ACHDOU Yves

We consider deterministic mean field games in which the agents control their acceleration and are constrained to remain in a domain of R n. We study relaxed equilibria in the Lagrangian setting. they are described by a probability measure on trajectories. The main results of the paper concern the existence of relaxed equilibria under suitable assumptions. The fact that the optimal trajectories of the related optimal control problem solved by the agents do not form a compact set brings a difficulty in the proof of existence. The proof also requires closed graph properties of the map which associates to initial conditions the set of optimal trajectories.

We consider deterministic mean field games in which the agents control their acceleration and are constrained to remain in a domain of R n. We study relaxed equilibria in the Lagrangian setting. they are described by a probability measure on trajectories. The main results of the paper concern the existence of relaxed equilibria under suitable assumptions. The fact that the optimal trajectories of the related optimal control problem solved by the agents do not form a compact set brings a difficulty in the proof of existence. The proof also requires closed graph properties of the map which associates to initial conditions the set of optimal trajectories.

We consider a class of closed loop stochastic optimal control problems in finite time horizon, in which the cost is an expectation conditional on the event that the process has not exited a given bounded domain. An important difficulty is that the probability of the event that conditionates the strategy decays as time grows. The optimality conditions consist of a system of partial differential equations, including a Hamilton-Jacobi-Bellman equation (backward w.r.t. time) and a (forward w.r.t. time) Fokker-Planck equation for the law of the conditioned process. The two equations are supplemented with Dirichlet conditions. Next, we discuss the asymptotic behavior as the time horizon tends to`8. This leads to a new kind of optimal control problem driven by an eigenvalue problem related to a continuity equation with Dirichlet conditions on the boundary. We prove existence for the latter. We also propose numerical methods and supplement the various theoretical aspects with numerical simulations.

No summary available.

This thesis deals with the theory of mean-field games. Most of it is devoted to mean-field games in which players can interact through their state and control law. We will use the terminology mean-field control game to designate such games. First, we make a structural assumption, which essentially consists in saying that the optimal dynamics depends on the control law in a lipschitzian way with a constant less than one. In this case, we prove several existence results for solutions to the mean control field game system, and a uniqueness result in short time. In a second step, we set up a numerical scheme and perform simulations for population motion models. In a third step, we show the existence and uniqueness when the control interaction satisfies a monotonicity condition. The last chapter concerns a numerical solution algorithm for mean-field games of variational type and without interaction via the control law. We use a preconditioning strategy by a multigrid method to obtain a fast convergence.

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.

The theory of mean field games aims at studying deterministic or stochastic differential games (Nash equilibria) as the number of agents tends to infinity. Since very few mean field games have explicit or semi-explicit solutions, numerical simulations play a crucial role in obtaining quantitative information from this class of models. They may lead to systems of evolutive partial differential equations coupling a backward Bellman equation and a forward Fokker-Planck equation. In the present survey, we focus on such systems. The forward-backward structure is an important feature of this system, which makes it necessary to design unusual strategies for mathematical analysis and numerical approximation. In this survey, several aspects of a finite difference method used to approximate the previously mentioned system of PDEs are discussed, including convergence, variational aspects and algorithms for solving the resulting systems of nonlinear equations. Finally, we discuss in details two applications of mean field games to the study of crowd motion and to macroeconomics, a comparison with mean field type control, and present numerical simulations.

No summary available.

No summary available.

In the present work, we study deterministic mean field games (MFGs) with finite time horizon in which the dynamics of a generic agent is controlled by the acceleration. They are described by a system of PDEs coupling a continuity equation for the density of the distribution of states (forward in time) and a Hamilton-Jacobi (HJ) equation for the optimal value of a representative agent (backward in time). The state variable is the pair $(x, v)\in R^N\times R^N$ where x stands for the position and v stands for the velocity. The dynamics is often referred to as the double integrator. In this case, the Hamiltonian of the system is neither strictly convex nor coercive, hence the available results on MFGs cannot be applied. Moreover, we will assume that the Hamiltonian is unbounded w.r.t. the velocity variable v. We prove the existence of a weak solution of the MFG system via a vanishing viscosity method and we characterize the distribution of states as the image of the initial distribution by the flow associated with the optimal control.

This volume provides an introduction to the theory of Mean Field Games, suggested by J.-M. Lasry and P.-L. Lions in 2006 as a mean-field model for Nash equilibria in the strategic interaction of a large number of agents. Besides giving an accessible presentation of the main features of mean-field game theory, the volume offers an overview of recent developments which explore several important directions: from partial differential equations to stochastic analysis, from the calculus of variations to modeling and aspects related to numerical methods. Arising from the CIME Summer School "Mean Field Games" held in Cetraro in 2019, this book collects together lecture notes prepared by Y. Achdou (with M.

This thesis deals with the study of the long time behavior of potential mean field games (MFG), independently of the convexity of the associated minimization problem. For the finite dimensional Hamiltonian system, similar problems have been treated by the weak KAM theory. We transpose many results of this theory to the context of potential mean field games. First, we characterize by ergodic approximation the boundary value associated with finite horizon MFG systems. We provide explicit examples in which this value is strictly greater than the energy level of the stationary solutions of the ergodic MFG system. This implies that the optimal trajectories of finite horizon MFG systems cannot converge to stationary configurations. Then, we prove the convergence of the minimization problem associated with finite horizon MFG to a solution of the critical Hamilton-Jacobi equation in the space of probability measures. Moreover, we show a mean field limit for the ergodic constant associated to the corresponding finite dimensional Hamilton-Jacobi equation. In the last part, we characterize the limit of the infinite horizon minimization problem that we used for the ergodic approximation in the first part of the manuscript.

Mean-field game systems (MFG) describe equilibrium configurations in differential games with an infinite number of infinitesimal agents. This thesis is structured around three different contributions to the theory of mean-field games. The main goal is to explore applications and extensions of this theory, and to propose new approaches and ideas to deal with the underlying mathematical issues. The first chapter first introduces the key concepts and ideas that we use throughout the thesis. We introduce the MFG problem and briefly explain the asymptotic connection with N-player differential games when N → ∞. We then present our main results and contributions. Chapter 2 explores an MFG model with a non-anticipatory interaction mode (myopic players). Unlike classical MFG models, we consider less rational agents who do not anticipate the evolution of the environment, but only observe the current state of the system, undergo changes, and take actions accordingly. We analyze the coupled PDE system resulting from this model, and establish the rigorous link with the corresponding N-Players game. We show that the population of agents can self-organize through a self-correcting process and converge exponentially fast to a well-known MFG equilibrium configuration. Chapters 3 and 4 concern the application of the MFG theory to the modeling of production and marketing processes of products with exhaustible resources (e.g. fossil fuels). In Chapter 3, we propose a variational approach for the study of the corresponding MFG system and analyze the deterministic limit (without demand fluctuations) in a regime where resources are renewable or abundant. In Chapter 4 we treat the MFG approximation by analyzing the asymptotic link between the N-player Cournot model and the MFG Cournot model when N is large. Finally, Chapter 5 considers an MFG model for the optimal execution of a portfolio of assets in a financial market. We explain our MFG model and analyze the resulting PDE system, then we propose a numerical method to compute the optimal execution strategy for an agent given its initial inventory, and present several simulations. Furthermore, we analyze the influence of trading activity on the intraday variation of the covariance matrix of asset returns. Next, we verify our findings and calibrate our model using historical trading data for a pool of 176 US stocks.

In the present work, we study deterministic mean field games (MFGs) with finite time horizon in which the dynamics of a generic agent is controlled by the acceleration. They are described by a system of PDEs coupling a continuity equation for the density of the distribution of states (forward in time) and a Hamilton-Jacobi (HJ) equation for the optimal value of a representative agent (backward in time). The state variable is the pair $(x, v)\in R^N\times R^N$ where x stands for the position and v stands for the velocity. The dynamics is often referred to as the double integrator. In this case, the Hamiltonian of the system is neither strictly convex nor coercive, hence the available results on MFGs cannot be applied. Moreover, we will assume that the Hamiltonian is unbounded w.r.t. the velocity variable v. We prove the existence of a weak solution of the MFG system via a vanishing viscosity method and we characterize the distribution of states as the image of the initial distribution by the flow associated with the optimal control.

We consider a class of closed loop stochastic optimal control problems in finite time horizon, in which the cost is an expectation conditional on the event that the process has not exited a given bounded domain. An important difficulty is that the probability of the event that conditionates the strategy decays as time grows. The optimality conditions consist of a system of partial differential equations, including a Hamilton-Jacobi-Bellman equation (backward w.r.t. time) and a (forward w.r.t. time) Fokker-Planck equation for the law of the conditioned process. The two equations are supplemented with Dirichlet conditions. Next, we discuss the asymptotic behavior as the time horizon tends to`8. This leads to a new kind of optimal control problem driven by an eigenvalue problem related to a continuity equation with Dirichlet conditions on the boundary. We prove existence for the latter. We also propose numerical methods and supplement the various theoretical aspects with numerical simulations.

We consider finite horizon stochastic mean field games in which the state space is a network. They are described by a system coupling a backward in time Hamilton-Jacobi-Bellman equation and a forward in time Fokker-Planck equation. The value function u is continuous and satisfies general Kirchhoff conditions at the vertices. The density m of the distribution of states satisfies dual transmission conditions: in particular, m is generally discontinuous across the vertices, and the values of m on each side of the vertices satisfy special compatibility conditions. The stress is put on the case when the Hamiltonian is Lipschitz continuous.

We consider a family of optimal control problems in the plane with dynamics and running costs possibly discontinuous across a two-scale oscillatory interface. Typically, the amplitude of the oscillations is of the order of ε while the period is of the order of ε 2. As ε → 0, the interfaces tend to a straight line Γ. We study the asymptotic behavior of the value function as ε → 0. We prove that the value function tends to the solution of Hamilton-Jacobi equations in the two half-planes limited by Γ, with an effective transmission condition on Γ keeping track of the oscillations.

We consider stochastic mean field games for which the state space is a network. In the ergodic case, they are described by a system coupling a Hamilton-Jacobi-Bellman equation and a Fokker-Planck equation, whose unknowns are the invariant measure m, a value function u, and the ergodic constant ρ. The function u is continuous and satisfies general Kirchhoff conditions at the vertices. The invariant measure m satisfies dual transmission conditions: in particular, m is discontinuous across the vertices in general, and the values of m on each side of the vertices satisfy special compatibility conditions.

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.

Mean-field games (MFG) are a class of differential games in which each agent is infinitesimal and interacts with a huge population of agents. In this thesis, we raise the question of the actual formation of the MFG equilibrium. Indeed, since the game is very complex, it is unrealistic to assume that agents can actually compute the equilibrium configuration. This suggests that if the equilibrium configuration arises, it is because the agents have learned to play the game. Thus, the main question is to find learning procedures in mean-field games and to analyze their convergences to an equilibrium. We were inspired by learning schemes in static games and tried to apply them to our dynamic MFG model. We focus particularly on applications of fictitious play and online mirror descent on different types of mean field games: Potential, Monotonic or Discrete.

No summary available.

We consider a class of systems of time dependent partial differential equations which arise in mean field type models with congestion. The systems couple a backward viscous Hamilton-Jacobi equation and a forward Kolmogorov equation both posed in (0, T) × (R N /Z N). Because of congestion and by contrast with simpler cases, the latter system can never be seen as the optimality conditions of an optimal control problem driven by a partial differential equation. The Hamiltonian vanishes as the density tends to +∞ and may not even be defined in the regions where the density is zero. After giving a suitable definition of weak solutions, we prove the existence and uniqueness results of the latter under rather general assumptions. No restriction is made on the horizon T .

This thesis deals with the study of Hamilton-Jacobi-Bellman equations associated with optimal control and mean-field games problems with the particularity that we place ourselves on a network (i.e., sets consisting of edges connected by junctions) in both problems, for which we allow different dynamics and different costs in each edge of a network. In the first part of this thesis, we consider an optimal control problem on networks in the spirit of the work of Achdou, Camilli, Cutrì & Tchou (2013) and Imbert, Moneau & Zidani (2013). The main novelty is that we add input (or output) costs to the vertices of the network leading to a possible discontinuity of the value function. This is characterized as the unique viscosity solution of a Hamilton-Jacobi equation for which a suitable join condition is established. The uniqueness is a consequence of a comparison principle for which we give two different proofs, one with arguments from optimal control theory, inspired by Achdou, Oudet & Tchou (2015) and the other based on partial differential equations, after Lions & Souganidis (2017). The second part concerns stochastic mean-field games on networks. In the ergodic case, they are described by a system coupling a Hamilton-Jacobi-Bellman equation and a Fokker- Planck equation, whose unknowns are the density m of the invariant measure that represents the distribution of players, the value function v that comes from an "average" optimal control problem, and the ergodic constant ρ. The value function v is continuous and satisfies very general Kirchhoff conditions at vertices in our problem. The function m satisfies two transmission conditions at vertices. In particular, due to the generality of Kirchhoff conditions, m is in general discontinuous at the vertices. The existence and uniqueness of a weak solution are proved for subquadratic Hamiltonians and very general assumptions on the coupling. Finally, in the last part, we study non-stationary stochastic mean field games on networks. The transition conditions for the value function v and the density m are similar to those given in the second part. Again, we prove the existence and uniqueness of a weak solution for sublinear Hamiltonians and couplings and in the case of a lower bounded regularizing nonlocal coupling. The main additional difficulty compared to the stationary case, which imposes more restrictive assumptions, is to establish the regularity of the solutions of the system on a lattice. Our approach consists in studying the solution of the Hamilton-Jacobi equation derived to gain regularity on the solution of the initial equation.

We consider a class of systems of time dependent partial differential equations which arise in mean field type models with congestion. The systems couple a backward viscous Hamilton-Jacobi equation and a forward Kolmogorov equation both posed in (0, T) × (R N /Z N). Because of congestion and by contrast with simpler cases, the latter system can never be seen as the optimality conditions of an optimal control problem driven by a partial differential equation. The Hamiltonian vanishes as the density tends to +∞ and may not even be defined in the regions where the density is zero. After giving a suitable definition of weak solutions, we prove the existence and uniqueness results of the latter under rather general assumptions. No restriction is made on the horizon T .

This paper introduces and analyzes some models in the framework of Mean Field Games describing interactions between two populations motivated by the studies on urban settlements and residential choice by Thomas Schelling. For static games, a large population limit is proved. For differential games with noise, the existence of solutions is established for the systems of partial differential equations of Mean Field Game theory, in the stationary and in the evolutive case. Numerical methods are proposed, with several simulations. In the examples and in the numerical results, particular emphasis is put on the phenomenon of segregation between the populations.

This thesis is composed of two parts in which we study on the one hand error estimates for numerical schemes associated with first order Hamilton-Jacobi equations. On the other hand, we study the stability and the exact indirect boundary controllability of coupled wave equations. First, using the Crandall-Lions technique, we establish an error estimate of a monotone finite difference numerical scheme for flux-limited boundary conditions, for a first order Hamilton-Jacobi equation. Then, we show that this numerical scheme can be generalized to general junction conditions. We then establish the convergence of the discretized solution to the viscosity solution of the continuous problem. Finally, we propose a new approach, à la Crandall-Lions, to improve the error estimates already obtained, for a class of well chosen Hamiltonians. This approach relies on the interpretation of the optimal control type of the considered Hamilton-Jacobi equation.In a second step, we study the stabilization and the exact indirect boundary controllability of a one-dimensional system of coupled wave equations. First, we consider the case of a coupling with velocity terms, and by a spectral method, we show that the system is exactly controllable with a single boundary control. The results depend on the arithmetic nature of the quotient of the propagation velocities and on the algebraic nature of the coupling term. Moreover, they are optimal. Next, we consider the case of zero-order coupling and establish an optimal polynomial rate of the energy decay. Finally, we show that the system is exactly controllable with a single boundary control.

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.

This work deals with a numerical method for solving a mean-field type control problem with congestion. It is the continuation of an article by the same authors, in which suitably defined weak solutions of the system of partial differential equations arising from the model were discussed and existence and uniqueness were proved. Here, the focus is put on numerical methods: a monotone finite difference scheme is proposed and shown to have a variational interpretation. Then an Alternating Direction Method of Multipliers for solving the variational problem is addressed. It is based on an augmented Lagrangian. Two kinds of boundary conditions are considered: periodic conditions and more realistic boundary conditions associated to state constrained problems. Various test cases and numerical results are presented.

We analyze some systems of partial differential equations arising in the theory of mean field type control with congestion effects. We look for weak solutions. Our main result is the existence and uniqueness of suitably defined weak solutions, which are characterized as the optima of two optimal control problems in duality.

This work deals with a numerical method for solving a mean-field type control problem with congestion. It is the continuation of an article by the same authors, in which suitably defined weak solutions of the system of partial differential equations arising from the model were discussed and existence and uniqueness were proved. Here, the focus is put on numerical methods: a monotone finite difference scheme is proposed and shown to have a variational interpretation. Then an Alternating Direction Method of Multipliers for solving the variational problem is addressed. It is based on an augmented Lagrangian. Two kinds of boundary conditions are considered: periodic conditions and more realistic boundary conditions associated to state constrained problems. Various test cases and numerical results are presented.

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.

eWe consider a family of optimal control problems in the plane with dynamics and running costs possibly discontinuous across an oscillatory interface $\Gamma_\epsilon$. The oscillations of the interface have small period and amplitude, both of the order of $\epsilon$, and the interfaces $\Gamma_\epsilon$ tend to a straight line $\Gamma$. We study the asymptotic behavior as $\epsilon\to 0$. We prove that the value function tends to the solution of Hamilton-Jacobi equations in the two half-planes limited by $\Gamma$, with an effective transmission condition on $\Gamma$ keeping track of the oscillations of $\Gamma_\epsilon$.

In this work, we study the homogenization of some front propagation problems in stationary and ergodic environments. In the first part, we study the stochastic homogenization of some non-local front propagation problems. In particular, a non-local version of the perturbed Evans test function method is given. The second part is devoted to the numerical approximation of the effective Hamiltonian which follows from the stochastic homogenization of the Hamilton-Jacobi equations. Error estimates between the numerical solutions and the effective Hamiltonian are established. In the third part, we are interested in the stochastic homogenization of problems of propagating fronts which evolve in the normal direction with a speed which can be unbounded. We show homogenization results in the case of i.i.d. media.

We consider a transmission problem in which the interior domain has infinitely ramified structures. Transmission between the interior and exterior domains occurs only at the fractal component of the interface between the interior and exterior domains. We also consider the sequence of the transmission problems in which the interior domain is obtained by stopping the self-similar construction after a finite number of steps. the transmission condition is then posed on a prefractal approximation of the fractal interface. We prove the convergence in the sense of Mosco of the energy forms associated with these problems to the energy form of the limit problem. In particular, this implies the convergence of the solutions of the approximated problems to the solution of the problem with fractal interface. The proof relies in particular on an extension property. Emphasis is put on the geometry of the ramified domain. The convergence result is obtained when the fractal interface has no self-contact, and in a particular geometry with self-contacts, for which an extension result is proved.

This paper introduces and analyzes some models in the framework of Mean Field Games describing interactions between two populations motivated by the studies on urban settlements and residential choice by Thomas Schelling. For static games, a large population limit is proved. For differential games with noise, the existence of solutions is established for the systems of partial differential equations of Mean Field Game theory, in the stationary and in the evolutive case. Numerical methods are proposed, with several simulations. In the examples and in the numerical results, particular emphasis is put on the phenomenon of segregation between the populations.

This thesis deals with the modeling, analysis and numerical analysis of nonlinear and nonlocal partial differential equations with applications to road traffic. Road traffic can be modeled at different scales. In particular, one can consider the microscopic scale which describes the dynamics of each vehicle individually and the macroscopic scale which sees traffic as a fluid and describes traffic using macroscopic quantities such as vehicle density and average speed. In this thesis, using viscosity solution theory, we make the transition from microscopic to macroscopic models. The interest of this switch is that microscopic models are more intuitive and easy to handle to simulate particular situations (junctions, traffic lights,.) but they are not adapted to large simulations (to simulate the traffic in a whole city for example). On the contrary, macroscopic models are less easy to modify (to simulate a particular situation) but they can be used for large scale simulations. The idea is therefore to find the macroscopic model equivalent to a microscopic model that describes a specific scenario (a junction, a fork, different types of drivers, a school zone,.). The first part of this thesis contains a homogenization and numerical homogenization result for a microscopic model with different types of drivers. In a second part, homogenization and numerical homogenization results are obtained for microscopic models with a local disturbance (speed bump, school zone,.). Finally, we present a homogenization result in the context of a bifurcation.

The authors review some important aspects of finance modeling involving partial differential equations and focus on numerical algorithms for the fast and accurate pricing of financial derivatives and for the calibration of parameters. This book explores the best numerical algorithms and discusses them in depth, from their mathematical analysis up to their implementation in C++ with efficient numerical libraries.

We analyze some systems of partial differential equations arising in the theory of mean field type control with congestion effects. We look for weak solutions. Our main result is the existence and uniqueness of suitably defined weak solutions, which are characterized as the optima of two optimal control problems in duality.

Mean field type models describing the limiting behavior of stochastic differential games as the number of players tends to +∞, have been recently introduced by J-M. Lasry and P-L. Lions. Under suitable assumptions, they lead to a system of two coupled partial differential equations, a forward Bellman equation and a backward Fokker-Planck equations. Finite difference schemes for the approximation of such systems have been proposed in previous works. Here, we prove the convergence of these schemes towards a weak solution of the system of partial differential equations.

We consider continuous-state and continuous-time control problems where the admissible trajectories of the system are constrained to remain on a network. A notion of viscosity solution of Hamilton-Jacobi equations on the network has been proposed in earlier articles. Here, we propose a simple proof of a comparison principle based on arguments from the theory of optimal control. We also discuss stability of viscosity solutions.

The work of this thesis focused on the development of numerical techniques for the homogenization of equations whose coefficients exhibit random small-scale heterogeneities. The difficulties related to the solution of such partial differential equations can be solved thanks to the theory of stochastic homogenization. The solution of an equation with random and oscillating coefficients at the finest scale of the problem is then substituted by the solution of an equation with constant coefficients. However, a difficulty remains: the calculation of these so-called homogenized coefficients are defined by an ergodic mean, which cannot be reached in practice. Only random approximations of these deterministic quantities can be computed, and the error committed during the approximation is important. These issues are developed in detail in Chapter 1 which serves as an introduction. The purpose of Chapter 2 of this thesis is to reduce the error of this approximation in a nonlinear case, by reducing the variance of the estimator by the method of antithetic variables. In Chapter 3, we show how to obtain a better variance reduction by the method of control variables. This approach is based on an approximate model, available in the case studied. It is more invasive and less generic, and we study it in a linear case. In Chapter 4, again in a linear case, we introduce a selection method to reduce the error. Finally, Chapter 5 deals with the analysis of an in-pour problem, where we search for parameters at the finest scale, knowing only a few macroscopic quantities, for example the homogenized coefficients of the model.

We consider a family of open star-shaped domains made of a finite number of non intersecting semi-infinite strips of small thickness and of a central region whose diameter is of the same order of thickness, that may be called the junction. When the thickness tends to 0, the domains tend to a union of half-lines sharing an endpoint. This set is termed "network". We study infinite horizon optimal control problems in which the state is constrained to remain in the star-shaped domains. In the above mentioned strips the running cost may have a fast variation w.r.t. the transverse coordinate. When the thickness tends to 0 we prove that the value function tends to the solution of a Hamilton-Jacobi equation on the network, which may also be related to an optimal control problem. One difficulty is to find the transmission condition at the junction node in the limit problem. For passing to the limit, we use the method of the perturbed test-functions of Evans, which requires constructing suitable correctors. This is another difficulty since the domain is unbounded.

We consider a family of optimal control problems in the plane with dynamics and running costs possibly discontinuous across an oscillatory interface Γ ε. The oscillations of the interface have small period and amplitude, both of the order of ε, and the interfaces Γ ε tend to a straight line Γ. We study the asymptotic behavior as ε → 0. We prove that the value function tends to the solution of Hamilton-Jacobi equations in the two half-planes limited by Γ, with an effective transmission condition on Γ keeping track of the oscillations of Γ ε .

This thesis deals with the study of optimal control problems on networks (i.e., sets of subregions connected by junctions), for which different dynamics and different instantaneous costs are allowed in each subregion of the network. As in the more classical cases, one would like to be able to characterize the value function of such a control problem through a Hamilton-Jacobi-Bellman equation. However, the geometrical singularities of the domain, as well as the discontinuities of the data, do not allow us to apply the classical theory of viscosity solutions. In the first part of this thesis we prove that the value functions of optimal control problems defined on 1-dimensional networks are characterized by such equations. In the second part the previous results are extended to the case of control problems defined on a 2-dimensional junction. Finally, in a last part, we use the previous results to treat a singular perturbation problem involving in-plane optimal control problems for which the dynamics and instantaneous costs can be discontinuous across an oscillating boundary.

We discuss the system of Fokker-Planck and Hamilton-Jacobi-Bellman equations arising from the finite horizon control of McKean-Vlasov dynamics.

We discuss the system of Fokker-Planck and Hamilton-Jacobi-Bellman equations arising from the finite horizon control of McKean-Vlasov dynamics.

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.

In the present article, we study the numerical approximation of a system of Hamilton-Jacobi and transport equations arising in geometrical optics. We consider a semi-Lagrangian scheme. We prove the well posedness of the discrete problem and the convergence of the approximated solution toward the viscosity-measure valued solution of the exact problem.

Hamilton-Jacobi equations on networks as limits of singularly perturbed problems.

This thesis addresses two topics: (i) The use of a new numerical method for the valuation of options on a basket of assets, (ii) Liquidity risk, order book modeling and market microstructure. First topic: A greedy algorithm and its applications to solve partial differential equations. The typical example in finance is the valuation of an option on a basket of assets, which can be obtained by solving the Black-Scholes PDE having as dimension the number of assets considered. We propose to study an algorithm that has been proposed and studied recently in [ACKM06, BLM09] to solve high dimensional problems and try to circumvent the curse of dimension. The idea is to represent the solution as a sum of tensor products and to iteratively compute the terms of this sum using a gluttonous algorithm. The solution of PDEs in high dimension is strongly related to the representation of functions in high dimension. In Chapter 1, we describe different approaches to represent high-dimensional functions and introduce the high-dimensional problems in finance that are addressed in this thesis work. The method selected in this manuscript is a nonlinear approximation method called Proper Generalized Decomposition (PGD). Chapter 2 shows the application of this method for the approximation of the solution of a linear PDE (the Poisson problem) and for the approximation of an integrable square function by a sum of tensor products. A numerical study of the latter problem is presented in Chapter 3. The Poisson problem and the approximation of an integrable square function will be used as a basis in Chapter 4 to solve the Black-Scholes equation using the PGD approach. In numerical examples, we have obtained results up to dimension 10. In addition to approximating the solution of the Black-Scholes equation, we propose a variance reduction method of classical Monte Carlo methods for pricing financial options. Second topic: Liquidity risk, order book modeling, market microstructure. Liquidity risk and market microstructure have become very important topics in financial mathematics. The deregulation of financial markets and the competition between them to attract more investors is one of the possible reasons. In this work, we study how to use this information to optimally execute the sale or purchase of orders. Orders can only be placed in a price grid. At each moment, the number of pending buy (or sell) orders for each price is recorded. In [AFS10], Alfonsi, Fruth and Schied proposed a simple model of the order book. In this model, it is possible to explicitly find the optimal strategy to buy (or sell) a given quantity of shares before a maturity. The idea is to split the buy (or sell) order into other smaller orders in order to find the balance between the acquisition of new orders and their price. This thesis work focuses on an extension of the order book model introduced by Alfonsi, Fruth and Schied. Here, the originality is to allow the depth of the order book to depend on time, which is a new feature of the order book that has been illustrated by [JJ88, GM92, HH95, KW96]. In this framework, we solve the optimal execution problem for discrete and continuous strategies. This gives us, in particular, sufficient conditions to exclude price manipulation in the sense of Huberman and Stanzl [HS04] or Transaction-Triggered Price Manipulation (see Alfonsi, Schied and Slynko).

No summary available.

This thesis is devoted to analytical issues upstream of the modeling of tree-like structures, such as the human lung. In particular, we focus our interest on a class of branched domains of the plane, whose boundary has a self-similar fractal part. We start with a study of function spaces in this class of domains. We first study the Sobolev regularity of the trace on the fractal part of the boundary of functions belonging to Sobolev spaces in the considered domains. We then study the existence of extension operators on the class of branched domains. We finally compare the notion of self-similar trace on the fractal part of the boundary with more classical definitions of trace. Finally, we consider a mixed transmission problem between the branched domain and the outer domain. The interface of the problem is the fractal part of the domain edge. We propose here a numerical approach, by approximating the fractal interface by a prefractal interface. The strategy proposed here is based on the coupling of a self-similar method for solving the inner problem and an integral method for solving the outer problem.

No summary available.

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. Numerical methods for the approximation of the stationary and evolutive versions of such models have been proposed by the authors in previous works . Convergence theorems for these methods are proved under various assumptions.

We consider a class of ramified bidimensional domains with a self-similar boundary, which is supplied with the self-similar probability measure. Emphasis is put on the case when the domain is not an epsilon-delta domain as defined by Jones and the fractal is not totally disconnected.We compare two notions of trace on the fractal boundary for functions in some Sobolev space, the classical one ( the strict definition ) and another one proposed in 2007 and heavily relying on self-similarity. We prove that the two traces coincide almost everywhere with respect to the self similar probability measure.

We consider continuous-state and continuous-time control problems where the admissible trajectories of the system are constrained to remain on a network. In our setting, the value function is continuous. We de ne a notion of constrained viscosity solution of Hamilton-Jacobi equations on the network and we study related comparison principles. Under suitable assumptions, we prove in particular that the value function is the unique constrained viscosity solution of the Hamilton-Jacobi equation on the network.

These Lecture Notes contain the material relative to the courses given at the CIME summer school held in Cetraro, Italy from August 29 to September 3, 2011. The topic was "Hamilton-Jacobi Equations: Approximations, Numerical Analysis and Applications". The courses dealt mostly with the following subjects: first order and second order Hamilton-Jacobi-Bellman equations, properties of viscosity solutions, asymptotic behaviors, mean field games, approximation and numerical methods, idempotent analysis. The content of the courses ranged from an introduction to viscosity solutions to quite advanced topics, at the cutting edge of research in the field. We believe that they opened perspectives on new and delicate issues. These lecture notes contain four contributions by Yves Achdou (Finite Difference Methods for Mean Field Games), Guy Barles (An Introduction to the Theory of Viscosity Solutions for First-order Hamilton-Jacobi Equations and Applications), Hitoshi Ishii (A Short Introduction to Viscosity Solutions and the Large Time Behavior of Solutions of Hamilton-Jacobi Equations) and Grigory Litvinov (Idempotent/Tropical Analysis, the Hamilton-Jacobi and Bellman Equations).

This thesis gathers several works related to the numerical solution of partial differential equations and integro-differential equations resulting from the stochastic modeling of financial products. The first part of the work is devoted to Sparse Grid methods applied to the numerical solution of equations in dimension greater than three. Two types of problems are addressed. The first one concerns the valuation of vanilla options in a jump model with multi-factor stochastic volatility. The numerical solution of the valuation equation, posed in dimension, is obtained using a sparse finite difference method and a collocation method for the discretization of the integral operator. The second problem deals with the valuation of products on a basket of several underlyings. It requires the use of a Galerkin method on a wavelet basis obtained with a sparse tensor product. The second part of the work concerns a posteriori error estimates for American options on a basket of several assets.

No summary available.

The first part of the thesis is devoted to the study of the reflection of electromagnetic waves by periodical structures arranged on various tones. We are first interested in the case where the wavelength of electromagnetic phenomena is of the same order as the period of the structure. We consider the problem from the point of view of shape optimization. Concretely, we try to optimize the interface between two dielectric layers in a polar cell, in order to increase its efficiency. This study is mainly numerical, and we consider the original problem and the relaxed problem, where this time we try to optimize a mixing coefficient. We are then interested in the case where the wavelength is greater than the size of the period. We can then use asymptotic expansion techniques to find an equivalent boundary condition, i. e. The second part of the paper is devoted to the analysis of the problem of the period structure. The second part is devoted to the numerical study of the boundary layer for incompressible viscous fluid flows with large reynolds numbers. Finally, we consider a mixed finite element method where the current function, which does not develop boundary layers, is discretized on a much coarser grid than the vorticity which has fast variations.