arXiv subjects

Yves Achdou

Publications and source records attributed to Yves Achdou.

At least 19 recordsLinked to original sources

arXiv2026

Mastering Stochastic OLG Models in Continuous Time

We propose a comprehensive framework for solving overlapping-generations (OLG) models in continuous time with both idiosyncratic and aggregate risk. Our general characterization of equilibrium through the master equation operates on the joint distribution over the continuous idiosyncratic states, age and wealth. Our computational strategy is to take a finite-dimensional representation of this distribution as an input of a neural net which in turn outputs a finite-difference representation of the (conditional) value function. This idea can be applied generally to heterogeneous agent models with aggregate risk, and we call it finite-difference neural operator. Our method combines advantages from modern neural nets and traditional finite-difference methods: It is grid-free in the high-dimensional distribution, and retains control on boundary conditions in low-dimensional state variables. Moreover, our method is able to enforce shape constraints. We showcase its flexibility by solving a continuous-time OLG model with aggregate risk alone where we characterize the distribution by its supporting function; and to an OLG model with both types of risk.

econ.GN↗

arXiv2026

Finite difference methods for a continuous-time heterogeneous agent model with recursive utility

We propose, analyze and test computational methods for solving a continuous-time heterogenous agent model with Epstein-Zin utility. Such recursive utilities allow the model to disentangle between risk aversion and intertemporal substitution. Having discretized the Hamilton-Jacobi-Bellman (HJB) equation arising in the model, we propose a Howard-Newton algorithm for the late resolution preference case, and a Howard-Tarski-Kantorovich algorithm for the early resolution preference case. We prove the convergence of the iterative algorithms. We obtain as a consequence the existence of solutions to the discretized HJB equations. In the late resolution case, we supply a priori estimates between the unique solutions of the continuous and discretized HJB equations.

math.OC↗

arXiv2026

Continuous-Time Heterogeneous Agent Models with Recursive Utility and Preference for Late Resolution

We consider continuous-time heterogeneous agent models with recursive utility (Epstein-Zin utility) cast as mean field games, in which agents prefer late resolution of uncertainty. The model leads to a system coupling a pair of Hamilton-Jacobi-Bellman equations with state constraints and Fokker-Planck-Kolmogorov equations. We investigate the existence of solutions to the mean field game system and discuss some important qualitative features of the model.

math.OC↗

arXiv2025

Deterministic Mean Field Games on Networks and Related Optimal Control Problems

We study a class of deterministic mean field games and related optimal control problems, with a finite time horizon and in which the state space is a network. An agent controls her velocity, and, when she occupies a vertex, she can either remain still or enter any adjacent edge. The running and terminal costs are assumed to be continuous in each edge, but may jump at the vertices. Compared to the companion paper [4], we make more general assumptions about the costs and consider networks with an arbitrary number of vertices; this higher degree of generality brings new difficulties. For the optimal control problems mentioned above, we obtain in particular the existence of optimal trajectories and regularity results concerning the optimal trajectories and the value function. These control theoretic results make it possible to address a class of mean field games on networks, with costs that do not depend separately on the control and on the distribution of states, and that are non-local with respect to the latter. Focusing on a Lagrangian formulation, we obtain the existence of relaxed equilibria consisting of probability measures on admissible trajectories. To any relaxed equilibrium corresponds a mild solution, i.e. a pair $(u, m)$ made of the value function $u$ of a related optimal control problem and a family $m = (m(t))_t$ of probability measures on the network. Given $m$, the value function $u$ is a viscosity solution of a Hamilton-Jacobi problem on the network. We then investigate the regularity properties of $u$ and a weak form of a Fokker-Planck equation satisfied by $m$.

math.OC↗

arXiv2025

State Constrained Optimal Control Problems With Control On The Acceleration. Applications To Kinetic Mean Field Games

Relying on the careful study of a related problem in the calculus of variations, we study a class of optimal control problems in which the control lies on the acceleration, with state constraints on the position variable. In dimension one, we find explicit formulas in the special case when the running cost is a power of the acceleration (in absolute value) and the terminal cost is zero. For more general costs or/and higher dimensions, we study the singularities of the value function. We also prove the closedness (in the C 1 topology) of the graph of the multivalued mapping which maps a point in the state space to the set of optimal trajectories which start from this point. A consequence of the latter is the existence, under general assumptions, of relaxed equilibria for a class of kinetic mean field games with state constraints.

math.OC↗

arXiv2024

Homogenization of Hamilton-Jacobi equations with defects leading to stratified problems

We study homogenization of a class of bidimensional stationary Hamilton-Jacobi equations where the Hamiltonian is obtained by perturbing near a half-line of the state space a Hamiltonian that either does not have fast variations with respect to the state variable, or depends on the latter in a periodic manner. We prove that the limiting problem belongs to the class of stratified problems introduced by A. Bressan and Y. Hong and later studied by G. Barles and E. Chasseigne. The related Whitney stratification is made of a submanifold of dimension zero, namely the endpoint of the half-line, a submanifold of dimension one, the open half-line, and the complement of the latter two sets which is a submanifold of dimension two. The limiting problem involves effective Hamiltonians that are associated to the above mentioned three submanifolds and keep track of the perturbation. Another example in which the Hamiltonian is perturbed in a tubular neighborhood of a line is studied.

math.AP↗

arXiv2022

Homogenization of some periodic Hamilton-Jacobi equations with defects

We study homogenization for a class of stationnary Hamilton-Jacobi equations in which the Hamiltonian is obtained by perturbing near the origin an otherwise periodic Hamiltonian. We prove that the limiting problem consists of a Hamilton-Jacobi equation outside the origin, with the same effective Hamiltonian as in periodic homogenization, supplemented at the origin with an effective Dirichlet condition that keeps track of the perturbation. Various comments and extensions are discussed.

math.AP↗

arXiv2022

Simulating numerically the Krusell-Smith model with neural networks

The celebrated Krusel-Smith growth model is an important example of a Mean Field Game with a common noise. The Mean Field Game is encoded in the master equation, a partial differential equation satisfied by the value of the game which depends on the whole distribution of states. The latter equation is therefore posed in an infinite dimensional space. This makes the numerical simulations quite challenging. However, Krusell and Smith conjectured that the value function of the game mostly depends on the state distribution through low dimensional quantities. In this paper, we wish to propose a numerical method for approximating the solutions of the master equation arising in Krusell-Smith model, and for adaptively identifying low-dimensional variables which retain an important part of the information. This new numerical framework is based on a semi-Lagrangian method and uses neural networks as an important ingredient.

math.AP↗

arXiv2022

A simple city equilibrium model with an application to teleworking

We propose a simple semi-discrete spatial model where rents, wages and the density of population in a city can be deduced from free-mobility and equilibrium conditions on the labour and residential housing markets. We prove existence and (under stronger assumptions) uniqueness of the equilibrium. We extend our model to the case where teleworking is introduced. We present numerical simulations which shed light on the effect of teleworking on the structure of the city at equilibrium.

math.OC↗

arXiv2022

First order Mean Field Games on networks

This paper is devoted to finite horizon deterministic mean field games in which the state space is a network. The agents control their velocity, and when they occupy a vertex, they can enter into any incident edge. The running and terminal costs are assumed to be continuous in each edge but not necessarily globally continuous on the network. A Lagrangian formulation is proposed and studied. It leads to relaxed equilibria consisting of probability measures on admissible trajectories. The existence of such relaxed equilibria is obtained. The proof requires the existence of optimal trajectories and a closed graph property for the map which associates to each point the set of optimal trajectories starting from that point. To any relaxed equilibrium corresponds a mild solution of the mean field game, i.e. a pair $(u,m)$ made of the value function $u$ of a related optimal control problem, and a family $m= (m(t))_t$ of probability measures on the network. Given $m$, the value function $u$ is characterized by a Hamilton-Jacobi problem on the network. Regularity properties of $u$ and a weak form of a Fokker-Planck equation satisfied by $m$ are investigated.

math.OC↗

arXiv2022

A mean field model for the interactions between firms on the markets of their inputs

We consider an economy made of competing firms which are heterogeneous in their capital and use several inputs for producing goods. Their consumption policy is fixed rationally by maximizing a utility and their capital cannot fall below a given threshold (state constraint). We aim at modeling the interactions between firms on the markets of the different inputs on the long term. The stationary equlibria are described by a system of coupled non-linear differential equations: a Hamilton-Jacobi equation describing the optimal control problem of a single atomistic firm; a continuity equation describing the distribution of the individual state variable (the capital) in the population of firms; the equilibria on the markets of the production factors. We prove the existence of equilibria under suitable assumptions.

math.AP↗

arXiv2021

Deterministic mean field games with control on the acceleration and state constraints

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.

math.AP↗

arXiv2020

A class of short-term models for the oil industry addressing speculative storage

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.

math.AP↗

arXiv2020

Deterministic mean field games with control on the acceleration

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.

math.AP↗

arXiv2020

Mean Field Games and Applications: Numerical Aspects

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.

math.OC↗

arXiv2019

Optimal control of conditioned processes with feedback controls

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 $+\infty$. 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.

math.OC↗

arXiv2019

Finite Horizon Mean Field Games on Networks

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. Existence and uniqueness are proven.

math.AP↗

arXiv2018

A Class of Infinite Horizon Mean Field Games on Networks

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. Existence and uniqueness are proven, under suitable assumptions.

math.AP↗