolio. The method we propose is very general as it can be applied to multi-asset portfolios with any form of execution costs, including a bid-ask spread component, even when participation constraints are imposed. Our method, based on convex duality, only requires Hamiltonian functions to have C^{1,1} regularity while classical methods require additional regularity and cannot be applied to all cases found in practice.

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The purpose of this article is to get mathematicians interested in studying a number of PDEs that naturally arise in macroeconomics. These PDEs come from models designed to study some of the most important questions in economics. At the same time they are highly interesting for mathematicians because their structure is often quite difficult. We present a number of examples of such PDEs, discuss what is known about their properties, and list some open questions for future research.

The purpose of this article is to get mathematicians interested in studying a number of partial differential equations (PDEs) that naturally arise in macroeconomics. These PDEs come from models designed to study some of the most important questions in economics. At the same time, they are highly interesting for mathematicians because their structure is often quite difficult. We present a number of examples of such PDEs, discuss what is known about their properties, and list some open questions for future research.

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We study the long time average, as the time horizon tends to infinity, of the solution of a mean field game system with a nonlocal coupling. We show an exponential convergence to the solution of the associated stationary ergodic mean field game. Proofs rely on semiconcavity estimates and smoothing properties of the linearized system. The research that led to the present paper was partially supported by a grant of the group GNAMPA of INdA.

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This work consists of two independent parts which consist in solving nonlinear functional differential systems. The first part concerns a dynamic equilibrium problem in a framework of incomplete markets. It highlights the appearance of macroeconomic fluctuations due to the incompleteness of the economy. It is devoted to the resolution of differential systems whose solutions provide an equilibrium for the economic model studied. The search for these solutions comes down to the search for fixed points for increasing applications or fixed points for applications on firm convexes. It requires very technical and specific constructions of the problem in question. The second part concerns an elliptic problem with mixed boundary conditions of the Dirichlet Neumann type, which originates from a model for semiconductors. We study the existence and regularity of a solution of this problem using the leray-schauder degree theory.

We present the different algorithms of robust control and more particularly those leading to the synthesis of the h-infinite control. The robust control problems are put in the form of a so-called standard problem which consists in augmenting the nominal system taking into consideration the required robustness specifications, then minimizing a criterion of a frequency nature of quadratic or infinite type. There are two different approaches to designing the h-infinite controller. Nehari's approach consists in finding the projection of an anti-stable system on the space of stable systems with the h-infinite norm. Glover's approach, also called ultra-modern, consists in solving two Riccati equations and then putting the controller in a descriptor form. The algorithm that is presented and implemented in the thesis is an improvement of Glover's method leading to more robust calculations. The robust control algorithms have been implemented in Basile, a software for automatic control. A library of Basile macros has been realized allowing to solve standard h-infinite problems and more generally, to study the robustness of linear systems. In order to validate the algorithms and to give examples of h-infinite problems, academic and industrial applications (aircraft, satellite) are treated in detail.

In the first part, valuation and risk analysis techniques for floating rate bonds are presented. Actuarial tools are applied to the valuation of illiquid securities. Another technique specific to these assets, based on arbitrage asset pricing models, is also developed. It allows us to measure the impact of a deformation of the yield curve on these products. In the second part, the concepts of the theory of viscosity solutions are applied to the solution of a stochastic optimal control problem with state constraints and unbounded domain.

The first part deals with dynamic equilibrium problems under debt constraints. More precisely, it presents two models, one in a one-country economy, the other in a two-country economy, in which agents are subject to stochastic variations in productivity against which they can only imperfectly insure themselves, because of the presence of debt constraints. The consequences are the appearance, in the first case, of fluctuations in aggregate variables, and in the second case, of correlations between the economic quantities of the two countries, phenomena that would be absent in a complete market. The second part deals with mathematical problems concerning the valuation of contingent assets. We solve two option pricing models that depend on the history of the underlying price. We also present a model of the yield curve under imperfect information.

The first part presents in detail the practical realization of a simple inverse pendulum and the feasibility study of a double inverse pendulum using the linear stochastic control theory. The originality of these pendulums is the choice of a low-end hardware, associated with a rather complex, multi-task and real time assembler programming. The power of the motors being limited, the tests on the simple pendulum show that the difficulty lies in the constraint on the control. After a general study of the feedbacks minimizing the norm of the control, on the set of stable commands, the study of the global stochastic system allows to estimate the minimal acceleration necessary for the double pendulum and to conclude that it is not feasible. The second part presents the theoretical and numerical study of a tracking game, more precisely the solution of the Isaacs equation of this differential game, thanks to the notion of viscosity solution. Moreover, it is a game modeling a pursuit in a given domain, i.e. with constraints on the edge of the domain. The value functions of this game verify the dynamic programming, are continuous, and are viscosity solutions of the same Isaacs equation. This equation with boundary conditions has a unique viscosity solution. The monotonic schemes, with differential form, and consistent with the equation, allow to approximate the solutions. The numerical codes then provide the value function of the set and thus the optimal trajectories for any initial condition.