arXiv subjects

Badr Missaoui

Publications and source records attributed to Badr Missaoui.

13 recordsLinked to original sources

arXiv2026

Initialization-driven neural generation and training for high-dimensional optimal control and first-order mean field games

This paper first introduces a method to approximate the value function of high-dimensional optimal control by neural networks. Based on the established relationship between Pontryagin's maximum principle (PMP) and the value function of the optimal control problem, which is characterized as being the unique solution to an associated Hamilton-Jacobi-Bellman (HJB) equation, we propose an approach that begins by using neural networks to provide a first rough estimate of the value function, which serves as initialization for solving the two point boundary value problem in the PMP and, as a result, generates reliable data. To train the neural network we define a loss function that takes into account this dataset and also penalizes deviations from the HJB equation. In the second part, we address the computation of equilibria in first-order Mean Field Game (MFG) problems by integrating our method with the fictitious play algorithm. These equilibria are characterized by a coupled system of a first-order HJB equation and a continuity equation. To approximate the solution to the continuity equation, we introduce a second neural network that learns the flow map transporting the initial distribution of agents. This network is trained on data generated by solving the underlying ODEs for a batch of initial conditions sampled from the initial distribution of agents. By combining this flow approximation, the previously described method for approximating the value function, and the fictitious play algorithm, we obtain an effective method to tackle high-dimensional deterministic MFGs.

math.OC↗

arXiv2026

Szego Theorem for Operator Orthogonal Polynomials

In this paper, we extend some classical results of the Szego theory of orthogonal polynomials on the unit circle to the infinite-dimensional case, and we establish the corresponding Szego limit theorem.

math.PR↗

arXiv2026

Mean Field Games with Reflected Dynamics: Penalization and Relaxed Control Approach

In this paper, we investigate a class of Mean Field Games (MFGs) in which the state dynamics are governed by multidimensional reflected stochastic differential equations (SDEs). We establish the existence of an equilibrium and show that it can be approximated by the equilibrium of MFGs with non-reflected SDE. This approximation is constructed via a penalization method combined with the relaxed control approach introduced in [21]. Under a uniform ellipticity condition, and by applying the penalization method together with the mimicking theorem, we prove the existence of a Markovian MFG. Furthermore, under an additional convexity assumption, we demonstrate the existence of a strict-Markovian MFG. In the general case, we prove that relaxed MFG solutions with reflected dynamics can be approximated by strict controls whose dynamics are governed by penalized SDEs.

math.PR↗

arXiv2026

Mean Field Games with Reflected Dynamics

This paper establishes an equilibrium existence result for a class of Mean Field Games involving Reflected Stochastic Differential Equations. The proof relies on the framework of relaxed controls and martingale problems.

math.PR↗

arXiv2025

Multidimensional McKean-Vlasov SDEs with mean reflection: well-posedness and existence of optimal control

In this work, we investigate the multidimensional Skorokhod problem for càdlàg processes, where the reflection is subject to a minimality condition depending on the law of the solution. We then apply these results to establish existence and uniqueness for multidimensional McKean-Vlasov stochastic differential equations with mean reflection. Finally, we address the existence of optimal relaxed controls for such equations.

math.PR↗

arXiv2025

Optimal control problem for reflected stochastic differential equation: existence

We establish the existence of both optimal relaxed controls and strict optimal controls for systems driven by Reflected Stochastic Differential Equations RSDEs. Our approach is based on weak convergence techniques for the associated RSDEs in the uniform convergence topology, along with an appropriate Skorokhod representation theorem.

math.PR↗

arXiv2024

GeoScatt-GNN: A Geometric Scattering Transform-Based Graph Neural Network Model for Ames Mutagenicity Prediction

This paper tackles the pressing challenge of mutagenicity prediction by introducing three ground-breaking approaches. First, it showcases the superior performance of 2D scattering coefficients extracted from molecular images, compared to traditional molecular descriptors. Second, it presents a hybrid approach that combines geometric graph scattering (GGS), Graph Isomorphism Networks (GIN), and machine learning models, achieving strong results in mutagenicity prediction. Third, it introduces a novel graph neural network architecture, MOLG3-SAGE, which integrates GGS node features into a fully connected graph structure, delivering outstanding predictive accuracy. Experimental results on the ZINC dataset demonstrate significant improvements, emphasizing the effectiveness of blending 2D and geometric scattering techniques with graph neural networks. This study illustrates the potential of GNNs and GGS for mutagenicity prediction, with broad implications for drug discovery and chemical safety assessment.

cs.LG↗

arXiv2024

Deep Policy Iteration for High-Dimensional Mean Field Games

This paper introduces Deep Policy Iteration (DPI), a novel approach that integrates the strengths of Neural Networks with the stability and convergence advantages of Policy Iteration (PI) to address high-dimensional stochastic Mean Field Games (MFG). DPI overcomes the limitations of PI, which is constrained by the curse of dimensionality to low-dimensional problems, by iteratively training three neural networks to solve PI equations and satisfy forward-backwards conditions. Our findings indicate that DPI achieves comparable convergence levels to the Mean Field Deep Galerkin Method (MFDGM), with additional advantages. Furthermore, deep learning techniques show promise in handling separable Hamiltonian cases where PI alone is less effective. DPI effectively manages high-dimensional problems, extending the applicability of PI to both separable and non-separable Hamiltonians.

math.OC↗

arXiv2024

Reflected Mckean-Vlasov stochastic differential equations with jumps in time-dependent domains

In this paper, we investigate the deterministic multidimensional Skorokhod problem with normal reflection in a family of time-dependent convex domains that are càdlàg with respect to the Hausdorff metric. We then show the existence and uniqueness of solutions to multidimensional McKean-Vlasov stochastic differential equations reflected in these time-dependent domains. Additionally, we derive stability properties with respect to the initial condition and the coefficients. Finally, we establish a propagation of chaos result.

math.PR↗

arXiv2024

Method of Successive Approximations for Stochastic Optimal Control: Contractivity and Convergence

The Method of Successive Approximations (MSA) is a fixed-point iterative method used to solve stochastic optimal control problems. It is an indirect method based on the conditions derived from the Stochastic Maximum Principle (SMP), an extension of the Pontryagin Maximum Principle (PMP) to stochastic control problems. In this study, we investigate the contractivity and the convergence of MSA for a specific and interesting class of stochastic dynamical systems (when the drift coefficient is one-sided-Lipschitz with a negative constant and the diffusion coefficient is Lipschitz continuous). Our analysis unfolds in three key steps: firstly, we prove the stability of the state process with respect to the control process. Secondly, we establish the stability of the adjoint process. Finally, we present rigorous evidence to prove the contractivity and then the convergence of MSA. This study contributes to enhancing the understanding of MSA's applicability and effectiveness in addressing stochastic optimal control problems.

math.OC↗

arXiv2024

Integrating Graph Neural Networks with Scattering Transform for Anomaly Detection

In this paper, we present two novel methods in Network Intrusion Detection Systems (NIDS) using Graph Neural Networks (GNNs). The first approach, Scattering Transform with E-GraphSAGE (STEG), utilizes the scattering transform to conduct multi-resolution analysis of edge feature vectors. This provides a detailed representation that is essential for identifying subtle anomalies in network traffic. The second approach improves node representation by initiating with Node2Vec, diverging from standard methods of using uniform values, thereby capturing a more accurate and holistic network picture. Our methods have shown significant improvements in performance compared to existing state-of-the-art methods in benchmark NIDS datasets.

cs.CR↗

arXiv2023

Deep Learning for Mean Field Games with non-separable Hamiltonians

This paper introduces a new method based on Deep Galerkin Methods (DGMs) for solving high-dimensional stochastic Mean Field Games (MFGs). We achieve this by using two neural networks to approximate the unknown solutions of the MFG system and forward-backward conditions. Our method is efficient, even with a small number of iterations, and is capable of handling up to 300 dimensions with a single layer, which makes it faster than other approaches. In contrast, methods based on Generative Adversarial Networks (GANs) cannot solve MFGs with non-separable Hamiltonians. We demonstrate the effectiveness of our approach by applying it to a traffic flow problem, which was previously solved using the Newton iteration method only in the deterministic case. We compare the results of our method to analytical solutions and previous approaches, showing its efficiency. We also prove the convergence of our neural network approximation with a single hidden layer using the universal approximation theorem.

cs.LG↗

arXiv2016

Nonparametric and arbitrage-free construction of call surfaces using l1-recovery

This paper is devoted to the application of an $l_1$ -minimisation technique to construct an arbitrage-free call-option surface. We propose a nononparametric approach to obtaining model-free call option surfaces that are perfectly consistent with market quotes and free of static arbitrage. The approach is inspired from the compressed-sensing framework that is used in signal processing to deal with under-sampled signals. We address the problem of fitting the call-option surface to sparse option data. To illustrate the methodology, we proceed to the construction of the whole call-price surface of the S\&P500 options, taking into account the arbitrage possibilities in the time direction. The resulting object is a surface free of both butterfly and calendar-spread arbitrage that matches the original market points. We then move on to an FX application, namely the HKD/USD call-option surface.

q-fin.CP↗