Publications

2021

• Monotone solutions for mean field games master equations : finite state space and optimal stopping
  
  • Bertucci Charles
  Journal de l'École polytechnique — Mathématiques, École polytechnique, 2021, 8, pp.1099-1132. We present a new notion of solution for mean field games master equations. This notion allows us to work with solutions which are merely continuous. We prove first results of uniqueness and stability for such solutions. It turns out that this notion is helpful to characterize the value function of mean field games of optimal stopping or impulse control and this is the topic of the second half of this paper. The notion of solution we introduce is only useful in the monotone case. We focus in this paper in the finite state space case. (10.5802/jep.167)
  DOI : 10.5802/jep.167
• Decentralized Stochastic Control of Heterogeneous Energy Systems
  
  • Gobet Emmanuel
  • Grangereau Maxime
  , 2021.
• Optimal potential functions for the interacting particle system method
  
  • Garnier Josselin
  • Chraibi Hassane
  • Dutfoy Anne
  • Galtier Thomas
  Monte Carlo Methods and Applications, De Gruyter, 2021, 27 (2), pp.137-152. Abstract The assessment of the probability of a rare event with a naive Monte Carlo method is computationally intensive, so faster estimation or variance reduction methods are needed. We focus on one of these methods which is the interacting particle system (IPS) method. The method is not intrusive in the sense that the random Markov system under consideration is simulated with its original distribution, but selection steps are introduced that favor trajectories (particles) with high potential values. An unbiased estimator with reduced variance can then be proposed. The method requires to specify a set of potential functions. The choice of these functions is crucial because it determines the magnitude of the variance reduction. So far, little information was available on how to choose the potential functions. This paper provides the expressions of the optimal potential functions minimizing the asymptotic variance of the estimator of the IPS method and it proposes recommendations for the practical design of the potential functions. (10.1515/mcma-2021-2086)
  DOI : 10.1515/mcma-2021-2086
• Insights from the Future for Continual Learning
  
  • Douillard Arthur
  • Valle Eduardo
  • Ollion Charles
  • Robert Thomas
  • Cord Matthieu
  , 2021, pp.3477-3486. Continual learning aims to learn tasks sequentially, with (often severe) constraints on the storage of old learning samples, without suffering from catastrophic forgetting. In this work, we propose prescient continual learning, a novel experimental setting, to incorporate existing information about the classes, prior to any training data. Usually, each task in a traditional continual learning setting evaluates the model on present and past classes, the latter with a limited number of training samples. Our setting adds future classes, with no training samples at all. We introduce Ghost Model, a representation-learning-based model for continual learning using ideas from zero-shot learning. A generative model of the representation space in concert with a careful adjustment of the losses allows us to exploit insights from future classes to constraint the spatial arrangement of the past and current classes. Quantitative results on the AwA2 and aP\&Y datasets and detailed visualizations showcase the interest of this new setting and the method we propose to address it. (10.1109/CVPRW53098.2021.00387)
  DOI : 10.1109/CVPRW53098.2021.00387
• refined-weissman
  
  • Allouche Michaël
  • El Methni Jonathan
  • Girard Stéphane
  , 2021. Weissman extrapolation methodology for estimating extreme quantiles from heavy-tailed distributions is based on two estimators: an order statistic to estimate an intermediate quantile and an estimator of the tail-index. The common practice is to select the same intermediate sequence for both estimators. In this work, we show how an adapted choice of two different intermediate sequences leads to a reduction of the asymptotic bias associated with the resulting refined Weissman estimator. The asymptotic normality of the latter estimator is established and a data-driven method is introduced for the practical selection of the intermediate sequences. Our approach is compared to Weissman estimator and to six bias reduced estimators of extreme quantiles on a large scale simulation study. It appears that the refined Weissman estimator outperforms its competitors in a wide variety of situations, especially in the challenging high bias cases. Finally, an illustration on an actuarial real data set is provided.
• Foreword to the Special Issue "Inverse problems and nonlinear phenomena
  
  • Hasanov Hasanoglu Alemdar
  • Novikov Roman
  • Shananin Alexander
  • Taimanov Iskander
  Journal of Inverse and Ill-posed Problems, De Gruyter, 2021, 29 (3), pp.317. (10.1515/jiip-2021-2078)
  DOI : 10.1515/jiip-2021-2078
• Mathematical modeling for market making and related problems of financial liquidity: a song of assets and traders.
  
  • Bergault Philippe
  , 2021. On a given market, a market maker is in charge of providing liquidity for one (or more) asset(s) by proposing at all time a price at which she is ready to buy the asset and a price at which she is ready to sell it. The market maker aims at maximzing her expected PnL, while controlling the risk to which she is exposed. Mathematically, this translates as a problem of optimal control of point processes. In particular, we propose a multi-asset market making model including stochastic transaction sizes (unlike existing models), solved using a verification approach, as well as a model taking into account the possibility for the market maker to liquidate its inventory on a dealer-to-dealer segment of the market, thereby exposing herself to transaction costs linked to her impact on the price; the resolution of this second problem uses the theory of viscosity solutions. Two techniques are proposed in order to approach the solutions of these models in high dimension: a factorial approach allowing to reduce the dimension of the problem, as well as a perturbative approach allowing to approximate the HJB equation by a system of Riccati equations. Finally, the last two chapters of this thesis focus on the one hand on the problem of an exchange seeking to set an optimal tick size to increase liquidity on its platform, and on the other hand on a problem of optimal liquidation of a portfolio of co-integrated assets.
• Stress minimization for lattice structures. Part I: Micro-structure design
  
  • Ferrer A.
  • Geoffroy-Donders P.
  • Allaire Grégoire
  Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, Royal Society, The, 2021, 379 (2201), pp.20200109. Lattice structures are periodic porous bodies which are becoming popular since they are a good compromise between rigidity and weight and can be built by additive manufacturing techniques. Their optimization has recently attracted some attention, based on the homogenization method, mostly for compliance minimization. The goal of our two-parts work is to extend lattice optimiza- tion to stress minimization problems in 2-d. The present first part is devoted to the choice of a parametrized periodicity cell that will be used for structural optimization in the second part of our work. In order to avoid stress concentration, we propose a square cell micro-structure with a super-ellipsoidal hole instead of the standard rectangular hole often used for compliance minimiza- tion. This type of cell is parametrized in 2-d by one orientation angle, two semi-axis and a corner smoothing parameter. We first analyse their influence on the stress amplification factor by perform- ing some numerical experiments. Second, we compute the optimal corner smoothing parameter for each possible micro-structure and macroscopic stress. Then, we average (with specific weights) the optimal smoothing exponent with respect to the macroscopic stress. Finally, to validate the results, we compare our optimal super-ellipsoidal hole with the Vigdergauz micro-structure which is known to be optimal for stress minimization in some special cases. (10.1098/rsta.2020.0109)
  DOI : 10.1098/rsta.2020.0109
• Estimation of the tail-index and extreme quantiles from a mixture of heavy-tailed distributions
  
  • Girard Stéphane
  • Gobet Emmanuel
  , 2021, pp.1. The estimation of extreme quantiles requires adapted methods to extrapolate beyond the largest observation of the sample. Extreme-value theory provides a mathematical framework to tackle this problem together with statistical procedures based on the estimation of the so-called tail-index describing the distribution tail. We focus on heavy-tailed distributions and consider the case where the shape of the distribution tail depends on unknown auxiliary variables. As a consequence, one has to deal with observations from a mixture of heavytailed distributions, and it is shown that, in such a situation, usual extreme-value estimators suffer from a strong bias. We propose several methods to mitigate this bias. Their asymptotic properties are established and their finite sample performance is illustrated both on simulated and real financial data This is joint work with Emmanuel Gobet.
• Energy and wave-action flows underlying Rayleigh-Jeans thermalization of optical waves propagating in a multimode fiber
  
  • Baudin K
  • Fusaro A
  • Garnier J
  • Berti N
  • Krupa K
  • Carusotto I
  • Rica S
  • Millot G
  • Picozzi Antonio
  EPL - Europhysics Letters, European Physical Society / EDP Sciences / Società Italiana di Fisica / IOP Publishing, 2021, 134 (1), pp.14001. The wave turbulence theory predicts that a conservative system of nonlinear waves can exhibit a process of condensation, which originates in the singularity of the Rayleigh-Jeans equilibrium distribution of classical waves. Considering light propagation in a multimode fiber, we show that light condensation is driven by an energy flow toward the higher-order modes, and a bi-directional redistribution of the wave-action (or power) to the fundamental mode and to higher-order modes. The analysis of the near-field intensity distribution provides experimental evidence of this mechanism. The kinetic equation also shows that the wave-action and energy flows can be inverted through a thermalization toward a negative temperature equilibrium state, in which the high-order modes are more populated than low-order modes. In addition, a Bogoliubov stability analysis reveals that the condensate state is stable. (10.1209/0295-5075/134/14001)
  DOI : 10.1209/0295-5075/134/14001
• High order time integration and mesh adaptation with error control for incompressible Navier-Stokes and scalar transport resolution on dual grids
  
  • N'Guessan Marc-Arthur
  • Massot Marc
  • Series Laurent
  • Tenaud Christian
  Journal of Computational and Applied Mathematics, Elsevier, 2021, 387, pp.112542. Relying on a building block developed by the authors in order to resolve the incompressible Navier-Stokes equation with high order implicit time stepping and dynamic mesh adaptation based on multiresolution analysis with collocated variables, the present contribution investigates the ability to extend such a strategy for scalar transport at relatively large Schmidt numbers using a finer level of refinement compared to the resolution of the hydrody-namic variables, while preserving space adaptation with error control. This building block is a key part of a strategy to construct a low-Mach number code based on a splitting strategy for combustion applications, where several spatial scales are into play. The computational efficiency and accuracy of the proposed strategy is assessed on a well-chosen three-vortex simulation. (10.1016/j.cam.2019.112542)
  DOI : 10.1016/j.cam.2019.112542
• Quantitative finance at the microstructure scale : algorithmic trading and regulation
  
  • Baldacci Bastien
  , 2021. This thesis is split into three parts. In the first part, we apply the Principal-Agent theory to some problems of market microstructure. First, we build an incentives mechanism to improve the market quality in the context of market-making activity in a lit and a dark pool managed by the same exchange. Then, we adapt the incentives design to the regulation of market-making activity when several market-makers compete in a liquidity platform. We also propose a form of incentives based on the choice of tick sizes on the bid and ask sides of a single asset. Next, we tackle the issue of designing a derivatives market, using a quantization method to select the options listed on the exchange and the Principal-Agent framework to create incentives for an option market-maker. Finally, we develop an incentives mechanism to increase the investment in green bonds, robust to model specification, and outperforming current tax-incentives policies of the governments.The second part of this thesis is dedicated to option market-making in high dimension. We first propose a framework a constant Greek assumption to deal with long-dated options. Then, we propose an approximation of the value function enabling to deal with time-varying Greeks and short-dated options. Finally, we develop a framework for the high-frequency dynamics of the implied volatility surface. Using multidimensional Hawkes processes, we show how this setting can reproduce easily well-known stylized facts such as the skew, smile and term structure of the surface.The last part of this thesis is devoted to optimal trading problems in high dimension. First, we develop a framework to tackle the smart order routing (SOR) problem taking into account non-stationarity of markets. For a large number of venues, we use a deep reinforcement learning approach to compute the optimal controls of the trader. Then, we present a methodology to solve approximately optimal trading problems without using stochastic control theory. We propose a framework in which a myopic agent exhibits approximately an optimal behavior if he uses the gradient of the high-level trajectory as short-term alpha. Finally, we present two new developments on the optimal execution literature. First, we show that we can obtain a closed-form solution for the Almgren-Chriss execution problem with geometric Brownian motion and quadratic penalty. Second, we propose an application of the latent order book model to the problem of optimal execution of a portfolio of assets, in the context of liquidity stress testing.
• Large fluctuations in multi-scale modeling for rest hematopoiesis
  
  • Bonnet Céline
  • Méléard Sylvie
  Journal of Mathematical Biology, Springer, 2021, 82 (6), pp.58. Hematopoiesis is a biological phenomenon (process) of production of mature blood cells by cellular differentiation. It is based on amplification steps due to an interplay between renewal and differentiation in the successive cell types from stem cells to mature blood cells. We will study this mechanism with a stochastic point of view to explain unexpected fluctuations on the mature blood cell number, as surprisingly observed by biologists and medical doctors in a rest hematopoiesis. We consider three cell types: stem cells, progenitors and mature blood cells. Each cell type is characterized by its own dynamics parameters, the division rate and the renewal and differentiation probabilities at each division event. We model the global population dynamics by a three-dimensional stochastic decomposable branching process. We show that the amplification mechanism is given by the inverse of the small difference between the differentiation and renewal probabilities. Introducing a parameter K which scales simultaneously the size of the first component, the differentiation and renewal probabilities and the mature blood cell death rate, we describe the asymptotic behavior of the process for large K. We show that each cell type has its own size and time scales. Focusing on the third component, we prove that the mature blood cell population size, conveniently renormalized (in time and size), is expanded in an unusual way inducing large fluctuations. The proofs are based on a fine study of the different scales involved in the model and on the use of different convergence and average techniques in the proofs. (10.1007/s00285-021-01611-4)
  DOI : 10.1007/s00285-021-01611-4
• Discrete-time mean field games with risk-averse agents
  
  • Bonnans Frédéric J
  • Lavigne Pierre
  • Pfeiffer Laurent
  ESAIM: Control, Optimisation and Calculus of Variations, EDP Sciences, 2021. We propose and investigate a discrete-time mean field game model involving risk-averse agents. The model under study is a coupled system of dynamic programming equations with a Kolmogorov equation. The agents' risk aversion is modeled by composite risk measures. The existence of a solution to the coupled system is obtained with a fixed point approach. The corresponding feedback control allows to construct an approximate Nash equilibrium for a related dynamic game with finitely many players. (10.1051/cocv/2021044)
  DOI : 10.1051/cocv/2021044
• Availability of the Molecular Switch XylR Controls Phenotypic Heterogeneity and Lag Duration during Escherichia coli Adaptation from Glucose to Xylose
  
  • Enjalbert Brice
  • Barthe Manon
  • Tchouanti Josué
  • Gomes Pédro Henrique
  • Bideaux Carine
  • Lestrade Delphine
  • Graham Carl
  • Steyer Jean-Philippe
  • Meleard Sylvie
  • Harmand Jérome
  • Gorret Nathalie
  • Cocaign-Bousquet Muriel
  , 2021.
• 蒙特卡罗方法与随机过程：从线性到非线性
  
  • Gobet Emmanuel
  , 2021. Translated by Xu Mingyu
• Machine learning for large observational datasets in healthcare
  
  • Morel Maryan
  , 2021. This thesis develops innovative tools and methods to leverage large observational databases (LODs) in healthcare, with a focus on the Système National des Données de Santé (SNDS), one of the largest healthcare claims database.These databases record administrative information supporting the care and its billing.As SNDS data was not initially designed for research but for accounting purposes, identifying patients' healthcare history requires costly transformations.The first chapter introduces SCALPEL3, an open-source framework easing medical concept extraction and cohort data manipulation, focusing on scalability and reproducibility.SCALPEL3 relies on distributed computing, data denormalization, and columnar storage.It is now used at the agency collecting SNDS data, at the French Ministry of Health, and soon at the National Health Data Hub in France.The following two chapters focus on adverse drug reactions (ADRs) detection using SNDS data. Chapter 2 introduces ConvSCCS, a new model based on conditional Poisson processes and penalization techniques. Using a convolution between step functions and temporal events, this model estimates readily interpretable longitudinal risks. Applied to a cohort of diabetic patients, it recovers a known association between a molecule use and bladder cancer from timestamped sequences of drug reimbursements and diagnoses.In Chapter 3, the same model is used to screen anxiolytic, hypnotic, antidepressant, and neuroleptic molecules for bone fracture risk among the elderly. This study reveals original patterns and SNDS-specific biases.Finally, Chapter 4 focuses on building reusable representation for health data. It presents extensive experiments evaluating several deep attention models and pre-training strategies. While the results are not yet satisfying, this work opens exciting tracks for future research.
• A novel association rule mining method for the identification of rare functional dependencies in Complex Technical Infrastructures from alarm data
  
  • Antonello Federico
  • Baraldi Piero
  • Shokry Ahmed
  • Zio Enrico
  • Gentile Ugo
  • Serio Luigi
  Expert Systems with Applications, Elsevier, 2021, 170, pp.114560. (10.1016/j.eswa.2021.114560)
  DOI : 10.1016/j.eswa.2021.114560
• On conditioning a self-similar growth-fragmentation by its intrinsic area
  
  • Bertoin Jean
  • Curien Nicolas
  • Kortchemski Igor
  Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, Institut Henri Poincaré (IHP), 2021, 57 (2). (10.1214/20-AIHP1110)
  DOI : 10.1214/20-AIHP1110
• Predicting the propagation of COVID-19 at an international scale: extension of an SIR model
  
  • Lavielle Marc
  • Faron Matthieu
  • Lefevre Jérémie H
  • Zeitoun Jean-David
  BMJ Open, BMJ Publishing Group, 2021, 11 (5), pp.e041472. Objectives Several epidemiological models have been published to forecast the spread of the COVID-19 pandemic, yet many of them have proven inaccurate for reasons that remain to be fully determined. We aimed to develop a novel model and implement it in a freely accessible web application. Design We built an SIR-type compartmental model with two additional compartments: D (deceased patients); L (individuals who will die but who will not infect anybody due to social or medical isolation) and integration of a time-dependent transmission rate and a periodical weekly component linked to the way in which cases and deaths are reported. Results The model was implemented in a web application (as of 2 June 2020). It was shown to be able to accurately capture the changes in the dynamics of the pandemic for 20 countries whatever the type of pandemic spread or containment measures: for instance, the model explains 97% of the variance of US data (daily cases) and predicts the number of deaths at a 2-week horizon with an error of 1%. Conclusions In early performance evaluation, our model showed a high level of accuracy between prediction and observed data. Such a tool might be used by the global community to follow the spread of the pandemic. (10.1136/bmjopen-2020-041472)
  DOI : 10.1136/bmjopen-2020-041472
• Association rules extraction for the identification of functional dependencies in complex technical infrastructures
  
  • Antonello Federico
  • Baraldi Piero
  • Shokry Ahmed
  • Zio Enrico
  • Gentile Ugo
  • Serio Luigi
  Reliability Engineering and System Safety, Elsevier, 2021, 209, pp.107305. (10.1016/j.ress.2020.107305)
  DOI : 10.1016/j.ress.2020.107305
• Asymptotic analysis of different covariance matrices estimation for minimum variance portfolio
  
  • Chamakh Linda
  • Gobet Emmanuel
  • Lemor Jean-Philippe
  , 2021. In dynamic minimum variance portfolio, we study the impact of the sequence of covariance matrices taken in inputs, on the realized variance of the portfolio computed along a sample market path. The allocation of the portfolio is adjusted on a regular basis (every H days) using an updated covariance matrix estimator. In a modelling framework where the covariance matrix of the asset returns evolves as an ergodic process, we quantify the probability of observing an underperformance of the optimal dynamic covariance matrix compared to any other choice. The bounds depend on the tails of the returns, on the adjustment period H, and on the total number of rebalancing times N. These results provide asset managers with new insights into the optimality of their choice of covariance matrix estimators, depending on the depth of the backtest N H and the investment period H. Experiments based on GARCH modelling support our theoretical results.
• On Real-time Management of On-board Ice Protection Systems by means of Machine Learning
  
  • Arizmendi Bárbara
  • Bellosta Tommaso
  • del Val Anabel
  • Gori Giulio
  • Reis João
  • Prazeres Mariana
  , 2021.
• Interpretable Random Forests via Rule Extraction
  
  • Bénard Clément
  • Biau Gérard
  • da Veiga Sébastien
  • Scornet Erwan
  , 2021, 130, pp.937-945. We introduce SIRUS (Stable and Interpretable RUle Set) for regression, a stable rule learning algorithm which takes the form of a short and simple list of rules. State-of-the-art learning algorithms are often referred to as "black boxes" because of the high number of operations involved in their prediction process. Despite their powerful predictivity, this lack of interpretability may be highly restrictive for applications with critical decisions at stake. On the other hand, algorithms with a simple structure-typically decision trees, rule algorithms, or sparse linear models-are well known for their instability. This undesirable feature makes the conclusions of the data analysis unreliable and turns out to be a strong operational limitation. This motivates the design of SIRUS, which combines a simple structure with a remarkable stable behavior when data is perturbed. The algorithm is based on random forests, the predictive accuracy of which is preserved. We demonstrate the efficiency of the method both empirically (through experiments) and theoretically (with the proof of its asymptotic stability). Our R/C++ software implementation sirus is available from CRAN. (10.48550/arXiv.2004.14841)
  DOI : 10.48550/arXiv.2004.14841
• AD Course Map charts Alzheimer’s disease progression
  
  • Koval Igor
  • Bône Alexandre
  • Louis Maxime
  • Lartigue Thomas
  • Bottani Simona
  • Marcoux Arnaud
  • Samper-González Jorge
  • Burgos Ninon
  • Charlier Benjamin
  • Bertrand Anne
  • Epelbaum Stéphane
  • Colliot Olivier
  • Allassonnière Stéphanie
  • Durrleman​ Stanley
  Scientific Reports, Nature Publishing Group, 2021, 11 (1). Alzheimer's disease (AD) is characterized by the progressive alterations seen in brain images which give rise to the onset of various sets of symptoms. The variability in the dynamics of changes in both brain images and cognitive impairments remains poorly understood. This paper introduces AD Course Map a spatiotemporal atlas of Alzheimer's disease progression. It summarizes the variability in the progression of a series of neuropsychological assessments, the propagation of hypometabolism and cortical thinning across brain regions and the deformation of the shape of the hippocampus. The analysis of these variations highlights strong genetic determinants for the progression, like possible compensatory mechanisms at play during disease progression. AD Course Map also predicts the progression of patient data in the future with a better accuracy than the 56 methods benchmarked in the open challenge TADPOLE. Finally, AD Course Map is used to simulate cohorts of virtual patients developing Alzheimer's disease. AD Course Map offers therefore new tools for exploring the progression of AD and personalizing patients care. (10.1038/s41598-021-87434-1)
  DOI : 10.1038/s41598-021-87434-1