Publications

2025

• Logarithmic Sobolev inequalities for non-equilibrium steady states
  
  • Monmarché Pierre
  • Wang Songbo
  Potential Analysis, Springer Verlag, 2025. We consider two methods to establish log-Sobolev inequalities for the invariant measure of a diffusion process when its density is not explicit and the curvature is not positive everywhere. In the first approach, based on the Holley-Stroock and Aida-Shigekawa perturbation arguments [J. Stat. Phys., 46(5-6):1159-1194, 1987, J. Funct. Anal., 126(2):448-475, 1994], the control on the (non-explicit) perturbation is obtained by stochastic control methods, following the comparison technique introduced by Conforti [Ann. Appl. Probab., 33(6A):4608-4644, 2023]. The second method combines the Wasserstein-2 contraction method, used in [Ann. Henri Lebesgue, 6:941-973, 2023] to prove a Poincaré inequality in some non-equilibrium cases, with Wang's hypercontractivity results [Potential Anal., 53(3):1123-1144, 2020]. (10.1007/s11118-025-10211-6)
  DOI : 10.1007/s11118-025-10211-6
• From Hyper Roughness to Jumps as H → -1/2
  
  • Abi Jaber Eduardo
  • Attal Elie
  • Rosenbaum Mathieu
  , 2025. We investigate the weak limit of the hyper-rough square-root process as the Hurst index H goes to -1/2 . This limit corresponds to the fractional kernel $t ^{H-1/2}$ losing integrability. We establish the joint convergence of the couple (X, M) , where X is the hyper-rough process and M the associated martingale, to a fully correlated Inverse Gaussian Lévy jump process. This unveils the existence of a continuum between hyper-rough continuous models and jump processes, as a function of the Hurst index. Since we prove a convergence of continuous to discontinuous processes, the usual Skorokhod J1 topology is not suitable for our problem. Instead, we obtain the weak convergence in the Skorokhod M1 topology for X and in the non-Skorokhod S topology for M .
• A Spectral Dominance Approach to Large Random Matrices
  
  • Bertucci Charles
  • Debbah Mérouane
  • Lasry Jean-Michel
  • Lions Pierre-Louis
  , 2021. This paper presents a novel approach to characterize the dynamics of the limit spectrum of large random matrices. This approach is based upon the notion we call "spectral dominance". In particular, we show that the limit spectral measure can be determined as the derivative of the unique viscosity solution of a partial integro-differential equation. This also allows to make general and "short" proofs for the convergence problem. We treat the cases of Dyson Brownian motions, Wishart processes and present a general class of models for which this characterization holds. (10.48550/arXiv.2105.08983)
  DOI : 10.48550/arXiv.2105.08983
• A class of short-term models for the oil industry addressing speculative storage
  
  • Achdou Yves
  • Bertucci Charles
  • Lasry Jean-Michel
  • Lions Pierre Louis
  • Rostand Antoine
  • Scheinkman Jose
  , 2020. This is a work in progress. The aim is to propose a plausible mechanism for the short term dynamics of the oil market based on the interaction of economic agents. This is a theoretical research which by no means aim at describing all the aspects of the oil market. In particular, we use the tools and terminology of game theory, but we do not claim that this game actually exists in the real world. In parallel, we are currently studying and calibrating a long term model for the oil industry, which addresses the interactions of a monopolists with a competitive fringe of small producers. It is the object of another paper that will be available soon. The present premiminary version does not contain all the economic arguments and all the connections with our long term model. It mostly addresses the description of the model, the equations and numerical simulations focused on the oil industry short term dynamics. A more complete version will be available soon. (10.48550/arXiv.2003.11790)
  DOI : 10.48550/arXiv.2003.11790
• On non-uniqueness of phase retrieval in multidimensions
  
  • Novikov Roman
  • Xu Tianli
  , 2025. We give a large class of examples of non-uniqueness for the phase retrieval problem in multidimensions. Our examples include the case of functions with strongly disconnected compact support. A numerical illustration is also given.
• Martingale property and moment explosions in signature volatility models
  
  • Abi Jaber Eduardo
  • Gassiat Paul
  • Sotnikov Dimitri
  , 2025. We study the martingale property and moment explosions of a signature volatility model, where the volatility process of the log-price is given by a linear form of the signature of a time-extended Brownian motion. Excluding trivial cases, we demonstrate that the price process is a true martingale if and only if the order of the linear form is odd and a correlation parameter is negative. The proof involves a fine analysis of the explosion time of a signature stochastic differential equation. This result is of key practical relevance, as it highlights that, when used for approximation purposes, the linear combination of signature elements must be taken of odd order to preserve the martingale property. Once martingality is established, we also characterize the existence of higher moments of the price process in terms of a condition on a correlation parameter.
• General reproducing properties in RKHS with application to derivative and integral operators
  
  • El-Boukkouri Fatima-Zahrae
  • Garnier Josselin
  • Roustant Olivier
  , 2025. In this paper, we consider the reproducing property in Reproducing Kernel Hilbert Spaces (RKHS). We establish a reproducing property for the closure of the class of combinations of composition operators under minimal conditions. This allows to revisit the sufficient conditions for the reproducing property to hold for the derivative operator, as well as for the existence of the mean embedding function. These results provide a framework of application of the representer theorem for regularized learning algorithms that involve data for function values, gradients, or any other operator from the considered class.
• Maximal Entropy Random Walks in Z: Random and non-random environments
  
  • Thibaut Duboux
  • Gerin Lucas
  • Offret Yoann
  , 2025. The Maximal Entropy Random Walk (MERW) is a natural process on a finite graph, introduced a few years ago with motivations from theoretical physics. The construction of this process relies on Perron-Frobenius theory for adjacency matrices. Generalizing to infinite graphs is rather delicate, and in this article, we treat in a fairly exhaustive manner the case of the MERW on Z with loops, for both random and nonrandom loops. Thanks to an explicit combinatorial representation of the corresponding Perron-Frobenius eigenvectors, we are able to precisely determine the asymptotic behavior of these walks. We show, in particular, that essentially all MERWs on Z with loops have positive speed.
• Quantifying the impact of climate risks on credit risk
  
  • Ndiaye Elisa
  , 2025. Stress-tests are forward-looking risk assessment exercises aimed at evaluating the robustness of financial institutions under adverse but plausible macroeconomic scenarios. These tests, regularly performed voluntarily or required by the financial regulators, provide computations of financial risks' metrics along the provided scenarios. Among these, credit risk stress-testing focuses on estimating the default probabilities (PD) of counterparts in a bank's credit portfolio. However, integrating climate risks into stress-tests introduces unique challenges, such as the need for granular modeling, dynamic adaptation of portfolios, and long-term scenario horizons. The typically used credit risk stress-testing model, known as the Asymptotic Single Risk Factor (ASRF) model, fails to capture the specific dynamics of climate scenarios, namely the effects of transition risks driven by policy changes, technological shifts, and consumer sentiment.This thesis addresses these challenges by developing novel methods to quantify credit risk under Climate Stress-Tests. First, it proposes a probabilistic framework to model corporate business models and their adaptation under energy transition scenarios.Second, it extends this framework to compute scenario-conditional PDs by integrating stochastic processes thanks to a structural and path-dependent credit risk model where both sides of the balance sheet are modeled as stochastic processes and using Nested Monte Carlo simulations.Finally, it explores the impact of a single firm's misaligned anticipations of transition scenarios on credit risk, introducing a model that accounts for a potential re-evaluation of the anticipations at a later stage.The findings demonstrate that cost-based approaches reduce credit risk more effectively than static or reactive strategies, with up to 9 times lower default probabilities for high-emission firms. Forward-looking strategies perform better than others in delayed transition scenarios, leading to 6 times lower PDs compared to cost-agnostic methods. Notably, wrong anticipations do not always increase credit risk. In particular, they may result in improved PDs if they lead to greater relative carbon emissions than the perfect anticipations. In the opposite case, firms with initial misaligned and unfavorable anticipations consistently benefit from reassessing their strategies, reducing PDs by up to 20 times when corrective measures are applied early. These results provide actionable insights and robust methodologies to enhance the reliability and precision of credit risk climate stress-tests.
• Capturing Smile Dynamics with the Quintic Volatility Model: SPX, Skew-Stickiness Ratio and VIX
  
  • Abi Jaber Eduardo
  • Li Shaun Xiaoyuan
  , 2025. We introduce the two-factor Quintic Ornstein-Uhlenbeck model, where volatility is modeled as a polynomial of degree five based on the sum of two Ornstein-Uhlenbeck processes driven by the same Brownian Motion, each mean-reverting at a different speed. We demonstrate that the Quintic model effectively captures the volatility surfaces of SPX and VIX while aligning with the skew-stickiness ratio (SSR) across maturities ranging from a few days to two over years. Furthermore, the Quintic model shows consistency with key empirical stylized facts, notably reproducing the Zumbach effect.
• Meta-modelling paths of simple climate models using Neural Networks and Dirichlet polynomials: An application to DICE
  
  • Gobet Emmanuel
  • Liu Yushan
  • Vermandel Gauthier
  , 2025. Our study focuses on climate models extensively employed in climate science and climate-economy research, which project temperature outcomes from carbon emission trajectories. Addressing the need for rapid evaluation in Integrated Assessment Models (IAMs) -- critical tools for carbon emission mitigation policy analysis -- we design a neural network (NN) meta-model as an efficient surrogate for mapping, in an infinite horizon setting, emission trajectories into temperature trajectories (usually modeled as coupled systems of differential equations). Our approach combines a projection on Generalized Dirichlet polynomials, whose coefficients are inputs of the NN and a suitable time change for handling infinite horizon: we prove that the quantity of interest is, under some assumptions, a smooth function of the inputs and therefore, is prone to accurate NN approximation. After a training with augmented Shared Socioeconomic Pathway scenarios, the NN achieves high-fidelity approximations of the original climate model. Additionally, we establish theoretical accuracy guarantees for both the encoding and neural network approximation. Our numerical experiments demonstrate the framework's accuracy, and computational efficiency is improved by a factor of 100 in comparison to traditional ODE solvers. As an application to Actuarial Sciences, we illustrate the use of the metamodel to quantify the distribution of future scorching days.
• Fredholm Approach to Nonlinear Propagator Models
  
  • Abi Jaber Eduardo
  • Bondi Alessandro
  • de Carvalho Nathan
  • Neuman Eyal
  • Tuschmann Sturmius
  , 2025. We formulate and solve an optimal trading problem with alpha signals, where transactions induce a nonlinear transient price impact described by a general propagator model, including power-law decay. Using a variational approach, we demonstrate that the optimal trading strategy satisfies a nonlinear stochastic Fredholm equation with both forward and backward coefficients. We prove the existence and uniqueness of the solution under a monotonicity condition reflecting the nonlinearity of the price impact. Moreover, we derive an existence result for the optimal strategy beyond this condition when the underlying probability space is countable. In addition, we introduce a novel iterative scheme and establish its convergence to the optimal trading strategy. Finally, we provide a numerical implementation of the scheme that illustrates its convergence, stability, and the effects of concavity on optimal execution strategies under exponential and power-law decay.
• The Volterra Stein-Stein model with stochastic interest rates
  
  • Abi Jaber Eduardo
  • Hainaut Donatien
  • Motte Edouard
  , 2025. <div><p>We introduce the Volterra Stein-Stein model with stochastic interest rates, where both volatility and interest rates are driven by correlated Gaussian Volterra processes. This framework unifies various wellknown Markovian and non-Markovian models while preserving analytical tractability for pricing and hedging financial derivatives. We derive explicit formulas for pricing zero-coupon bond and interest rate cap or floor, along with a semi-explicit expression for the characteristic function of the log-forward index using Fredholm resolvents and determinants. This allows for fast and efficient derivative pricing and calibration via Fourier methods. We calibrate our model to market data and observe that our framework is flexible enough to capture key empirical features, such as the humped-shaped term structure of ATM implied volatilities for cap options and the concave ATM implied volatility skew term structure (in loglog scale) of the S&amp;P 500 options. Finally, we establish connections between our characteristic function formula and expressions that depend on infinite-dimensional Riccati equations, thereby making the link with conventional linear-quadratic models.</p></div>
• Complex discontinuities of the square root of Fredholm determinants in the Volterra Stein-Stein model
  
  • Abi Jaber Eduardo
  • Guellil Maxime
  , 2025. <div><p>Fourier-based methods are central to option pricing and hedging when the Fourier–Laplace transform of the log-price and integrated variance is available semi-explicitly. This is the case for the Volterra Stein–Stein stochastic volatility model, where the characteristic function is known analytically. However, naive evaluation of this formula can produce discontinuities due to the complex square root of a Fredholm determinant, particularly when the determinant crosses the negative real axis, leading to severe numerical instabilities. We analyze this phenomenon by characterizing the determinant’s crossing behavior for the joint Fourier–Laplace transform of integrated variance and log-price. We then derive an expression for the transform to account for such crossings and develop efficient algorithms to detect and handle them. Applied to Fourier-based pricing in the rough Stein–Stein model, our approach significantly improves accuracy while drastically reducing computational cost relative to existing methods.</p></div>
• On the strong law of large numbers and Llog L condition for supercritical general branching processes
  
  • Bansaye Vincent
  • Berah Tresnia
  • Cloez Bertrand
  , 2025. We consider branching processes for structured populations: each individual is characterized by a type or trait which belongs to a general measurable state space. We focus on the supercritical recurrent case, where the population may survive and grow and the trait distribution converges. The branching process is then expected to be driven by the positive triplet of first eigenvalue problem of the first moment semigroup. Under the assumption of convergence of the renormalized semigroup in weighted total variation norm, we prove strong convergence of the normalized empirical measure and non-degeneracy of the limiting martingale. Convergence is obtained under an Llog L condition which provides a Kesten-Stigum result in infinite dimension and relaxes the uniform convergence assumption of the renormalized first moment semigroup required in the work of Asmussen and Hering in 1976. The techniques of proofs combine families of martingales and contraction of semigroups and the truncation procedure of Asmussen and Hering. We also obtain L^1 convergence of the renormalized empirical measure and contribute to unifying different results in the literature. These results greatly extend the class of examples where a law of large numbers applies, as we illustrate it with absorbed branching diffusion, the house of cards model and some growth-fragmentation processes.
• Sufficient dimension reduction for regression with spatially correlated errors: application to prediction
  
  • Forzani Liliana
  • Arancibia Rodrigo García
  • Gieco Antonella
  • Llop Pamela
  • Yao Anne-Françoise
  , 2025. In this paper, we address the problem of predicting a response variable in the context of both, spatially correlated and high-dimensional data. To reduce the dimensionality of the predictor variables, we apply the sufficient dimension reduction (SDR) paradigm, which reduces the predictor space while retaining relevant information about the response. To achieve this, we impose two different spatial models on the inverse regression: the separable spatial covariance model (SSCM) and the spatial autoregressive error model (SEM). For these models, we derive maximum likelihood estimators for the reduction and use them to predict the response via nonparametric rules for forward regression. Through simulations and real data applications, we demonstrate the effectiveness of our approach for spatial data prediction. (10.48550/arXiv.2502.02781)
  DOI : 10.48550/arXiv.2502.02781
• Interior Point Methods Are Not Worse than Simplex
  
  • Allamigeon Xavier
  • Dadush Daniel
  • Loho Georg
  • Natura Bento
  • Végh László
  SIAM Journal on Computing, Society for Industrial and Applied Mathematics, 2025, 54 (5), pp.FOCS22-178-FOCS22-264. (10.1137/23M1554588)
  DOI : 10.1137/23M1554588
• A mesh-independent method for second-order potential mean field games
  
  • Liu Kang
  • Pfeiffer Laurent
  IMA Journal of Numerical Analysis, Oxford University Press (OUP), 2025, 45 (2), pp.1226-1266. This article investigates the convergence of the Generalized Frank-Wolfe (GFW) algorithm for the resolution of potential and convex second-order mean field games. More specifically, the impact of the discretization of the mean-field-game system on the effectiveness of the GFW algorithm is analyzed. The article focuses on the theta-scheme introduced by the authors in a previous study. A sublinear and a linear rate of convergence are obtained, for two different choices of stepsizes. These rates have the mesh-independence property: the underlying convergence constants are independent of the discretization parameters. (10.1093/imanum/drae061)
  DOI : 10.1093/imanum/drae061
• Ancestral lineages and sampling in populations with density-dependent interactions
  
  • Kubasch Madeleine
  , 2025. We study a density-dependent Markov jump process describing a population where each individual is characterized by a type, and reproduces at rates depending both on its type and on the population type distribution. First, using an appropriate change in probability, we exhibit a time-inhomogeneous Markov process, the auxiliary process, which allows to capture the behavior of a sampled lineage in the population process. This is achieved through a many-to-one formula, which relates the average of a function over ancestral lineages sampled in the population processes to its average over the auxiliary process, yielding a direct interpretation of the underlying survivorship bias. In addition, this construction allows for more general sampling procedures than what was previously obtained in the literature, such as sampling restricted to subpopulations. Second, we consider the large population regime, when the population size grows to infinity. Under classical assumptions, the population type distribution can then be approached by a diffusion approximation, which captures the fluctuations of the population process around its deterministic large population limit. We establish a many-to-one formula allowing to sample in the diffusion approximation, and quantify the associated approximation error.
• Applications of new boundary conditions for the Boltzmann equation derived from a kinetic model of gas-surface interaction
  
  • Kosuge Shingo
  • Aoki Kazuo
  • Giovangigli Vincent
  • Golse François
  , 2024. Recently, new models of the boundary condition for the Boltzmann equation were proposed on the basis of a kinetic model of gas-surface interactions [K. Aoki et.al., Phys. Rev. E 106(3), 035306 (2022)]. In the present paper, the kernel representations of the models are given, and the models are applied to some basic problems of a rarefied gas between two parallel plates. To be more specific, the heat-transfer between the plates with different temperatures, plane Couette flow, and plane Poiseuille flow driven by an external force are numerically investigated by using the Bhatnagar-Gross-Krook (BGK) model of the Boltzmann equation and the new models of the boundary condition. The results are compared with those based on the conventional Maxwell-type boundary conditions.
• Wave dispersion and bifurcation analyses of eikonal gradient-enhanced isotropic damage models
  
  • Ribeiro Nogueira Breno
  • Rastiello Giuseppe
  • Giry Cédric
  • Gatuingt Fabrice
  European Journal of Mechanics - A/Solids, Elsevier, 2025, pp.105643. (10.1016/j.euromechsol.2025.105643)
  DOI : 10.1016/j.euromechsol.2025.105643
• A two spaces extension of Cauchy-Lipschitz Theorem
  
  • Bertucci Charles
  • Lions Pierre-Louis
  Journal of Differential Equations, Elsevier, 2025, 421, pp.524-530. We adapt the classical theory of local well-posedness of evolution problems to cases in which the nonlinearity can be accurately quantified by two different norms. For ordinary differential equations, we consider ẋ = f (x, x) for a function f : V × E → E where E is a Banach space and V → E a normed vector space. This structure allows us to distinguish between the two dependencies of f in x and allows to generalize classical results. We also prove a similar results for partial differential equations. (10.1016/j.jde.2024.12.031)
  DOI : 10.1016/j.jde.2024.12.031
• Open Problem: Two Riddles in Heavy-Ball Dynamics
  
  • Goujaud Baptiste
  • Taylor Adrien
  • Dieuleveut Aymeric
  , 2025. This short paper presents two open problems on the widely used Polyak's Heavy-Ball algorithm. The first problem is the method's ability to exactly \textit{accelerate} in dimension one exactly. The second question regards the behavior of the method for parameters for which it seems that neither a Lyapunov nor a cycle exists. For both problems, we provide a detailed description of the problem and elements of an answer.
• CoHiRF: A Scalable and Interpretable Clustering Framework for High-Dimensional Data
  
  • Belucci Bruno
  • Lounici Karim
  • Meziani Katia
  , 2025. Clustering high-dimensional data poses significant challenges due to the curse of dimensionality, scalability issues, and the presence of noisy and irrelevant features. We propose Consensus Hierarchical Random Feature (CoHiRF), a novel clustering method designed to address these challenges effectively. CoHiRF leverages random feature selection to mitigate noise and dimensionality effects, repeatedly applies K-Means clustering in reduced feature spaces, and combines results through a unanimous consensus criterion. This iterative approach constructs a cluster assignment matrix, where each row records the cluster assignments of a sample across repetitions, enabling the identification of stable clusters by comparing identical rows. Clusters are organized hierarchically, enabling the interpretation of the hierarchy to gain insights into the dataset. CoHiRF is computationally efficient with a running time comparable to K-Means, scalable to massive datasets, and exhibits robust performance against state-of-the-art methods such as SC-SRGF, HDBSCAN, and OPTICS. Experimental results on synthetic and real-world datasets confirm the method's ability to reveal meaningful patterns while maintaining scalability, making it a powerful tool for high-dimensional data analysis.
• An inverse problem in cell dynamics: Recovering an initial distribution of telomere lengths from measurements of senescence times
  
  • Olayé Jules
  , 2025. Telomeres are repetitive sequences situated at both ends of the chromosomes of eukaryotic cells. At each cell division, they are eroded until they reach a critical length that triggers a state in which the cell stops to divide: the senescent state. In this work, we are interested in the link between the initial distribution of telomere lengths and the distribution of senescence times. We propose a method to retrieve the initial distribution of telomere lengths, using only measurements of senescence times. Our approach relies on approximating our models with transport equations, which provide natural estimators for the initial telomere lengths distribution. We investigate this method from a theoretical point of view by providing bounds on the errors of our estimators, pointwise and in all Lebesgue spaces. We also illustrate it with estimations on simulations, and discuss its limitations related to the curse of dimensionality.