Research Papers in Optimal Control and Mean Field Games
Optimal control on finite graphs: asymptotic optimal controls and ergodic constant in the case of entropic costs, submitted
Abstract: For optimal control problems on finite graphs in continuous time, the dynamic programming principle leads to value functions characterized by systems of nonlinear ordinary differential equations. In this paper, we consider the case of entropic costs for which the nonlinear differential equations can be transformed into linear ones thanks to a change of variables linked to the classical duality between entropy and exponential. When the graph is connected, we show that the asymptotic optimal control and the ergodic constant can be computed very easily with classical tools of matrix analysis.
Optimal control on graphs: existence, uniqueness, and long-term behavior, with I. Manziuk, ESAIM: Control, Optimisation and Calculus of Variations (ESAIM: COCV), Volume 26, 2020
Abstract: The literature on continuous-time stochastic optimal control seldom deals with the case of discrete state spaces. In this paper, we provide a general framework for the optimal control of continuous-time Markov chains on finite graphs. In particular, we provide results on the long-term behavior of value functions and optimal controls, along with results on the associated ergodic Hamilton-Jacobi equation.
Existence and uniqueness result for mean field games with congestion effect on graphs, Applied Mathematics and Optimization, Volume 72, Issue 2, October 2015
Abstract: This paper presents a general existence and uniqueness result for mean field games equations on graphs. In particular, our setting allows to take into account congestion effects of almost any form. These general congestion effects are particularly relevant in graphs in which the cost to move from one node to another may for instance depend on the proportion of players in both the source node and the target node. Existence is proved using a priori estimates and a fixed point argument à la Schauder. We propose a new criterion to ensure uniqueness in the case of Hamiltonian functions with a complex (non-local) structure. This result generalizes the discrete counterpart of existing uniqueness results.
New numerical methods for mean field games with quadratic costs, Networks and Heterogenous Media, Volume 7, Number 2, June 2012
Abstract: Mean field games have been introduced by J.-M. Lasry and P.-L. Lions as the limit case of stochastic differential games when the number of players goes to infinity. In the case of quadratic costs, we present two changes of variables that allow to transform the mean field games (MFG) equations into two simpler systems of equations. The first change of variables, already introduced in a preceding paper, leads to two heat equations with nonlinear source terms. The second change of variables, which is introduced for the first time in this paper, leads to two Hamilton-Jacobi-Bellman equations. Numerical methods based on these equations are presented and numerical experiments are carried out.
Mean field games equations with quadratic Hamiltonian: a specific approach, Mathematical Models and Methods in Applied Sciences (M3AS), Volume 22, Issue 9, September 2012
Abstract: Mean field games models describing the limit of a large class of stochastic differential games, as the number of players goes to infinity, have been introduced by J.-M. Lasry and P.-L. Lions. We use a change of variables to transform the mean field games (MFG) equations into a system of simpler coupled partial differential equations, in the case of a quadratic Hamiltonian. This system is then used to exhibit a monotonic scheme to build solutions of the MFG equations. Effective numerical methods based on this constructive scheme are presented and numerical experiments are carried out.
Mean Field Games with a Quadratic Hamiltonian: A Constructive Scheme, in Advances in Dynamic Games, Annals of the International Society of Dynamic Games, 2013.
Abstract: Mean field games models describing the limit case of a large class of stochastic differential games, as the number of players goes to infinity, were introduced by Lasry and Lions. We use a change of variables to transform the mean field games equations into a system of simpler coupled partial differential equations in the case of a quadratic Hamiltonian. This system is then used to exhibit a monotonic scheme to build solutions of the mean field games equations.
Mean Field Games and Applications, with J.-M. Lasry and P.-L. Lions, in Paris-Princeton Lectures on Mathematical Finance 2010, Ed. Springer, January 2011
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 developed by the authors. Proofs were not presented during the conferences and are now available. So are some issues that were only rapidly tackled during class. The content of this text is therefore far more important than the actual "Cours Bachelier" conferences, though the guiding principle is the same and consists in a progressive introduction of the concepts, methodologies and mathematical tools of mean field games theory.
Mean Field Games and Oil Production, with J.-M. Lasry and P.-L. Lions, in The Economics of Sustainable Development, Ed. Economica, 2010
Abstract: In this paper we study the evolution of oil production in the long run. A first optimization model is presented, that can be solved using Euler-Lagrange tools. Because these classical tools are not the best suited to the model, we adopt a mean field games approach based on two partial differential equations. An extended model is then presented to analyze the influence of new competitors which might enter the market with energy from renewable sources. The usefulness of a subsidy to potential entrants is discussed.
A reference case for mean field games, Journal de Mathématiques Pures et Appliquées, Volume 92, Issue 3, September 2009
Abstract: In this article, we present a reference case of mean field games. This case can be seen as a reference for two main reasons. First, the case is simple enough to allow for explicit resolution: Bellman functions are quadratic, stationary measures are normal and stability can be dealt with explicitly using Hermite polynomials. Second, despite its simplicity, the case is rich enough in terms of mathematics to be generalized and to inspire the study of more complex models that may not be as tractable as this one.
Tournament-induced risk-shifting: A mean field games approach, Risk and Decision Analysis, Volume 4, Issue 2, 2013
Abstract: The agency problem between an investor and his mutual funds managers has long been studied in the economic literature. Because the very business of mutual funds managers is not only to manage money but also, and rather, to increase the money under management, one of the numerous agency problems is the implicit incentive induced by the relationship between inflows and performance. If the consequences of incentives, be they implicit or explicit – as for compensation schemes of individual asset managers – are well known in terms of risk-shifting when the incentives are linked to a benchmark, the very fact that the mutual fund market is a tournament does not seem to be modeled properly in the literature. In this paper, we propose a mean field games model to quantify the risk-shifting induced by a tournament-like competition between mutual funds.
Abstract: For optimal control problems on finite graphs in continuous time, the dynamic programming principle leads to value functions characterized by systems of nonlinear ordinary differential equations. In this paper, we consider the case of entropic costs for which the nonlinear differential equations can be transformed into linear ones thanks to a change of variables linked to the classical duality between entropy and exponential. When the graph is connected, we show that the asymptotic optimal control and the ergodic constant can be computed very easily with classical tools of matrix analysis.
Optimal control on graphs: existence, uniqueness, and long-term behavior, with I. Manziuk, ESAIM: Control, Optimisation and Calculus of Variations (ESAIM: COCV), Volume 26, 2020
Abstract: The literature on continuous-time stochastic optimal control seldom deals with the case of discrete state spaces. In this paper, we provide a general framework for the optimal control of continuous-time Markov chains on finite graphs. In particular, we provide results on the long-term behavior of value functions and optimal controls, along with results on the associated ergodic Hamilton-Jacobi equation.
Existence and uniqueness result for mean field games with congestion effect on graphs, Applied Mathematics and Optimization, Volume 72, Issue 2, October 2015
Abstract: This paper presents a general existence and uniqueness result for mean field games equations on graphs. In particular, our setting allows to take into account congestion effects of almost any form. These general congestion effects are particularly relevant in graphs in which the cost to move from one node to another may for instance depend on the proportion of players in both the source node and the target node. Existence is proved using a priori estimates and a fixed point argument à la Schauder. We propose a new criterion to ensure uniqueness in the case of Hamiltonian functions with a complex (non-local) structure. This result generalizes the discrete counterpart of existing uniqueness results.
New numerical methods for mean field games with quadratic costs, Networks and Heterogenous Media, Volume 7, Number 2, June 2012
Abstract: Mean field games have been introduced by J.-M. Lasry and P.-L. Lions as the limit case of stochastic differential games when the number of players goes to infinity. In the case of quadratic costs, we present two changes of variables that allow to transform the mean field games (MFG) equations into two simpler systems of equations. The first change of variables, already introduced in a preceding paper, leads to two heat equations with nonlinear source terms. The second change of variables, which is introduced for the first time in this paper, leads to two Hamilton-Jacobi-Bellman equations. Numerical methods based on these equations are presented and numerical experiments are carried out.
Mean field games equations with quadratic Hamiltonian: a specific approach, Mathematical Models and Methods in Applied Sciences (M3AS), Volume 22, Issue 9, September 2012
Abstract: Mean field games models describing the limit of a large class of stochastic differential games, as the number of players goes to infinity, have been introduced by J.-M. Lasry and P.-L. Lions. We use a change of variables to transform the mean field games (MFG) equations into a system of simpler coupled partial differential equations, in the case of a quadratic Hamiltonian. This system is then used to exhibit a monotonic scheme to build solutions of the MFG equations. Effective numerical methods based on this constructive scheme are presented and numerical experiments are carried out.
Mean Field Games with a Quadratic Hamiltonian: A Constructive Scheme, in Advances in Dynamic Games, Annals of the International Society of Dynamic Games, 2013.
Abstract: Mean field games models describing the limit case of a large class of stochastic differential games, as the number of players goes to infinity, were introduced by Lasry and Lions. We use a change of variables to transform the mean field games equations into a system of simpler coupled partial differential equations in the case of a quadratic Hamiltonian. This system is then used to exhibit a monotonic scheme to build solutions of the mean field games equations.
Mean Field Games and Applications, with J.-M. Lasry and P.-L. Lions, in Paris-Princeton Lectures on Mathematical Finance 2010, Ed. Springer, January 2011
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 developed by the authors. Proofs were not presented during the conferences and are now available. So are some issues that were only rapidly tackled during class. The content of this text is therefore far more important than the actual "Cours Bachelier" conferences, though the guiding principle is the same and consists in a progressive introduction of the concepts, methodologies and mathematical tools of mean field games theory.
Mean Field Games and Oil Production, with J.-M. Lasry and P.-L. Lions, in The Economics of Sustainable Development, Ed. Economica, 2010
Abstract: In this paper we study the evolution of oil production in the long run. A first optimization model is presented, that can be solved using Euler-Lagrange tools. Because these classical tools are not the best suited to the model, we adopt a mean field games approach based on two partial differential equations. An extended model is then presented to analyze the influence of new competitors which might enter the market with energy from renewable sources. The usefulness of a subsidy to potential entrants is discussed.
A reference case for mean field games, Journal de Mathématiques Pures et Appliquées, Volume 92, Issue 3, September 2009
Abstract: In this article, we present a reference case of mean field games. This case can be seen as a reference for two main reasons. First, the case is simple enough to allow for explicit resolution: Bellman functions are quadratic, stationary measures are normal and stability can be dealt with explicitly using Hermite polynomials. Second, despite its simplicity, the case is rich enough in terms of mathematics to be generalized and to inspire the study of more complex models that may not be as tractable as this one.
Tournament-induced risk-shifting: A mean field games approach, Risk and Decision Analysis, Volume 4, Issue 2, 2013
Abstract: The agency problem between an investor and his mutual funds managers has long been studied in the economic literature. Because the very business of mutual funds managers is not only to manage money but also, and rather, to increase the money under management, one of the numerous agency problems is the implicit incentive induced by the relationship between inflows and performance. If the consequences of incentives, be they implicit or explicit – as for compensation schemes of individual asset managers – are well known in terms of risk-shifting when the incentives are linked to a benchmark, the very fact that the mutual fund market is a tournament does not seem to be modeled properly in the literature. In this paper, we propose a mean field games model to quantify the risk-shifting induced by a tournament-like competition between mutual funds.
