arXiv subjects

Mao Fabrice Djete

Publications and source records attributed to Mao Fabrice Djete.

16 recordsLinked to original sources

arXiv2026

W-BSDEs with Common Noise: Well-posedness, Characterization, and Consequences

We study a notion of backward stochastic differential equation on the Wasserstein space, called a W-BSDE. This work extends to the common noise setting recent results of the first author. The equation is formulated in Markovian form along admissible flows of conditional laws generated by McKean-Vlasov dynamics with common noise. A solution is a pair of fields $(Y,Z)$, where $Y$ is a functional on the space of probability measures and $Z$ is the corresponding intrinsic Wasserstein gradient. Our main result shows that the well-posedness of the W-BSDE can be reduced to a characteristic equation for a field $p$. Whenever this equation admits a suitable solution, we construct a W-BSDE solution and identify $Z$ with both the characteristic field $p$ and the intrinsic gradient of $Y$. This gives an existence result for the W-BSDE and, at the same time, a probabilistic construction of a functional on the Wasserstein space together with its intrinsic gradient. We then develop several consequences of the theory. We obtain verification principles for McKean-Vlasov control problems and for mean field games with common noise, without relying on convexity or separability assumptions. We also show that the characteristic field admits a dynamic representation through a McKean-Vlasov forward-backward SDE. The first component $Y$ solves the associated semilinear PDE on the space of probability measures in the viscosity sense, including the second-order term induced by common noise. Under additional smoothness, $Y$ is a classical solution. Finally, we prove that W-BSDEs arise as intrinsic mean-field limits of classical finite-dimensional BSDEs written on empirical measures, with convergence of both the value component and the rescaled martingale integrands.

math.PR↗

arXiv2026

A McKean--Vlasov semigroup and its application to a martingale representation problem

We introduce a McKean--Vlasov semigroup in the presence of common noise and study its first-order differential structure. Given a terminal functional $G$, the semigroup is defined by propagating the initial law through a reference conditional McKean--Vlasov flow. Using a finite-particle approximation and a diagonal/off-diagonal decomposition of the tangent processes, we identify the Lions derivative of the propagated functional and derive a backward representation along arbitrary admissible conditional law flows. This formula involves only first-order derivatives in the measure variable and yields both a Feynman--Kac representation for linear equations on the Wasserstein space and, in the driftless case, a martingale representation problem whose unique solution is given by the McKean--Vlasov heat semigroup and its Lions derivative. We also discuss mixed finite-dimensional/Wasserstein representations and discrete path-dependent extensions.

math.PR↗

arXiv2026

Non--exchangeable mean field games with moderate interactions and common noise

We study mean field games for large non--exchangeable populations with moderate local interactions and common noise. The finite--player system is driven by two complementary interaction mechanisms : a graphon--type structure, which encodes heterogeneous large--scale interactions between agents, and a rescaled local kernel, which produces a density-dependent interaction term in the limit. The limiting model is a non--exchangeable mean field game in which the representative player is indexed by a label $u\in[0,1]$, interacts through a graphon--weighted local density, and is affected by a graphon--induced environment law. We introduce a relaxed formulation of the limiting mean field game, adapted to the presence of common noise, and prove existence under general continuity and non--degeneracy assumptions. Under additional convexity assumptions, relaxed equilibria can be realized in strict form. In the deterministic case without common noise, we obtain deterministic equilibria and provide a probabilistic characterization of strict equilibria through a nonlinear Feynman--Kac representation. We then establish the asymptotic connection with the finite--player game. We prove that every limit point of approximate closed--loop Nash equilibria is a relaxed solution of the limiting mean field game, and that the corresponding averaged equilibrium payoffs converge. Conversely, every relaxed mean field game equilibrium can be approximated by Markovian approximate Nash equilibria of the finite--player systems. These results give a complete asymptotic characterization of equilibrium behavior for non--exchangeable games with moderate interactions and common noise.

math.OC↗

arXiv2026

On Approximate Nash Equilibria in Mean Field Games

In the context of large population symmetric games, approximate Nash equilibria are introduced through equilibrium solutions of the corresponding mean field game in the sense that the individual gain from optimal unilateral deviation under such strategies converges to zero in the large population size asymptotic. We show that these strategies satisfy an $Ł^\infty$ notion of approximate Nash equilibrium which guarantees that the individual gain from optimal unilateral deviation is small uniformly among players and uniformly on their initial characteristics. We establish these results in the context of static models and in the dynamic continuous time setting, and we cover situations where the agents' criteria depend on the conditional law of the controlled state process.

cs.GT↗

arXiv2025

A non-exchangeable mean field control problem with controlled interactions

This paper introduces and analyzes a new class of mean-field control (\textsc{MFC}) problems in which agents interact through a \emph{fixed but controllable} network structure. In contrast with the classical \textsc{MFC} framework -- where agents are exchangeable and interact only through symmetric empirical distributions -- we consider systems with heterogeneous and possibly asymmetric interaction patterns encoded by a structural kernel, typically of graphon type. A key novelty of our approach is that this interaction structure is no longer static: it becomes a genuine \emph{control variable}. The planner therefore optimizes simultaneously two distinct components: a \emph{regular control}, which governs the local dynamics of individual agents, and an \emph{interaction control}, which shapes the way agents connect and influence each other through the fixed structural kernel. \medskip We develop a generalized notion of relaxed (randomized) control adapted to this setting, prove its equivalence with the strong formulation, and establish existence, compactness, and continuity results for the associated value function under minimal regularity assumptions. Moreover, we show that the finite $n$-agent control problems with general (possibly asymmetric) interaction matrices converge to the mean-field limit when the corresponding fixed step-kernels converge in cut-norm, with asymptotic consistency of the optimal values and control strategies. Our results provide a rigorous framework in which the \emph{interaction structure itself is viewed and optimized as a control object}, thereby extending mean-field control theory to non-exchangeable populations and controlled network interactions.

math.OC↗

arXiv2025

A notion of BSDE on the Wasserstein space and its applications to control problems and PDEs

We introduce a class of backward stochastic differential equations (BSDEs) on the Wasserstein space of probability measures. This formulation extends the classical correspondence between BSDEs, stochastic control, and partial differential equations (PDEs) to the mean--field (McKean--Vlasov) setting, where the dynamics depend on the law of the state process. The standard BSDE framework becomes inadequate in this context, motivating a new definition in terms of measure--dependent solutions. Under suitable assumptions, we demonstrate that this formulation is in correspondence with both mean--field control problems and partial differential equations defined on the Wasserstein space. A comparison principle is established to ensure uniqueness, and existence results are obtained for generators that are linear or quadratic in the $z$--variable. This framework provides a probabilistic approach to control and analysis on the space of probability measures.

math.PR↗

arXiv2024

Three connected problems: principal with multiple agents in cooperation, Principal--Agent with Mckean--Vlasov dynamics and multitask Principal--Agent

In this paper, we address three Principal--Agent problems in a moral hazard context and show that they are connected. We start by studying the problem of Principal with multiple Agents in cooperation. The term cooperation is manifested here by the fact that the agents optimize their criteria through Pareto equilibria. We show that as the number of agents tends to infinity, the principal's value function converges to the value function of a McKean--Vlasov control problem. Using the solution to this McKean--Vlasov control problem, we derive a constructive method for obtaining approximately optimal contracts for the principal's problem with multiple agents in cooperation. In a second step, we show that the problem of Principal with multiple Agents turns out to also converge, when the number of agents goes to infinity, towards a new Principal--Agent problem which is the Principal--Agent problem with Mckean--Vlasov dynamics. This is a Principal--Agent problem where the agent--controlled production follows a Mckean-Vlasov dynamics and the contract can depend of the distribution of the production. The value function of the principal in this setting is equivalent to that of the same McKean--Vlasov control problem from the multi--agent scenario. Furthermore, we show that an optimal contract can be constructed from the solution to this McKean--Vlasov control problem. We conclude by discussing, in a simple example, the connection of these problems with the multitask Principal--Agent problem which is a situation when a principal delegates multiple tasks that can be correlated to a single agent.

math.OC↗

arXiv2024

Non--regular McKean--Vlasov equations and calibration problem in local stochastic volatility models

In order to deal with the question of the existence of a calibrated local stochastic volatility model in finance, we investigate a class of McKean--Vlasov equations where a minimal continuity assumption is imposed on the coefficients. Namely, the drift coefficient and, in particular, the volatility coefficient are not necessarily continuous in the measure variable for the Wasserstein topology. In this paper, we provide an existence result and show an approximation by $N$--particle system or propagation of chaos for this type of McKean--Vlasov equations. As a direct result, we are able to deduce the existence of a calibrated local stochastic volatility model for an appropriate choice of stochastic volatility parameters. The associated propagation of chaos result is also proved.

math.PR↗

arXiv2024

Mean Field Game of Mutual Holding with common noise

We consider the mean field game of cross--holding introduced in \citeauthor*{DjeteTouzi} \cite{DjeteTouzi} in the context where the equity value dynamics are affected by a common noise. In contrast with \cite{DjeteTouzi}, the problem exhibits the standard paradigm of mean--variance trade off. Our crucial observation is to search for equilibrium solutions of our mean field game among those models which satisfy an appropriate notion of no--arbitrage. Under this condition, it follows that the representative agent optimization step is reduced to a standard portfolio optimization problem with random endowment.

math.PR↗

arXiv2023

Stackelberg Mean Field Games: convergence and existence results to the problem of Principal with multiple Agents in competition

In a situation of moral hazard, this paper investigates the problem of Principal with $n$ Agents when the number of Agents $n$ goes to infinity. There is competition between the Agents expressed by the fact that they optimize their utility functions through a Nash equilibrium criterion. Each Agent is offered by the Principal a contract which is divided into a Markovian part involving the state/production of the Agent and a non--Markovian part involving the states/productions of all the other Agents. The Agents are in interactions. These interactions are characterized by common noise, the empirical distribution of states/productions and controls, and the contract which is not assumed to be a map of the empirical distribution. By the help of the mean field games theory, we are able to formulate an appropriate $limit$ problem involving a Principal with a $representative$ Agent. We start by solving the problem of both the Principal and the $representative$ Agent in this $limit$ problem. Then, when $n$ goes to infinity, we show that the problem of Principal with $n$ Agents converges to the $limit$ problem of Principal with a $representative$ Agent. A notable result is that, despite allowing a general type of contracts, it is approximately optimal for the Principal to offer contracts to the $n$ Agents that are maps of the empirical distribution of states/productions and controls of the Agents.

math.OC↗

arXiv2023

Large population games with interactions through controls and common noise: convergence results and equivalence between $open$--$loop$ and $closed$--$loop$ controls

In the presence of a common noise, we study the convergence problems in mean field game (MFG) and mean field control (MFC) problem where the cost function and the state dynamics depend upon the joint conditional distribution of the controlled state and the control process. In the first part, we consider the MFG setting. We start by recalling the notions of $measure$--$valued$ MFG equilibria and of approximate $closed$--$loop$ Nash equilibria associated to the corresponding $N$--player game. Then, we show that all convergent sequences of approximate $closed$--$loop$ Nash equilibria, when $N \to \infty,$ converge to $measure$--$valued$ MFG equilibria. And conversely, any $measure$--$valued$ MFG equilibrium is the limit of a sequence of approximate $closed$--$loop$ Nash equilibria. In other words, $measure$--$valued$ MFG equilibria are the accumulation points of the approximate $closed$--$loop$ Nash equilibria. Previous work has shown that $measure$--$valued$ MFG equilibria are the accumulation points of the approximate $open$--$loop$ Nash equilibria. Therefore, we obtain that the limits of approximate $closed$--$loop$ Nash equilibria and approximate $open$--$loop$ Nash equilibria are the same. In the second part, we deal with the MFC setting. After recalling the $closed$--$loop$ and $open$--$loop$ formulations of the MFC problem, we prove that they are equivalent. We also provide some convergence results related to approximate $closed$--$loop$ Pareto equilibria.

math.PR↗

arXiv2023

Mean field game of mutual holding with defaultable agents, and systemic risk

We introduce the possibility of default in the mean field game of mutual holding of Djete and Touzi [11]. This is modeled by introducing absorption at the origin of the equity process. We provide an explicit solution of this mean field game. Moreover, we provide a particle system approximation, and we derive an autonomous equation for the time evolution of the default probability, or equivalently the law of the hitting time of the origin by the equity process. The systemic risk is thus described by the evolution of the default probability.

math.PR↗

arXiv2022

Extended mean field control problem: a propagation of chaos result

In this paper, we study the $extended$ mean field control problem, which is a class of McKean-Vlasov stochastic control problem where the state dynamics and the reward functions depend upon the joint (conditional) distribution of the controlled state and the control process. By considering an appropriate controlled Fokker-Planck equation, we can formulate an optimization problem over a space of measure-valued processes and, under suitable assumptions, prove the equivalence between this optimization problem and the $extended$ mean-field control problem. Moreover, with the help of this new optimization problem, we establish the associated limit theory i.e. the $extended$ mean field control problem is the limit of a large population control problem where the interactions are achieved via the empirical distribution of state and control processes.

math.PR↗

arXiv2022

Mean Field Game of Mutual Holding

We introduce a mean field model for optimal holding of a representative agent of her peers as a natural expected scaling limit from the corresponding $N-$agent model. The induced mean field dynamics appear naturally in a form which is not covered by standard McKean-Vlasov stochastic differential equations. We study the corresponding mean field game of mutual holding in the absence of common noise. Our first main result provides an explicit equilibrium of this mean field game, defined by a bang--bang control consisting in holding those competitors with positive drift coefficient of their dynamic value. We next use this mean field game equilibrium to construct (approximate) Nash equilibria for the corresponding $N$--player game. We also provide some numerical illustrations of our mean field game equilibrium which highlight some unexpected effects induced by our results.

math.OC↗

arXiv2021

Mean Field Games of Controls: on the convergence of Nash equilibria

In this paper, we investigate a class of mean field games where the mean field interactions are achieved through the joint (conditional) distribution of the controlled state and the control process. The strategies are of $open\;loop$ type, and the volatility coefficient $σ$ can be controlled. Using (controlled) Fokker-Planck equations, we introduce a notion of measure-valued solution of mean-field games of controls, and through convergence results, prove a relation between these solutions on the one hand, and the $ε_N$--Nash equilibria on the other hand. It is shown that $ε_N$--Nash equilibria in the $N$--player games have limits as $N$ tends to infinity, and each limit is a measure-valued solution of the mean-field games of controls. Conversely, any measure-valued solution can be obtained as the limit of a sequence of $ε_N$--Nash equilibria in the $N$--player games. In other words, the measure-valued solutions are the accumulating points of $ε_N$--Nash equilibria. Similarly, by considering an $ε$--strong solution of mean field games of controls which is the classical strong solution where the optimality is obtained by admitting a small error $ε,$ we prove that the measure-valued solutions are the accumulating points of this type of solutions when $ε$ goes to zero. Finally, existence of measure-valued solution of mean-field games of controls are proved in the case without common noise.

math.PR↗

arXiv2020

McKean-Vlasov optimal control: the dynamic programming principle

We study the McKean-Vlasov optimal control problem with common noise in various formulations, namely the strong and weak formulation, as well as the Markovian and non-Markovian formulations, and allowing for the law of the control process to appear in the state dynamics. By interpreting the controls as probability measures on an appropriate canonical space with two filtrations, we then develop the classical measurable selection, conditioning and concatenation arguments in this new context, and establish the dynamic programming principle under general conditions.

math.OC↗