Publications

2025

• The quest for the GRAph Level autoEncoder (GRALE)
  
  • Krzakala Paul
  • Melo Gabriel
  • Laclau Charlotte
  • d'Alché-Buc Florence
  • Flamary Rémi
  , 2025. Although graph-based learning has attracted a lot of attention, graph representation learning is still a challenging task whose resolution may impact key application fields such as chemistry or biology. To this end, we introduce GRALE, a novel graph autoencoder that encodes and decodes graphs of varying sizes into a shared embedding space. GRALE is trained using an Optimal Transport-inspired loss that compares the original and reconstructed graphs and leverages a differentiable node matching module, which is trained jointly with the encoder and decoder. The proposed attention-based architecture relies on Evoformer, the core component of AlphaFold, which we extend to support both graph encoding and decoding. We show, in numerical experiments on simulated and molecular data, that GRALE enables a highly general form of pre-training, applicable to a wide range of downstream tasks, from classification and regression to more complex tasks such as graph interpolation, editing, matching, and prediction.
• Thresholds for sensitive optimality and Blackwell optimality in stochastic games
  
  • Gaubert Stéphane
  • Grand-Clément Julien
  • Katz Ricardo D.
  , 2025. We investigate refinements of the mean-payoff criterion in two-player zero-sum perfect-information stochastic games. A strategy is Blackwell optimal if it is optimal in the discounted game for all discount factors sufficiently close to $1$. The notion of $d$-sensitive optimality interpolates between mean-payoff optimality (corresponding to the case $d=-1$) and Blackwell optimality ($d=+\infty$). The Blackwell threshold $α_{\sf Bw} \in [0,1[$ is the discount factor above which all optimal strategies in the discounted game are guaranteed to be Blackwell optimal. The $d$-sensitive threshold $α_{\sf d} \in [0,1[$ is defined analogously. Bounding $α_{\sf Bw}$ and $α_{\sf d}$ are fundamental problems in algorithmic game theory, since these thresholds control the complexity for computing Blackwell and $d$-sensitive optimal strategies, by reduction to discounted games which can be solved in $O\left((1-α)^{-1}\right)$ iterations. We provide the first bounds on the $d$-sensitive threshold $α_{\sf d}$ beyond the case $d=-1$, and we establish improved bounds for the Blackwell threshold $α_{\sf Bw}$. This is achieved by leveraging separation bounds on algebraic numbers, relying on Lagrange bounds and more advanced techniques based on Mahler measures and multiplicity theorems.
• A holographic non-uniqueness for the Helmholtz equation
  
  • Novikov Roman
  • Sivkin Vladimir
  , 2025. We consider a plane wave, a radiation solution, and the sum of these solutions (total solution) for the Helmholtz equation in $\mathbb{R}^d$ , $d \geq 2$, outside the origin. We consider a sphere $S$ centered at the origin in $\mathbb{R}^d$ . We show that the radiation solution on $S$ is not uniquely determined by the intensity of the total solution on $S$. Extensions of this result to the case of other surfaces in place of $S$ are also mentioned. Our construction involves and develops technique of scattering on obstacles with Dirichlet boundary condition.
• Design of experiments based on a low fidelity model for seismic fragility curves estimation
  
  • Van Biesbroeck Antoine
  • Gauchy Clément
  • Feau Cyril
  • Garnier Josselin
  ESAIM: Proceedings and Surveys, EDP Sciences, 2025, 79, pp.96-109. Seismic fragility curves are key quantities of interest for Seismic Probabilistic Risk Assessment studies. They express the probability of failure of a mechanical structure of interest conditional to a scalar value derived from the ground motion signal coined Intensity Measure. In the literature, Bayesian approaches have emerged to enable their estimation within the difficult context of limited data availability. Yet, the log-normal modeling over which most of them are based requires the use of computationally expensive Markov chain Monte Carlo methods for providing Bayesian estimators. In this work, we propose an efficient modeling for the estimation of fragility curves in the Bayesian context, based on a low fidelity model of the structure’s response to the ground motion signal and an objective prior. The analytical expression of our modeling allows fast generation of estimates. Also, the representative bias arisen by the modeling choice is handled with a judicious design of experiments methodology. Finally, our method is evaluated on a real case study, and the results highlight its efficiency and its ability to robustly overcome any bias when coupled with the design of experiments we propose. (10.1051/proc/202579096)
  DOI : 10.1051/proc/202579096
• Relu and softplus neural nets as zero-sum turn-based games
  
  • Gaubert Stephane
  • Vlassopoulos Yiannis
  , 2025. We show that the output of a ReLU neural network can be interpreted as the value of a zero-sum, turn-based, stopping game, which we call the ReLU net game. The game runs in the direction opposite to that of the network, and the input of the network serves as the terminal reward of the game. In fact, evaluating the network is the same as running the Shapley-Bellman backward recursion for the value of the game. Using the expression of the value of the game as an expected total payoff with respect to the path measure induced by the transition probabilities and a pair of optimal policies, we derive a discrete Feynman-Kac-type path-integral formula for the network output. This game-theoretic representation can be used to derive bounds on the output from bounds on the input, leveraging the monotonicity of Shapley operators, and to verify robustness properties using policies as certificates. Moreover, training the neural network becomes an inverse game problem: given pairs of terminal rewards and corresponding values, one seeks transition probabilities and rewards of a game that reproduces them. Finally, we show that a similar approach applies to neural networks with Softplus activation functions, where the ReLU net game is replaced by its entropic regularization.
• The Nullstellensatz and Positivstellensatz for Sparse Tropical Polynomial Systems
  
  • Akian Marianne
  • Béreau Antoine
  • Gaubert Stéphane
  Foundations of Computational Mathematics, Springer Verlag, 2025. Grigoriev and Podolskii (2018) have established a tropical analogue of the effective Nullstellensatz, showing that a system of tropical polynomial equations is solvable if and only if a linearized system obtained from a truncated Macaulay matrix is solvable. They provided an upper bound of the minimal admissible truncation degree, as a function of the degrees of the tropical polynomials. We establish a tropical Nullstellensatz adapted to sparse tropical polynomial systems. Our approach is inspired by a construction of Canny-Emiris (1993), refined by Sturmfels (1994). This leads to an improved bound of the truncation degree, which coincides with the classical Macaulay degree in the case of n + 1 equations in n unknowns. We also establish a tropical Positivstellensatz, allowing one to decide the inclusion of tropical basic semialgebraic sets. This allows one to reduce decision problems for tropical semi-algebraic sets to the solution of systems of tropical linear equalities and inequalities. (10.1007/s10208-025-09708-8)
  DOI : 10.1007/s10208-025-09708-8
• Do you precondition on the left or on the right?
  
  • Spillane Nicole
  • Matalon Pierre
  • Szyld Daniel B
  , 2025. This work is a follow-up to a poster that was presented at the DD29 conference. Participants were asked the question: “Do you precondition on the left or on the right?”. Here we report on the results of this social experiment. We also provide context on left, right and split preconditioning, share our literature review on the topic, and analyze some of the finer points. Two examples illustrate that convergence bounds can sometimes lead to misleading conclusions.
• Signature approach for pricing and hedging path-dependent options with frictions
  
  • Abi Jaber Eduardo
  • Hainaut Donatien
  • Motte Edouard
  , 2025. We introduce a novel signature approach for pricing and hedging path-dependent options with instantaneous and permanent market impact under a mean-quadratic variation criterion. Leveraging the expressive power of signatures, we recast an inherently nonlinear and non-Markovian stochastic control problem into a tractable form, yielding hedging strategies in (possibly infinite) linear feedback form in the time-augmented signature of the control variables, with coefficients characterized by non-standard infinite-dimensional Riccati equations on the extended tensor algebra. Numerical experiments demonstrate the effectiveness of these signature-based strategies for pricing and hedging general path-dependent payoffs in the presence of frictions. In particular, market impact naturally smooths optimal trading strategies, making low-truncated signature approximations highly accurate and robust in frictional markets, contrary to the frictionless case.
• Multi-source and test-time domain adaptation on multivariate signals using Spatio-Temporal Monge Alignment
  
  • Gnassounou Theo
  • Collas Antoine
  • Flamary Rémi
  • Lounici Karim
  • Gramfort Alexandre
  , 2024. Machine learning applications on signals such as computer vision or biomedical data often face significant challenges due to the variability that exists across hardware devices or session recordings. This variability poses a Domain Adaptation (DA) problem, as training and testing data distributions often differ. In this work, we propose Spatio-Temporal Monge Alignment (STMA) to mitigate these variabilities. This Optimal Transport (OT) based method adapts the cross-power spectrum density (cross-PSD) of multivariate signals by mapping them to the Wasserstein barycenter of source domains (multi-source DA). Predictions for new domains can be done with a filtering without the need for retraining a model with source data (test-time DA). We also study and discuss two special cases of the method, Temporal Monge Alignment (TMA) and Spatial Monge Alignment (SMA). Non-asymptotic concentration bounds are derived for the mappings estimation, which reveals a bias-plus-variance error structure with a variance decay rate of $\mathcal{O}(n_\ell^{-1/2})$ with $n_\ell$ the signal length. This theoretical guarantee demonstrates the efficiency of the proposed computational schema. Numerical experiments on multivariate biosignals and image data show that STMA leads to significant and consistent performance gains between datasets acquired with very different settings. Notably, STMA is a pre-processing step complementary to state-of-the-art deep learning methods. (10.48550/arXiv.2407.14303)
  DOI : 10.48550/arXiv.2407.14303
• PSDNORM: TEST-TIME TEMPORAL NORMALIZATION FOR DEEP LEARNING IN SLEEP STAGING
  
  • Gnassounou Theo
  • Collas Antoine
  • Flamary Rémi
  • Gramfort Alexandre
  , 2025. <div><p>Distribution shift poses a significant challenge in machine learning, particularly in biomedical applications using data collected across different subjects, institutions, and recording devices, such as sleep data. While existing normalization layers, BatchNorm, LayerNorm and InstanceNorm, help mitigate distribution shifts, when applied over the time dimension they ignore the dependencies and auto-correlation inherent to the vector coefficients they normalize. In this paper, we propose PSDNorm that leverages Monge mapping and temporal context to normalize feature maps in deep learning models for signals. Notably, the proposed method operates as a test-time domain adaptation technique, addressing distribution shifts without additional training. Evaluations with architectures based on U-Net or transformer backbones trained on 10K subjects across 10 datasets, show that PSDNorm achieves state-of-the-art performance on unseen left-out datasets while being 4-times more data-efficient than BatchNorm.</p></div> (10.48550/arXiv.2503.04582)
  DOI : 10.48550/arXiv.2503.04582
• Certificates of nonnegativity of multivariate polynomials with zero-dimensional gradient ideal and infimum attained
  
  • Bender Matías
  • Tsigaridas Elias
  • Zhu Chaoping
  , 2025. We study the problem of certifying nonnegativity of multivariate polynomials with rational coefficients, under the assumptions that the polynomial attains its infimum and its gradient ideal is zero-dimensional. We introduce a new certificate, based on Rational Univariate Representations (RUR), that is both perfectly complete, it certifies every nonnegative polynomial, and perfectly sound, it correctly identifies any polynomial that is not nonnegative. The certificate reduces the problem to checking whether a univariate polynomial admits a weighted sum-of-squares decomposition modulo an ideal defined by the RUR, thereby avoiding the radicality assumption that limits other existing approaches. We present a Monte Carlo algorithm, \sosrur, that produces such a certificate or returns a rational witness point where the polynomial takes a negative value. For a polynomial in $n$ variables of degree~$d$ and maximum coefficient bitsize~$\tau$, \sosrur runs in bit complexity $\widetilde{\mathcal{O}}_B(e^{\omega n + \mathcal{O}(\lg n)} d^{(\omega+2)n}(d+\tau))$, and the polynomials in the certificate have coefficients of bitsize at most $\widetilde{\mathcal{O}}(d^{2n+1}(d+n\tau))$, when we ignore polylogarithmic factors. If the polynomial can take negative values, then the algorithm outputs a rational witness point with coordinates of bitsize $\widetilde{\mathcal{O}}(nd^{2n+2}(d+n\tau))$. In the special case where the gradient ideal is radical, in addition to the improvement in the bit size, our certificate specializes in a representation with improved degree bounds, better by an additive factor of $d$, and lower bit complexity compared to previous results, together with an explicit validation of the RUR, which had previously been overlooked. We also consider sparse certificates, that their structure, complexity, and (bitsize) bounds depend on the Newton polytope~$Q$ of the input polynomial and not on the total degree $d$. To compute them, we rely on a Monte Carlo algorithm to obtain a sparse RUR for unmixed systems, which is of independent interest. In particular, we compute the sparse certificate in {$\widetilde{\mathcal{O}}_B( n^{6n} \, \vol(Q)^6 \, \tau)$} bit operations, where $\vol(Q)$ is the volume of $Q$. The certificate involves sparse polynomials of degree at most $2^{n+2} \, n!^2 \, \vol(Q)^2$ and coefficient bitsize {$\widetilde{\mathcal{O}}(n^{4n} \, \vol(Q)^4 \, \tau)$}. These bounds provide sharper guaranties in the presence of sparsity, as they are input sensitive. This result contributes to the ongoing effort to encode polynomial optimization problems through compact descriptions, in our case by exploiting the sparsity of the input, thereby extending the current frontier of tractability from a computational point of view. Overall, our results extend the scope of certificates modulo the gradient ideal by eliminating the radicality requirement, introduce a variant that exploits the sparsity of the input polynomial, improve existing bitsize complexity bounds by at least a factor of~$d^n$, and provide efficient, verifiable algebraic certificates of nonnegativity for polynomial optimization.
• Optimal exit from Uniswap v3 and best expected return for a liquidity provider
  
  • Agarwal Ankush
  • Gobet Emmanuel
  , 2025. We analyze the profitability of liquidity providers’ (LPs) positions in Uniswap v3 by aggregating fee income and impermanent loss within an optimal stopping framework. Our first result shows that the liquidity burn should be optimized over one range at a time, rather than simultaneously. Second, without discounting future fees, there is no finite optimal liquidity burn time and indefinite liquidity provision is optimal. In this case, we derive closed-form expressions for the value of LP positions according to different price levels of liquidity burn. Third, with a discount factor, we introduce an equivalent rate of return and demonstrate that under a Black-Scholes model with volatility σ, the optimal return is approximately 0.425·σ2 (i.e. about 10% return for 50% volatility), and it is achieved by choosing the at-the-money range of liquidity. These results provide explicit formulas and strategic insights for LPs in Uniswap v3, and complement recent works on Uniswap v2 and fee modelling by highlighting the distinct impact of concentrated liquidity.
• AdaCap: An Adaptive Contrastive Approach for Small-Data Neural Networks
  
  • Belucci Bruno
  • Lounici Karim
  • Meziani Katia
  , 2025. Neural networks struggle on small tabular datasets, where tree-based models remain dominant. We introduce Adaptive Contrastive Approach (AdaCap ), a training scheme that combines a permutationbased contrastive loss with a Tikhonov-based closed-form output mapping. Across 85 real-world regression datasets and multiple architectures, AdaCap yields consistent and statistically significant improvements in the small-sample regime, particularly for residual models. A meta-predictor trained on dataset characteristics (size, skewness, noise) accurately anticipates when AdaCap is beneficial. These results show that AdaCap acts as a targeted regularization mechanism, strengthening neural networks precisely where they are most fragile. All results and code are publicly available at https://github.com/BrunoBelucci/adacap.
• A Distributionally Robust Optimization Approach to Model Risk in Mathematical Finance
  
  • Sauldubois Nathan
  , 2025. This thesis addresses a series of problems related to model risk.In the first part, we conduct a sensitivity analysis of Wasserstein distributionally robust optimization (W-DRO) and adapted Wasserstein distributionally robust optimization (AW-DRO), in the presence of infinitely many constraints, such as conditional moment restrictions.We begin by examining W-DRO and AW-DRO formulations that incorporate martingale constraints.Building upon the literature on martingale optimal transport, we then move to the case of the martingale coupling constraint.To address this latter case, we introduce a novel family of martingale couplings, derived through an application of the Implicit Function Theorem.It turns out that this method is applicable to other constrained versions of W-DRO and AW-DRO problems.Subsequently, we investigate higher-order expansion of the W-DRO and AW-DRO problems, by proving that it can be reduce to a functional optimization problem.This reformulation enables more efficient initialization of neural networks when numerically solving both W-DRO and AW-DRO.Finally, we turn our attention to the sensitivity analysis of non-markovian stochastic control and optimal stopping problems.
• New dimensional bounds for a branched transport problem
  
  • Cosenza Alessandro
  • Goldman Michael
  • Koser Melanie
  , 2024. We consider a branched transport problem with weakly imposed boundary conditions. This problem arises as a reduced model for pattern formation in type-I superconductors. For this model, it is conjectured that the dimension of the boundary measure is non-integer. We prove this conjecture in a simplified 2D setting, under the (strong) assumption of Ahlfors regularity of the irrigated measure. This work is the first rigorous proof of a singular behaviour for irrigated measures resulting from minimality.
• Gaussian process regression for non-Euclidean inputs : graphs and categorical variables
  
  • Carpintero Perez Raphaël
  , 2025. In the design phases of aircraft engine parts, numerical simulation is traditionally used to identify the right characteristics (geometry, materials, etc.) to meet desired specifications. Meta-models like Gaussian processes can achieve this objective at lower cost by learning from physics-based simulations, with the added benefit of uncertainty quantification. The goal of this thesis is to improve the predictivity of Gaussian process meta-models for non-Euclidean spaces. We propose the Sliced Wasserstein Weisfeiler-Lehman (SWWL) graph kernel combining continuous Weisfeiler-Lehman iterations and an optimal transport for inputs that are graphs with continuous node attributes. This new kernel is used to learn scalar outputs with Gaussian processes when the inputs are finite element meshes. In the context of numerical simulations, it is also customary to work with output fields defined on meshes, which are challenging to handle due to changes between input geometries in terms of both size and adjacency structure. We develop a new methodology to address such limitations by relying on the combination of regularized optimal transport, dimension reduction techniques, and the use of Gaussian processes indexed by graphs. We illustrate the efficiency of the method to solve real problems in fluid dynamics and solid mechanics with uncertainty quantification. A last axis focuses on Gaussian process regression with categorical input variables. We carry out a reproducible comparative study of existing categorical kernels with novel evaluation metrics. We also introduce a new clustering-based strategy using target encodings of categorical variables in order to identify groups of levels when they are unknown.
• Path-dependent processes from signatures
  
  • Abi Jaber Eduardo
  • Gérard Louis-Amand
  • Huang Yuxing
  , 2024. We provide explicit series expansions to certain stochastic path-dependent integral equations in terms of the path signature of the time augmented driving Brownian motion. Our framework encompasses a large class of stochastic linear Volterra and delay equations and in particular the fractional Brownian motion with a Hurst index H ∈ (0, 1). Our expressions allow to disentangle an infinite dimensional Markovian structure and open the door to straightforward and simple approximation schemes, that we illustrate numerically. A key application is the derivation of explicit series representations for both conditional and unconditional moments.
• Continuous-time modeling and bootstrap for chain ladder reserving
  
  • Baradel Nicolas
  , 2024. We revisit the famous Mack's model which gives an estimate for the mean square error of prediction of the chain ladder claims reserves. We introduce a stochastic dierential equation driven by a Brownian motion to model accumulated total claims amount for the chain ladder method. Within this continuous-time framework, we propose a bootstrap technique for estimating the distribution of claims reserves. It turns out that our approach leads to inherently capturing asymmetry and non-negativity, eliminating the necessity for additional assumptions. We conclude with a case study and comparative analysis against alternative methodologies based on Mack's model. (10.1016/j.insmatheco.2025.103176)
  DOI : 10.1016/j.insmatheco.2025.103176
• From Kinetic Theory to Hyperbolic-Parabolic Fluid Systems of Equations
  
  • Giovangigli Vincent
  , 2025. Fluid systems of equations can be derived from the kinetic theory by using the Chapman-Enskog asymptotic method. We investigate the links between the hyperbolic-parabolic structure of such systems and the kinetic framework. We also discuss the mathematical structure of linear systems associated with transport property evaluation.
• Mathematical modeling of fluids from the kinetic theory
  
  • Giovangigli Vincent
  , 2025. A synthesis of mathematical fluid models derived from the kinetic theory of gases is presented. Kinetic theory yields well structured set of partial equations with entropies compatible with convection, diffusion and sources. It also yields properly structured nonlinear source terms as well as the transport coefficients. We first address multicomponent flow models including detailed transport and chemical reactions. Symmetrization, hyperbolic-parabolic structure and asymptotic stability of constant equilibrium states are established. We next investigate relaxation issues for a two temperature model that leads to symmetrized hyperbolic-parabolic systems with stiff source terms. Local existence of solutions is established and the apparition of the bulk viscosity coefficient is justified in the fast relaxation limit. We finally consider nonideal fluids and their cohesive properties that lead to diffuse interfaces models with capillary effects. Using an augmented formulation-with density gradient added as an extra unknown-a new type of hyperbolic-parabolic-dispersive structure is obtained with antisymmetric second order coupling terms arising from capillarity. Local existence of solutions is established as well as asymptotic stability of constant states.
• Asymptotic Stability of Equilibrium States for Cohesive Fluids
  
  • Giovangigli Vincent
  , 2025. <div><p>We investigate asymptotic stability of constant equilibrium states for compressible nonisothermal cohesive fluids also termed capillary fluids or diffuse interface fluids. The density gradient is added as an extra variable and the augmented system of equations is recast into a normal form with symmetric transport first order terms, symmetric dissipative second order terms and antisymmetric cohesive second order terms. Global existence and asymptotic stability of constant equilibrium states are established by using new dissipative conditions for such augmented hyperbolic-parabolic-dispersive systems of equations. Decay estimates are obtained in all spatial dimensions by using the augmented formulation as well as estimates in Fourier spaces.</p></div>
• Inf-sup stable non-conforming finite elements on tetrahedra with second and third order accuracy
  
  • Balazi Loïc
  • Allaire Grégoire
  • Jolivet Pierre
  • Omnes Pascal
  , 2025. We introduce a family of scalar non-conforming finite elements with second and third order accuracy with respect to the $H^1$-norm on tetrahedra. Their vector-valued versions generate, together with discontinuous pressure approximations of order one and two respectively, inf-sup stable finite element pairs with convergence order two and three for the Stokes problem in energy norm.
• High-order multistep coupling: convergence, stability and PDE application
  
  • Simon Antoine
  • François Laurent
  • Massot Marc
  Comptes Rendus. Mécanique, Académie des sciences (Paris), 2025, 353 (G1), pp.1159-1184. Designing coupling schemes for specialized advanced mono-physics solvers in order to conduct accurate and efficient multiphysics simulations is a key issue that has recently received a lot of attention. A novel high-order adaptive multistep coupling strategy has shown potential to improve the efficiency and accuracy of such simulations, but requires further analysis. The purpose of the present contribution is to conduct the numerical analysis of convergence of the explicit and implicit variants of the method and to provide a first analysis of its absolute stability. A simplified coupled problem is constructed to assess the stability of the method along the lines of the Dahlquist’s test equation for ODEs. We propose a connection with the stability analysis of other methods such as splitting and ImEx schemes. A stability analysis on a representative conjugate heat transfer case is also presented. This work constitutes a first building block to an a priori analysis of the stability of coupled PDEs. (10.5802/crmeca.333)
  DOI : 10.5802/crmeca.333
• A sequential Bayesian approach to Gaussian process quantile regression for optimization
  
  • Nicolas Hugo
  • Le Maître Olivier
  , 2025, pp.115. Quantile regression extends the classical least-squares regression to the estimation of conditional quantiles of a response variable. In the frequentist approach, the quantile regression problem is cast as the minimization of a loss function, possibly complemented with regularization terms. The Bayesian counterpart formulates the problem as posterior inference over a function space. Of particular interest is Gaussian process quantile regression, which formulates the regression problem as posterior inference over the latent conditional quantiles, where prior knowledge is encoded in the form of a Gaussian process. Existing approaches to Gaussian process quantile regression either perform the regression directly on observed data [1] or resort to sparse approximations to mitigate computational costs [2]. However, in the latter case, the approximation is typically defined over a small set of latent auxiliary variables that act as compact representations of the quantile, but whose locations are fixed in advance, ultimately resulting in suboptimal predictive performance. In this work, we introduce an adaptive strategy that exploits the Gaussian process predictive variance to infill the set of auxiliary variable locations. Inference of the posterior distribution over these auxiliary variables is recast as its Laplace approximation. The impact of finite training data on the auxiliary variables is estimated through bootstrap resampling. Building on this, we introduce an active learning strategy that acquires new observations of the response variable via rejection sampling, with the sampling density guided by the uncertainties in the auxiliary estimates. Our algorithm combines adaptive auxiliary variable allocation and active learning, leading to a sequential approach that ensures a rich and well-balanced representation of the quantile function. Finally, we extend our quantile regression method and its enrichment criteria to the quantile minimization problem within a Bayesian optimization framework.
• Efficient Simulation of Hawkes Processes using their Affine Volterra Structure
  
  • Abi Jaber Eduardo
  • Attal Elie
  • Sotnikov Dimitri
  , 2025. We introduce a novel and efficient simulation scheme for Hawkes processes on a fixed time grid, leveraging their affine Volterra structure. The key idea is to first simulate the integrated intensity and the counting process using Inverse Gaussian and Poisson distributions, from which the jump times can then be easily recovered. Unlike conventional exact algorithms based on sampling jump times first, which have random computational complexity and can be prohibitive in the presence of high activity or singular kernels, our scheme has deterministic complexity which enables efficient large-scale Monte Carlo simulations and facilitates vectorization. Our method applies to any nonnegative, locally integrable kernel, including singular and non-monotone ones. By reformulating the scheme as a stochastic Volterra equation with a measure-valued kernel, we establish weak convergence to the target Hawkes process in the Skorokhod J1topology. Numerical experiments confirm substantial computational gains while preserving high accuracy across a wide range of kernels, with remarkably improved performance for a variant of our scheme based on the resolvent of the kernel.