arXiv subjects

Roxana Dumitrescu

Publications and source records attributed to Roxana Dumitrescu.

At least 19 recordsLinked to original sources

arXiv2026

Fast and slow mean-field games

We propose a framework for constructing approximate Nash equilibria in mean-field games (MFG) with common noise based on a two-time-scale structure. In our model, the common noise is carried by a fast variable evolving under ergodic dynamics, while the slow variable is either optimally controlled or stopped. The fast variable enters the dynamics of the slow one through the drift coefficient. The key idea is to construct an approximation to the solution of the full MFG with common noise using an ``effective'' MFG without common noise, where the coefficients are averaged with respect to the stationary measure of the fast-scale process. We construct an explicit $\varepsilon$-MFG equilibrium for the full MFG from the equilibrium for the effective MFG with randomized control and stopping. To this end, we obtain new results on existence of MFG equilibria with randomized stopping. We rely on strong convergence results for two-scale diffusions under structural assumptions on the MFG, and show that the time-scale separation parameter controls the error in the Nash equilibrium condition.

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arXiv2026

A new probabilistic approach for mean field games of optimal stopping

We propose a novel probabilistic formulation for optimal stopping mean field games (OS-MFGs) with randomized strategies. We characterize mean field equilibria through a new class of coupled forward-backward systems, termed coupled reflected forward-backward McKean--Vlasov stochastic differential equations (MKV-RFBSDEs). An equilibrium is represented by a quintuple $(X,Y,Z,A,L)$, where $L$ is an adapted, $[0,1]$-valued, non-increasing càdlàg process representing the randomized stopping strategy. The optimality of randomized stopping strategies is characterized through two novel Skorokhod-type conditions involving $L$. This characterization is new even for classical optimal stopping problems without mean field interactions. We rigorously prove an equivalence between solutions of the MKV-RFBSDE system and OS-MFG equilibria in randomized strategies. We establish the existence of equilibria by applying the Kakutani--Fan--Glicksberg fixed-point theorem to a set-valued best-response correspondence, relying on new stability, compactness, and continuity results for the coupled MKV-RFBSDE system. We also prove uniqueness under suitable conditions. Under alternative monotonicity assumptions, we develop a new order-theoretic approach based on Tarski's fixed-point theorem, yielding the existence of extremal equilibria and constructive schemes for the minimal and maximal solutions. We further show that a mean field equilibrium induces an approximate Nash equilibrium for the associated $N$-player stopping game. Finally, we connect our probabilistic formulation with the analytical approach characterized by a coupled system of constrained partial differential equations.

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arXiv2026

A Rank-Based Reward between a Principal and a Field of Agents: Application to Energy Savings

In this paper, we consider the problem of a Principal aiming at designing a reward function for a population of heterogeneous agents. We construct an incentive based on the ranking of the agents, so that a competition among the latter is initiated. We place ourselves in the limit setting of mean-field type interactions and prove the existence and uniqueness of the equilibrium distribution for a given reward, for which we can find an explicit representation. Focusing first on the homogeneous setting, we characterize the optimal reward function using a convex reformulation of the problem and provide an interpretation of its behaviour. We then show that this characterization still holds for a sub-class of heterogeneous populations. For the general case, we propose a convergent numerical method which fully exploits the characterization of the mean-field equilibrium. We develop a case study related to the French market of Energy Saving Certificates based on the use of realistic data, which shows that the ranking system allows to achieve the sobriety target imposed by the regulation.

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arXiv2026

Law-invariant BSDEs and dynamic risk measures: new characterizations

We provide a new characterization of law-invariant backward stochastic differential equations (i.e. BSDEs) with quadratic growth. This answers the open question raised in Xu--Xu--Zhou (2022) on necessary conditions for law-invariance of g-expectations, and extends the analysis to general (possibly non-deterministic) generators. We also introduce and compare several dynamic notions of law-invariance in continuous time, establishing precise relationships among them. As an application, we study dynamic risk measures. For cash-additive, normalized risk measures, we recover and extend to continuous time the Kupper--Schachermayer (2009) characterization obtained in discrete time, showing that law-invariance and strong time-consistency force an entropic structure. We further obtain a new characterization of cash non-additive law-invariant risk measures generated by BSDEs via a time-dependent certainty equivalent representation.

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arXiv2026

Forward Performance Processes under Multiple Default Risks

This article constructs a forward exponential utility in a market with multiple defaultable risks. Using the Jacod-Pham decomposition for random fields, we first characterize forward performance processes in a defaultable market under the default-free filtration. We then construct a forward utility via a system of recursively defined, indexed infinite-horizon backward stochastic differential equations (BSDEs) with discounting, and establish the existence, uniqueness, and boundedness of their solutions. To verify the required (super)martingale property of the performance process, we develop a rigorous characterization of this property with respect to the general filtration in terms of a set of (in)equalities relative to the default-free filtration. We further extend the analysis to a stochastic factor model with ergodic dynamics. In this setting, we derive uniform bounds for the Markovian solutions of the infinite-horizon BSDEs, overcoming technical challenges arising from the special structure of the system of BSDEs in the defaultable setting. Passing to the ergodic limit, we identify the limiting BSDE and relate its constant to the risk-sensitive long-run growth rate of the optimal wealth process.

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arXiv2025

Entropy Regularization in Mean-Field Games of Optimal Stopping

We study mean-field games of optimal stopping (OS-MFGs) and introduce an entropy-regularized framework to enable learning-based solution methods. By utilizing randomized stopping times, we reformulate the OS-MFG as a mean-field game of singular stochastic controls (SC-MFG) with entropy regularization. We establish the existence of equilibria and prove their stability as the entropy parameter vanishes. Fictitious play algorithms tailored for the regularized setting are introduced, and we show their convergence under both Lasry-Lions monotonicity and supermodular assumptions on the reward functional. Our work lays the theoretical foundation for model-free learning approaches to OS-MFGs.

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arXiv2024

Price impact and long-term profitability of energy storage

We study the price impact of storage facilities in electricity markets and analyze the long-term profitability of these facilities in prospective scenarios of energy transition. To this end, we begin by characterizing the optimal operating strategy for a stylized storage system, assuming an arbitrary exogenous price process. Following this, we determine the equilibrium price in a market comprising storage systems (acting as price takers), renewable energy producers, and conventional producers with a defined supply function, all driven by an exogenous demand process. The price process is characterized as a solution to a fully coupled system of forward-backward stochastic differential equations, for which we establish existence and uniqueness under appropriate assumptions. We finally illustrate the impact of storage on intraday electricity prices through numerical examples and show how the revenues of storage agents may evolve in prospective energy transition scenarios from RTE, the French energy electricity network operator, taking into account both the increasing penetration of renewable energies and the self-cannibalization effect of growing storage capacity. We find that both the average revenues and the interquantile ranges increase in all scenarios, highlighting higher expected profits and higher risk for storage assets.

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arXiv2024

Deep learning algorithms for FBSDEs with jumps: Applications to option pricing and a MFG model for smart grids

In this paper, we introduce various machine learning solvers for (coupled) forward-backward systems of stochastic differential equations (FBSDEs) driven by a Brownian motion and a Poisson random measure. We provide a rigorous comparison of the different algorithms and demonstrate their effectiveness in various applications, such as cases derived from pricing with jumps and mean-field games. In particular, we show the efficiency of the deep-learning algorithms to solve a coupled multi-dimensional FBSDE system driven by a time-inhomogeneous jump process with stochastic intensity, which describes the Nash equilibria for a specific mean-field game (MFG) problem for which we also provide the complete theoretical resolution. More precisely, we develop an extension of the MFG model for smart grids introduced in Alasseur, Campi, Dumitrescu and Zeng (Annals of Operations Research, 2023) to the case when the random jump times correspond to the jump times of a doubly Poisson process. We first provide an existence result of an equilibria and derive its semi-explicit characterization in terms of a system of FBSDEs in the linear-quadratic setting. We then compare the MFG solution to the optimal strategy of a central planner and provide several numerical illustrations using the deep-learning solvers presented in the first part of the paper.

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arXiv2023

A mean-field game model of electricity market dynamics

We develop a model for the long-term dynamics of electricity market, based on mean-field games of optimal stopping. Our paper extends the recent contribution [Aïd, René, Roxana Dumitrescu, and Peter Tankov, ``The entry and exit game in the electricity markets: A mean-field game approach." Journal of Dynamics \& Games 8.4 (2021): 331] in several ways, making the model much more realistic, especially for describing the medium-term impacts of energy transition on electricity markets. In particular, we allow for an arbitrary number of technologies with endogenous fuel prices and enable the agents to both invest and divest. This makes it possible to describe the role of gas generation as a medium-term substitute for coal, to be replaced by renewable generation in the long term, and enables us to model the events like the 2022 energy price crisis.

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arXiv2023

Energy transition under scenario uncertainty: a mean-field game of stopping with common noise

We study the impact of transition scenario uncertainty, namely that of future carbon price and electricity demand, on the pace of decarbonization of the electricity industry. To this end, we develop a theory of optimal stopping mean-field games with non-Markovian common noise and partial observation. For mathematical tractability, the theory is formulated in discrete time and with common noise restricted to a finite probability space. We prove the existence of Nash equilibria for this game using the linear programming approach. We then apply the general theory to build a discrete time model for the long-term dynamics of the electricity market subject to common random shocks affecting the carbon price and the electricity demand. We consider two classes of agents: conventional producers and renewable producers. The former choose an optimal moment to exit the market and the latter choose an optimal moment to enter the market by investing into renewable generation. The agents interact through the market price determined by a merit order mechanism with an exogenous stochastic demand. We illustrate our model by an example inspired by the UK electricity market, and show that scenario uncertainty leads to significant changes in the speed of replacement of conventional generators by renewable production.

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arXiv2023

A new Mertens decomposition of $\mathscr{Y}^{g,ξ}$-submartingale systems. Application to BSDEs with weak constraints at stopping times

We first introduce the concept of $\mathscr{Y}^{g,ξ}$-submartingale systems, where the nonlinear operator $\mathscr{Y}^{g,ξ}$ corresponds to the first component of the solution of a reflected BSDE with generator $g$ and lower obstacle $ξ$. We first show that, in the case of a left-limited right-continuous obstacle, any $\mathscr{Y}^{g,ξ}$-submartingale system can be aggregated by a process which is right-lower semicontinuous. We then prove a \textit{Mertens decomposition}, by using an original approach which does not make use of the standard penalization technique. These results are in particular useful for the treatment of control/stopping game problems and, to the best of our knowledge, they are completely new in the literature. As an application, we introduce a new class of \textit{Backward Stochastic Differential Equations (in short BSDEs) with weak constraints at stopping times}, which are related to the partial hedging of American options. We study the wellposedness of such equations and, using the $\mathscr{Y}^{g,ξ}$-Mertens decomposition, we show that the family of minimal time-$t$-values $Y_t$, with $(Y,Z)$ a supersolution of the BSDE with weak constraints, admits a representation in terms of a reflected backward stochastic differential equation.

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arXiv2023

Linear Programming Fictitious Play algorithm for Mean Field Games with optimal stopping and absorption

We develop the fictitious play algorithm in the context of the linear programming approach for mean field games of optimal stopping and mean field games with regular control and absorption. This algorithm allows to approximate the mean field game population dynamics without computing the value function by solving linear programming problems associated with the distributions of the players still in the game and their stopping times/controls. We show the convergence of the algorithm using the topology of convergence in measure in the space of subprobability measures, which is needed to deal with the lack of continuity of the flows of measures. Numerical examples are provided to illustrate the convergence of the algorithm.

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arXiv2022

A propagation of chaos result for weakly interacting nonlinear Snell envelopes

In this article, we establish a propagation of chaos result for weakly interacting nonlinear Snell envelopes which converge to a class of mean-field reflected backward stochastic differential equations (BSDEs) with jumps and right-continuous and left-limited obstacle, where the mean-field interaction in terms of the distribution of the $Y$-component of the solution enters both the driver and the lower obstacle. Under mild Lipschitz and integrability conditions on the coefficients, we prove existence and uniqueness of the solution to both the mean-field reflected BSDEs with jumps and the corresponding system of weakly interacting particles and provide a propagation of chaos result for the whole solution $(Y,Z,U,K)$, which requires new technical results due to the dependence of the obstacle on the solution and the presence of jumps.

math.PR↗

arXiv2022

Zero-sum mean-field Dynkin games: characterization and convergence

We introduce a zero-sum game problem of mean-field type as an extension of the classical zero-sum Dynkin game problem to the case where the payoff processes might depend on the value of the game and its probability law. We establish sufficient conditions under which such a game admits a value and a saddle point. Furthermore, we provide a characterization of the value of the game in terms of a specific class of doubly reflected backward stochastic differential equations (BSDEs) of mean-field type, for which we derive an existence and uniqueness result. We then introduce a corresponding system of weakly interacting zero-sum Dynkin games and show its well-posedness. Finally, we provide a propagation of chaos result for the value of the zero-sum mean-field Dynkin game.

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arXiv2022

An optional decomposition of $\mathscr{Y}^{g,ξ}-submartingales$ and applications to the hedging of American options in incomplete markets

In the recent paper \cite{DESZ}, the notion of $\mathscr{Y}^{g,ξ}$-submartingale processes has been introduced. Within a jump-diffusion model, we prove here that a process $X$ which satisfies the simultaneous $\mathscr{Y}^{\mathbb{Q},g,ξ}$ -submartingale property under a suitable family of equivalent probability measures $\mathbb{Q}$, admits a \textit{nonlinear optional decomposition}. This is an analogous result to the well known optional decomposition of simultaneous (classical and $\mathscr{E}^g$-)supermartingales. We then apply this decomposition to the super-hedging problem of an American option in a jump-diffusion model, from the buyer's point of view. We obtain an \textit{infinitesimal characterization} of the buyer's superhedging price, this result being completely new in the literature. Indeed, it is well known that the seller's superheding price of an American option admits an infinitesimal representation in terms of the \textit{minimal supersolution of a constrained reflected BSDE}. To the best of our knowledge, no analogous result has been established for the buyer of the American option in an incomplete market. Our results fill this gap, and show that the buyer's super-hedging price admits an infinitesimal charcaterization in terms of the \textit{maximal subsolution of a constrained reflected BSDE}.

q-fin.MF↗

arXiv2021

MFG model with a long-lived penalty at random jump times: application to demand side management for electricity contracts

We consider an energy system with $n$ consumers who are linked by a Demand Side Management (DSM) contract, i.e. they agreed to diminish, at random times, their aggregated power consumption by a predefined volume during a predefined duration. Their failure to deliver the service is penalised via the difference between the sum of the $n$ power consumptions and the contracted target. We are led to analyse a non-zero sum stochastic game with $n$ players, where the interaction takes place through a cost which involves a delay induced by the duration included in the DSM contract. When $n \to \infty$, we obtain a Mean-Field Game (MFG) with random jump time penalty and interaction on the control. We prove a stochastic maximum principle in this context, which allows to compare the MFG solution to the optimal strategy of a central planner. In a linear quadratic setting we obtain an semi-explicit solution through a system of decoupled forward-backward stochastic differential equations with jumps, involving a Riccati Backward SDE with jumps. We show that it provides an approximate Nash equilibrium for the original $n$-player game for $n$ large. Finally, we propose a numerical algorithm to compute the MFG equilibrium and present several numerical experiments.

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arXiv2020

Control and optimal stopping Mean Field Games: a linear programming approach

We develop the linear programming approach to mean-field games in a general setting. This relaxed control approach allows to prove existence results under weak assumptions, and lends itself well to numerical implementation. We consider mean-field game problems where the representative agent chooses both the optimal control and the optimal time to exit the game, where the instantaneous reward function and the coefficients of the state process may depend on the distribution of the other agents. Furthermore, we establish the equivalence between mean-field games equilibria obtained by the linear programming approach and the ones obtained via the controlled/stopped martingale approach, another relaxation method used in a few previous papers in the case when there is only control.

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arXiv2020

Mean-field games of optimal stopping: a relaxed solution approach

We consider the mean-field game where each agent determines the optimal time to exit the game by solving an optimal stopping problem with reward function depending on the density of the state processes of agents still present in the game. We place ourselves in the framework of relaxed optimal stopping, which amounts to looking for the optimal occupation measure of the stopper rather than the optimal stopping time. This framework allows us to prove the existence of the relaxed Nash equilibrium and the uniqueness of the associated value of the representative agent under mild assumptions. Further, we prove a rigorous relation between relaxed Nash equilibria and the notion of mixed solutions introduced in earlier works on the subject, and provide a criterion, under which the optimal strategies are pure strategies, that is, behave in a similar way to stopping times. Finally, we present a numerical method for computing the equilibrium in the case of potential games and show its convergence.

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