arXiv subjects

Agnès Sulem

Publications and source records attributed to Agnès Sulem.

18 recordsLinked to original sources

arXiv2026

Extended Graphon Mean-Field Games in Discrete Time

In this paper, we study games involving a continuum of heterogeneous players in the discrete-time setting with finite state spaces and continuous action spaces. We introduce a new model that incorporates joint state-action interactions within the graphon-weighted aggregate, described by a coupled forward-backward system. We establish the existence of graphon mean-field equilibria and characterize them through this forward-backward system. Additionally, we provide uniqueness results under monotonicity and contraction conditions. To illustrate the practical relevance of our framework, we solve an example of portfolio liquidation with price impact. We provide numerical results for three different graphons and two different initial distributions, illustrating the impact of the network's structure and the distribution heterogeneity on the distribution and the policy.

math.OC↗

arXiv2025

Stochastic Graphon Games with Jumps and Approximate Nash Equilibria

We study continuous stochastic games with heterogeneous mean field interactions and jumps on large networks and explore their limit counterparts. We introduce the graphon game model based on a controlled graphon mean field stochastic differential equation system with jumps, which can be regarded as the limiting case of a finite game dynamic system as the number of players goes to infinity. We examine the case of controlled dynamics, with control terms present in the drift, diffusion, and jump components. We focus on the study of Markovian controls and concentrate on the limit theory. We provide convergence results on the state trajectories and their laws, transitioning from finite game systems to graphon systems. We also study approximate equilibria for finite games on large networks, using the graphon equilibrium as a benchmark. The rates of convergence are analyzed under various underlying graphon models and regularity assumptions.

math.PR↗

arXiv2023

Ruin Probabilities for Risk Processes in Stochastic Networks

We study multidimensional Cramér-Lundberg risk processes where agents, located on a large sparse network, receive losses form their neighbors. To reduce the dimensionality of the problem, we introduce classification of agents according to an arbitrary countable set of types. The ruin of any agent triggers losses for all of its neighbours. We consider the case when the loss arrival process induced by the ensemble of ruined agents follows a Poisson process with general intensity function that scales with the network size. When the size of the network goes to infinity, we provide explicit ruin probabilities at the end of the loss propagation process for agents of any type. These limiting probabilities depend, in addition to the agents' types and the network structure, on the loss distribution and the loss arrival process. For a more complex risk processes on open networks, when in addition to the internal networked risk processes the agents receive losses from external users, we provide bounds on ruin probabilities.

math.PR↗

arXiv2017

BSDEs with default jump

We study the properties of nonlinear Backward Stochastic Differential Equations (BSDEs) driven by a Brownian motion and a martingale measure associated with a default jump with intensity process $(λ_t)$. We give a priori estimates for these equations and prove comparison and strict comparison theorems. These results are generalized to drivers involving a singular process. The special case of a $λ$-linear driver is studied, leading to a representation of the solution of the associated BSDE in terms of a conditional expectation and an adjoint exponential semi-martingale. We then apply these results to nonlinear pricing of European contingent claims in an imperfect financial market with a totally defaultable risky asset. The case of claims paying dividends is also studied via a singular process.

q-fin.PR↗

arXiv2017

American options in an imperfect market with default

We study pricing and (super)hedging for American options in an imperfect market model with default, where the imperfections are taken into account via the nonlinearity of the wealth dynamics. The payoff is given by an RCLL adapted process $(ξ_t)$. We define the {\em seller's superhedging price} of the American option as the minimum of the initial capitals which allow the seller to build up a superhedging portfolio. We prove that this price coincides with the value function of an optimal stopping problem with nonlinear expectations induced by BSDEs with default jump, which corresponds to the solution of a reflected BSDE with lower barrier. Moreover, we show the existence of a superhedging portfolio strategy. We then consider the {\em buyer's superhedging price}, which is defined as the supremum of the initial wealths which allow the buyer to select an exercise time $τ$ and a portfolio strategy $φ$ so that he/she is superhedged. Under the additional assumption of left upper semicontinuity along stopping times of $(ξ_t)$, we show the existence of a superhedge $(τ, φ)$ for the buyer, as well as a characterization of the buyer's superhedging price via the solution of a nonlinear reflected BSDE with upper barrier.

q-fin.PR↗

arXiv2017

Game options in an imperfect market with default

We study pricing and superhedging strategies for game options in an imperfect market with default. We extend the results obtained by Kifer in \cite{Kifer} in the case of a perfect market model to the case of an imperfect market with default, when the imperfections are taken into account via the nonlinearity of the wealth dynamics. We introduce the {\em seller's price} of the game option as the infimum of the initial wealths which allow the seller to be superhedged. We {prove} that this price coincides with the value function of an associated {\em generalized} Dynkin game, recently introduced in \cite{DQS2}, expressed with a nonlinear expectation induced by a nonlinear BSDE with default jump. We moreover study the existence of superhedging strategies. We then address the case of ambiguity on the model, - for example ambiguity on the default probability - and characterize the robust seller's price of a game option as the value function of a {\em mixed generalized} Dynkin game. We study the existence of a cancellation time and a trading strategy which allow the seller to be super-hedged, whatever the model is.

q-fin.MF↗

arXiv2017

Stochastic control of mean-field SPDEs with jumps

We study the problem of optimal control for mean-field stochastic partial differential equations (stochastic evolution equations) driven by a Brownian motion and an independent Poisson random measure, in the case of \textit{partial information} control. One important novelty of our problem is represented by the introduction of \textit{general mean-field} operators, acting on both the controlled state process and the control process. We first formulate a sufficient and a necessary maximum principle for this type of control. We then prove existence and uniqueness of the solution of such general forward and backward mean-field stochastic partial differential equations. We finally apply our results to find the explicit optimal control for an optimal harvesting problem.

math.OC↗

arXiv2017

A stochastic HJB equation for optimal control of forward-backward SDEs

We study optimal stochastic control problems of general coupled systems of forward-backward stochastic differential equations with jumps. By means of the Itô-Ventzell formula the system is transformed to a controlled backward stochastic partial differential equation (BSPDE) with jumps. Using a comparison principle for such BSPDEs we obtain a general stochastic Hamilton-Jacobi- Bellman (HJB) equation for such control problems. In the classical Markovian case with optimal control of jump diffusions, the equation reduces to the classical HJB equation. The results are applied to study risk minimization in financial markets.

math.OC↗

arXiv2016

Mixed generalized Dynkin game and stochastic control in a Markovian framework

We introduce a mixed {\em generalized} Dynkin game/stochastic control with ${\cal E}^f$-expectation in a Markovian framework. We study both the case when the terminal reward function is supposed to be Borelian only and when it is continuous. We first establish a weak dynamic programming principle by using some refined results recently provided in \cite{DQS} and some properties of doubly reflected BSDEs with jumps (DRBSDEs). We then show a stronger dynamic programming principle in the continuous case, which cannot be derived from the weak one. In particular, we have to prove that the value function of the problem is continuous with respect to time $t$, which requires some technical tools of stochastic analysis and some new results on DRBSDEs. We finally study the links between our mixed problem and generalized Hamilton Jacobi Bellman variational inequalities in both cases.

math.OC↗

arXiv2016

A Weak Dynamic Programming Principle for Combined Optimal Stopping and Stochastic Control with $\mathcal{E}^f$- expectations

We study a combined optimal control/stopping problem under a nonlinear expectation ${\cal E}^f$ induced by a BSDE with jumps, in a Markovian framework. The terminal reward function is only supposed to be Borelian. The value function $u$ associated with this problem is generally irregular. We first establish a {\em sub- (resp. super-) optimality principle of dynamic programming} involving its {\em upper- (resp. lower-) semicontinuous envelope} $u^*$ (resp. $u_*$). This result, called {\em weak} dynamic programming principle (DPP), extends that obtained in \cite{BT} in the case of a classical expectation to the case of an ${\cal E}^f$-expectation and Borelian terminal reward function. Using this {\em weak} DPP, we then prove that $u^*$ (resp. $u_*$) is a {\em viscosity sub- (resp. super-) solution} of a nonlinear Hamilton-Jacobi-Bellman variational inequality.

math.OC↗

arXiv2015

Dynamic robust duality in utility maximization

A celebrated financial application of convex duality theory gives an explicit relation between the following two quantities: (i) The optimal terminal wealth $X^*(T) : = X_{φ^*}(T)$ of the problem to maximize the expected $U$-utility of the terminal wealth $X_φ(T)$ generated by admissible portfolios $φ(t), 0 \leq t \leq T$ in a market with the risky asset price process modeled as a semimartingale; (ii) The optimal scenario $\frac{dQ^*}{dP}$ of the dual problem to minimize the expected $V$-value of $\frac{dQ}{dP}$ over a family of equivalent local martingale measures $Q$, where $V$ is the convex conjugate function of the concave function $U$. In this paper we consider markets modeled by Itô-Lévy processes. In the first part we use the maximum principle in stochastic control theory to extend the above relation to a \emph{dynamic} relation, valid for all $t \in [0,T]$. We prove in particular that the optimal adjoint process for the primal problem coincides with the optimal density process, and that the optimal adjoint process for the dual problem coincides with the optimal wealth process, $0 \leq t \leq T$. In the terminal time case $t=T$ we recover the classical duality connection above. We get moreover an explicit relation between the optimal portfolio $φ^*$ and the optimal measure $Q^*$. We also obtain that the existence of an optimal scenario is equivalent to the replicability of a related $T$-claim. In the second part we present robust (model uncertainty) versions of the optimization problems in (i) and (ii), and we prove a similar dynamic relation between them. In particular, we show how to get from the solution of one of the problems to the other. We illustrate the results with explicit examples.

q-fin.PM↗

arXiv2015

Optimal control of predictive mean-field equations and applications to finance

We study a coupled system of controlled stochastic differential equations (SDEs) driven by a Brownian motion and a compensated Poisson random measure, consisting of a forward SDE in the unknown process $X(t)$ and a \emph{predictive mean-field} backward SDE (BSDE) in the unknowns $Y(t), Z(t), K(t,\cdot)$. The driver of the BSDE at time $t$ may depend not just upon the unknown processes $Y(t), Z(t), K(t,\cdot)$, but also on the predicted future value $Y(t+δ)$, defined by the conditional expectation $A(t):= E[Y(t+δ) | \mathcal{F}_t]$. \\ We give a sufficient and a necessary maximum principle for the optimal control of such systems, and then we apply these results to the following two problems:\\ (i) Optimal portfolio in a financial market with an \emph{insider influenced asset price process.} \\ (ii) Optimal consumption rate from a cash flow modeled as a geometric It\^ o-L\' evy SDE, with respect to \emph{predictive recursive utility}.

math.OC↗

arXiv2014

Generalized Dynkin Games and Doubly Reflected BSDEs with Jumps

We introduce a generalized Dynkin game problem with non linear conditional expectation ${\cal E}$ induced by a Backward Stochastic Differential Equation (BSDE) with jumps. Let $ξ, ζ$ be two RCLL adapted processes with $ξ\leq ζ$. The criterium is given by \begin{equation*} {\cal J}_{τ, σ}= {\cal E}_{0, τ\wedge σ} \left(ξ_τ\textbf{1}_{\{ τ\leq σ\}}+ζ_σ\textbf{1}_{\{σ<τ\}}\right) \end{equation*} where $τ$ and $ σ$ are stopping times valued in $[0,T]$. Under Mokobodski's condition, we establish the existence of a value function for this game, i.e. $\inf_σ\sup_τ {\cal J}_{τ, σ} = \sup_τ \inf_σ {\cal J}_{τ, σ}$. This value can be characterized via a doubly reflected BSDE. Using this characterization, we provide some new results on these equations, such as comparison theorems and a priori estimates. When $ξ$ and $ζ$ are left upper semicontinuous along stopping times, we prove the existence of a saddle point. We also study a generalized mixed game problem when the players have two actions: continuous control and stopping. We then address the generalized Dynkin game in a Markovian framework and its links with parabolic partial integro-differential variational inequalities with two obstacles.

math.PR↗

arXiv2014

Optimal stopping for dynamic risk measures with jumps and obstacle problems

We study the optimal stopping problem for a monotonous dynamic risk measure induced by a BSDE with jumps in the Markovian case. We show that the value function is a viscosity solution of an obstacle problem for a partial integro-differential variational inequality, and we provide an uniqueness result for this obstacle problem.

math.OC↗

arXiv2014

Singular mean-field control games with applications to optimal harvesting and investment problems

This paper studies singular mean field control problems and singular mean field stochastic differential games. Both sufficient and necessary conditions for the optimal controls and for the Nash equilibrium are obtained. Under some assumptions the optimality conditions for singular mean-field control are reduced to a reflected Skorohod problem, whose solution is proved to exist uniquely. Applications are given to optimal harvesting of stochastic mean-field systems, optimal irreversible investments under uncertainty and to mean-field singular investment games. In particular, a simple singular mean-field investment game is studied where the Nash equilibrium exists but is not unique.

math.OC↗

arXiv2014

Risk minimization in financial markets modeled by Itô-Lévy processes

This paper is mainly a survey of recent research developments regarding methods for risk minimization in financial markets modeled by Itô-Lévy processes, but it also contains some new results on the underlying stochastic maximum principle. The concept of a convex risk measure is introduced, and two representations of such measures are given, namely: (i) the dual representation and (ii) the representation by means of backward stochastic differential equations (BSDEs) with jumps. Depending on the representation, the corresponding risk minimal portfolio problem is studied, either in the context of stochastic differential games or optimal control of forward-backward SDEs. The related concept of recursive utility is also introduced, and corresponding recursive utility maximization problems are studied. In either case the maximum principle for optimal stochastic control plays a crucial role, and in the paper we prove a version of this principle which is stronger than what was previously known. The theory is illustrated by examples, showing explicitly the risk minimizing portfolio in some cases.

math.OC↗

arXiv2014

A comparison theorem for backward SPDEs with jumps

In this paper we obtain a comparison theorem for backward stochastic partial differential equation (SPDEs) with jumps. We apply it to introduce space-dependent convex risk measures as a model for risk in large systems of interacting components.

math.PR↗

arXiv2013

Market viability and martingale measures under partial information

We consider a financial market model with a single risky asset whose price process evolves according to a general jump-diffusion with locally bounded coefficients and where market participants have only access to a partial information flow. For any utility function, we prove that the partial information financial market is locally viable, in the sense that the optimal portfolio problem has a solution up to a stopping time, if and only if the (normalised) marginal utility of the terminal wealth generates a partial information equivalent martingale measure (PIEMM). This equivalence result is proved in a constructive way by relying on maximum principles for stochastic control problems under partial information. We then characterize a global notion of market viability in terms of partial information local martingale deflators (PILMDs). We illustrate our results by means of a simple example.

q-fin.PM↗