Publications

2024

• Optimal Stopping of Branching Diffusion Processes
  
  • Kharroubi Idris
  • Ocello Antonio
  , 2024. This article explores an optimal stopping problem for branching diffusion processes. It consists in looking for optimal stopping lines, a type of stopping time that maintains the branching structure of the processes under analysis. By using a dynamic programming approach, we characterize the value function for a multiplicative cost, which may depend on the particle's label. We reduce the problem's dimensionality by setting a branching property and defining the problem in a finite-dimensional context. Within this framework, we focus on the value function, establishing uniform continuity and boundedness properties, together with an innovative dynamic programming principle. This outcome leads to an analytical characterization with the help of a nonlinear elliptic PDE. We conclude by showing that the value function serves as the unique viscosity solution for this PDE, generalizing the comparison principle to this setting.
• State spaces of multifactor approximations of nonnegative Volterra processes
  
  • Abi Jaber Eduardo
  • Bayer Christian
  • Breneis Simon
  , 2024. We show that the state spaces of multifactor Markovian processes, coming from approximations of nonnegative Volterra processes, are given by explicit linear transformation of the nonnegative orthant. We demonstrate the usefulness of this result for applications, including simulation schemes and PDE methods for nonnegative Volterra processes.
• A randomisation method for mean-field control problems with common noise
  
  • Denkert Robert
  • Kharroubi Idris
  • Pham Huyên
  , 2024. We study mean-field control (MFC) problems with common noise using the control randomisation framework, where we substitute the control process with an independent Poisson point process, controlling its intensity instead. To address the challenges posed by the mean-field interactions in this randomisation approach, we reformulate the admissible control as L 0 -valued processes adapted only to the common noise. We then construct the randomised control problem from this reformulated control process, and show its equivalence to the original MFC problem. Thanks to this equivalence, we can represent the value function as the minimal solution to a backward stochastic differential equation (BSDE) with constrained jumps. Finally, using this probabilistic representation, we derive a randomised dynamic programming principle (DPP) for the value function, expressed as a supremum over equivalent probability measures.
• Mean Field Optimization Problem Regularized by Fisher Information
  
  • Claisse Julien
  • Conforti Giovanni
  • Ren Zhenjie
  • Wang Songbo
  , 2024. Recently there is a rising interest in the research of mean field optimization, in particular because of its role in analyzing the training of neural networks. In this paper by adding the Fisher Information as the regularizer, we relate the regularized mean field optimization problem to a so-called mean field Schrodinger dynamics. We develop an energy-dissipation method to show that the marginal distributions of the mean field Schrodinger dynamics converge exponentially quickly towards the unique minimizer of the regularized optimization problem. Remarkably, the mean field Schrodinger dynamics is proved to be a gradient flow on the probability measure space with respect to the relative entropy. Finally we propose a Monte Carlo method to sample the marginal distributions of the mean field Schrodinger dynamics.
• Convergence estimates in tropical geometry and application in optimization
  
  • Vandame Nicolas
  , 2024. We establish convergence estimates for parametric families of semialgebraic sets to their logarithmic limit, or tropicalization. We apply these methods from tropical geometry to obtain complexity results in optimization.We begin by establishing a universality result stating that no self-concordant interior point method is strongly polynomial. To do so, we give lower bounds on the complexity of these methods using the geometry of an associated tropical curve: the tropical central path. We give an explicit counterexample à la Klee-Minty by constructing an n-combinatorial cube for which the number of iterations is lower bounded by Ω(2^n).We then consider parametric families of matrices whose tropicalization is nonsingular. We define different condition numbers for tropical matrices that enable us to estimate the parameters for which the combinatorial type of a polyhedron coincides with that of its tropicalization. We identify a special class of linear programs with sufficiently separated coefficients, called layered feasibility problems, which can be solved in strongly quasi-polynomial time.In the final chapters, we study the log-convergence of parametric families of semialgebraic sets to their valuation. We obtain explicit metric estimates under genericity conditions for basic semialgebraic sets. In the general case, we establish a multivariate extension of the Denef--Pas cell decomposition. Using Smale's α-theory, we give a quantitative version of this decomposition that allows us to prove a metric lifting theorem for semialgebraic sets. It yields a constructive bound on the one-sided Hausdorff convergence of the parametric family to its valuation. (10.70675/33f66093za39cz40edzafb1z46e6d4e65dc9)
  DOI : 10.70675/33f66093za39cz40edzafb1z46e6d4e65dc9
• Geometric shape optimization for reflective optics
  
  • Gorini Luca
  , 2024. In this thesis, we are interested in the shape optimization of telescopes mirrors based on the Hadamard's boundary variation method. Two models are proposed in order to described propagation of light in the telescope. The first model we consider is a theoretical scattering problem at finite frequency around a metallic mirror. We propose a cost function involving the electromagnetic field on the focal plane and whose limit corresponds to some optical criteria. To this aim, a pseudo-differential operator on the focal plane is used. Its symbol allows to represent phase space regions where we want to concentrate rays.Once a cost defined, we obtain a shape derivative of the cost with the Hadamard's method. Using limit measures of Wigner transforms of direct and adjoint states, high frequency limits of shape derivatives are expressed.The second model we consider is a ray tracing model. Two entrant random variables model position and direction at the telescope entrance. Once propagated according to Snell-Descartes laws, two random variables representing position and direction of rays on the focal plane are obtained. We adapt the Hadamard's method to the case of random variables depending on mirrors shapes. Shape derivatives for rather general costs are obtained.This second approach is useful to characterize more precisely the limit measures involved in the high frequency limit of shape derivatives obtained in the Maxwell's case. In those cases, an adjoint state is derived and a new practical expression of the shape derivative is given.Numerical simulations are performed on simple optical examples in order to validate the shape derivative. (10.70675/dc23ab91z83d5z43c4zafd3z303693d30009)
  DOI : 10.70675/dc23ab91z83d5z43c4zafd3z303693d30009
• Statistical error bounds for weighted mean and median, with application to robust aggregation of cryptocurrency data
  
  • Allouche Michaël
  • Echenim Mnacho
  • Gobet Emmanuel
  • Maurice Anne-Claire
  , 2023. We study price aggregation methodologies applied to crypto-currency prices with quotations fragmented on different platforms. An intrinsic difficulty is that the price returns and volumes are heavytailed, with many outliers, making averaging and aggregation challenging. While conventional methods rely on Volume-Weighted Average Prices (called VWAPs), or Volume-Weighted Median prices (called VWMs), we develop a new Robust Weighted Median (RWM) estimator that is robust to price and volume outliers. Our study is based on new probabilistic concentration inequalities for weighted means and weighted quantiles under different tail assumptions (heavy tails, sub-gamma tails, sub-Gaussian tails). This justifies that fluctuations of VWAP and VWM are statistically important given the heavy-tailed properties of volumes and/or prices. We show that our RWM estimator overcomes this problem and also satisfies all the desirable properties of a price aggregator. We illustrate the behavior of RWM on synthetic data (within a parametric model close to real data): our estimator achieves a statistical accuracy twice as good as its competitors, and also allows to recover realized volatilities in a very accurate way. Tests on real data are also performed and confirm the good behavior of the estimator on various use cases.
• Linear convergence of evolution strategies with covariance matrix adaptation
  
  • Gissler Armand
  , 2024. Standing as the state-of-the-art algorithm among the evolution strategies, CMA-ES is a derivative-freeoptimization algorithm with many applications. However, the mathematical proof of its convergenceremains an open problem for more than 20 years. The main goal of this thesis is therefore to bringtheoretical guarantees of convergence of CMA-ES. More precisely, we prove that CMA-ES approachesthe minimum of an ellipsoidal function at a geometric rate. Furthermore, we confirm the conjecture thatthe covariance matrix in CMA-ES approximates the inverse Hessian of a convex-quadratic function.Our proof relies on the analysis of stochastic processes and is decomposed in several steps. Indeed,we define a stochastic process by normalizing the state variables of CMA-ES. This is inspired fromprevious works analyzing step-size adaptive ES: the normalization of the mean variable (translated bythe optimum) by the step-size yields to the definition of a Markov chain, assuming that the objectivefunction is scaling-invariant. Under additional assumptions the chain is geometrically ergodic and by limittheorems we find that the algorithm converges. For CMA-ES, we have to include the covariance matrixto obtain a Markov chain. First we define a normalization function R on the space of positive definitematrices, and by normalizing the mean by the stepsize and the function R applied to the covariancematrix, we aim to find a stationary process. With a normalized covariance matrix and possibly normalizedevolution paths, this process forms a time-homogeneous Markov chain when the objective functionis scaling-invariant. Proving that this normalized Markov chain converges to a stationary probabilitydistribution is the key to our proof of convergence of CMA-ES and will occupy Chapters 2 and 4.However first in Chapter 1, we give a methodology to establish the irreducibility, aperiodicityand topological properties of time-homogeneous Markov chains valued in manifolds and possibly withnonsmooth updates. These tools are the generalization of a previously developed analysis of nonlinearstate-space models, but which only includes Euclidean state spaces and continuously differentiable updatefunctions. By using results from the nonsmooth analysis and the theory of measures on topologicalmanifolds, we were able to extend this work.These preliminary results unlock a convergence proof of CMA-ES based on the stability analysis of underlying Markov chains, since the normalization of the covariance matrix transforms the state space ofthe algorithm into a manifold, and standard step-size adaptations include nonsmooth updates. Chapter 2explains how to use the fore-mentioned methodology and prove that the normalized Markov chain is anirreducible and aperiodic T-chain.We then prove its ergodicity by means of a Foster-Lyapunov method. In Chapter 4 we derive apotential function for which a state-dependent drift condition holds outside of a compact. Since the chain is a T-chain, compact sets are small, and since it is irreducible and aperiodic, a geometric driftcondition outside of a small set proves the geometric ergodicity of the chain. Yet the complexity of thechain (several state variables including a normalized covariance matrix) imposes us to restrict our proofto ellipsoidal objective functions.The final step of our convergence proof is shown in Chapter 5. We use an ergodic theorem and a Lawof Large Numbers to deduce the linear convergence of CMA-ES. Moreover, we use the affine-invarianceof the algorithm to find that the covariance matrix in CMA-ES learns second-order information and thatthe convergence rate is independent of which ellipsoidal objective function is minimized. (10.70675/820c617ezb5e5z4d7czbbdez942a5fd738a9)
  DOI : 10.70675/820c617ezb5e5z4d7czbbdez942a5fd738a9
• dles14: large-eddy simulation of solid/fluid heat and mass transfer applied to the thermal degradation of composite material
  
  • Grenouilloux Adrien
  • Letournel Roxane
  • Dellinger Nicolas
  • Bioche Kevin
  • Moureau Vincent
  , 2024. The ongoing trend towards improved aircraft efficiency in-volves the usage of higher-strength materials. In this context, carbon fiber reinforced polymers (CFRP), and more generally composite assemblies, have increasingly been used for the fuselage and nacelle’s fairing. They have gradually replaced heavier metallic alloys, thus improving the overall performance. Yet, a critical part of the design phase remains the fire certification of all components. Current international standards, such as the FAR25.856(b):2003 and ISO2685:1998(e), ensure the thermal resistance of these lighter materials when submitted to high-heat loads. Still, certification test campaigns are costly and often require a long time for their set-up. The introduction of numerical tools for the prediction of the degraded material properties could provide supplementary inputs and thus improve the design process. For the past decades, Large-Eddy Simulation (LES) has become a valuable tool for the simulation of unsteady reactive flows [1, 5, 11]. Several Conjugate Heat-Transfer (CHT) approaches have been performed to address the unsteady interactions between a fluid and a solid solver [5, 6]. These efforts have allowed to estimate the impact of the flame on the temperature distribution of solid geometries. However, the number of studies addressing the interaction of a flame leading to a composite plate degradation is limited [4]. In the present paper, a methodology for the coupling between a fluid, radiation and a solid solver, capable of respectively solving for the reactive, radiative heat losses and the thermal degradation of composite material, is presented. The procedure is first validated under simplified test-bed conditions on a so-called BLADE test [7]. In this context, a high-intensity laser beam replaces the external heat source from a flame.
• Development and analysis of efficient multi-scale numerical methods, with applications to plasma discharge simulations relying on multi-fluid models
  
  • Reboul Louis
  , 2024. Our main focus is the design and analysis of multi-scale numerical schemes for the simulation of multi-fluid models applied to low-temperature low-pressure plasmas. Our typical configuration of interest includes the onset of instabilities and sheaths, i.e. micrometric charged boundary layers that form at the plasma chamber walls. Our prototypical plasma model is the isothermal Euler-Poisson system of equations, but we also consider simpler models, the hyperbolic heat equations and the isothermal Euler-friction Equations, for the development and analysis of numerical methods. In a first axis, we develop and analyze a uniformly asymptotic-preserving second-order time-space coupling implicit-explicit method for the hyperbolic heat equations (linear case). We provide theoretical results on flux limiters for asymptotic-preserving methods, and a new well-balanced strategy. In a second axis, we propose several methods for the Euler-Poisson system of equations, to improve the accuracy of simulations of configurations featuring sheaths. In a third axis, we use these methods to conduct a parametric study of a 2D (rectangular) isothermal non-magnetized plasma discharge with sheaths, at various collisional regimes and aspect-ratios. We compare our result to PIC simulations and reference solutions. We show that simulating a fluid model with a tailored numerical method substantially reduces the time of simulation and improves the accuracy of the obtained solution. A discussion on the extensions of the multi-scale methods for the full non-isothermal Euler equations and to highly-magnetized cases is provided in the perspectives of our work. (10.70675/860f553az7ef2z4bb0z9118ze834fdcc9368)
  DOI : 10.70675/860f553az7ef2z4bb0z9118ze834fdcc9368
• A study of common noise in mean field games
  
  • Meynard Charles
  • Bertucci Charles
  , 2024. This paper is concerned with the study of mean field games master equations involving an additional variable modelling common noise. We address cases in which the dynamics of this variable can depend on the state of the game, which requires in general additional monotonicity assumptions on the coefficients. We explore the link between such a common noise and more traditional ones, as well as the links between different monotone regimes for the master equation.
• Impact of curved elements for flows over orography with a Discontinuous Galerkin scheme
  
  • Orlando Giuseppe
  • Benacchio Tommaso
  • Bonaventura Luca
  Journal of Computational Physics, Elsevier, 2024, 519, pp.113445. We present a quantitative assessment of the impact of high-order mappings on the simulation of flows over complex orography. Curved boundaries were not used in early numerical methods, whereas they are employed to an increasing extent in state of the art computational fluid dynamics codes, in combination with high-order methods, such as the Finite Element Method and the Spectral Element Method. Here we consider a specific Discontinuous Galerkin (DG) method implemented in the framework of the deal.II library, which natively supports high-order mappings. A number of numerical experiments based on classical benchmarks over idealized orographic profiles demonstrate the positive impact of curved boundaries on the accuracy of the results, with no significantly adverse effect on the computational cost of the simulation. These findings are also supported by results of the application of this approach to non-smooth and realistic orographic profiles. (10.1016/j.jcp.2024.113445)
  DOI : 10.1016/j.jcp.2024.113445
• Simulation of square-root processes made simple: applications to the Heston model
  
  • Abi Jaber Eduardo
  Risk Magazine (Cutting Edge Section), 2024. We introduce a simple, efficient and accurate nonnegative preserving numerical scheme for simulating the square-root process. The novel idea is to simulate the integrated squareroot process first instead of the square-root process itself. Numerical experiments on realistic parameter sets, applied for the integrated process and the Heston model, display high precision with a very low number of time steps. As a bonus, our scheme yields the exact limiting Inverse Gaussian distributions of the integrated square-root process with only one single time-step in two scenarios: (i) for high mean-reversion and volatility-of-volatility regimes, regardless of maturity; and (ii) for long maturities, independent of the other parameters.
• Modeling frequency distribution above a priority in presence of IBNR
  
  • Baradel Nicolas
  Scandinavian Actuarial Journal, Taylor & Francis (Routledge), 2024, pp.1-16. In reinsurance, Poisson and Negative binomial distributions are employed for modeling frequency. However, the incomplete data regarding reported incurred claims above a priority level presents challenges in estimation. This paper focuses on frequency estimation using Schnieper's framework for claim numbering. We demonstrate that Schnieper's model is consistent with a Poisson distribution for the total number of claims above a priority at each year of development, providing a robust basis for parameter estimation. Additionally, we explain how to build an alternative assumption based on a Negative binomial distribution, which yields similar results. The study includes a bootstrap procedure to manage uncertainty in parameter estimation and a case study comparing assumptions and evaluating the impact of the bootstrap approach. (10.1080/03461238.2024.2439815)
  DOI : 10.1080/03461238.2024.2439815
• Unravelling in Collaborative Learning
  
  • Capitaine Aymeric
  • Boursier Etienne
  • Scheid Antoine
  • Moulines Eric
  • Jordan Michael I.
  • El-Mhamdi El-Mahdi
  • Durmus Alain
  , 2024. Collaborative learning offers a promising avenue for leveraging decentralized data. However, collaboration in groups of strategic learners is not a given. In this work, we consider strategic agents who wish to train a model together but have sampling distributions of different quality. The collaboration is organized by a benevolent aggregator who gathers samples so as to maximize total welfare, but is unaware of data quality. This setting allows us to shed light on the deleterious effect of adverse selection in collaborative learning. More precisely, we demonstrate that when data quality indices are private, the coalition may undergo a phenomenon known as unravelling, wherein it shrinks up to the point that it becomes empty or solely comprised of the worst agent. We show how this issue can be addressed without making use of external transfers, by proposing a novel method inspired by probabilistic verification. This approach makes the grand coalition a Nash equilibrium with high probability despite information asymmetry, thereby breaking unravelling.
• Learning to Mitigate Externalities: the Coase Theorem with Hindsight Rationality
  
  • Scheid Antoine
  • Capitaine Aymeric
  • Boursier Etienne
  • Moulines Eric
  • Jordan Michael I
  • Durmus Alain
  , 2024. In economic theory, the concept of externality refers to any indirect effect resulting from an interaction between players that affects the social welfare. Most of the models within which externality has been studied assume that agents have perfect knowledge of their environment and preferences. This is a major hindrance to the practical implementation of many proposed solutions. To address this issue, we consider a two-player bandit setting where the actions of one of the players affect the other player and we extend the Coase theorem [Coase, 1960]. This result shows that the optimal approach for maximizing the social welfare in the presence of externality is to establish property rights, i.e., enable transfers and bargaining between the players. Our work removes the classical assumption that bargainers possess perfect knowledge of the underlying game. We first demonstrate that in the absence of property rights, the social welfare breaks down. We then design a policy for the players which allows them to learn a bargaining strategy which maximizes the total welfare, recovering the Coase theorem under uncertainty.
• Scaling limit of first passage percolation geodesics on planar maps
  
  • Kammerer Emmanuel
  , 2024. We establish the scaling limit of the geodesics to the root for the first passage percolation distance on random planar maps. We first describe the scaling limit of the number of faces along the geodesics. This result enables to compare the metric balls for the first passage percolation and the dual graph distance. It also enables to upperbound the diameter of large random maps. Then, we describe the scaling limit of the tree of first passage percolation geodesics to the root via a stochastic coalescing flow of pure jump diffusions. This stochastic flow also enables us to construct some random metric spaces which we conjecture to be the scaling limit of random planar maps with high degrees. The main tool in this work is a time-reversal of the uniform peeling exploration.
• Shapes analysis for time series
  
  • Germain Thibaut
  • Gruffaz Samuel
  • Truong Charles
  • Oudre Laurent
  • Durmus Alain
  , 2024. Analyzing inter-individual variability of physiological functions is particularly appealing in medical and biological contexts to describe or quantify health conditions. Such analysis can be done by comparing individuals to a reference one with time series as biomedical data. This paper introduces an unsupervised representation learning (URL) algorithm for time series tailored to inter-individual studies. The idea is to represent time series as deformations of a reference time series. The deformations are diffeomorphisms parameterized and learned by our method called TS-LDDMM. Once the deformations and the reference time series are learned, the vector representations of individual time series are given by the parametrization of their corresponding deformation. At the crossroads between URL for time series and shape analysis, the proposed algorithm handles irregularly sampled multivariate time series of variable lengths and provides shape-based representations of temporal data. In this work, we establish a representation theorem for the graph of a time series and derive its consequences on the LDDMM framework. We showcase the advantages of our representation compared to existing methods using synthetic data and real-world examples motivated by biomedical applications.
• Laplace transform based low-complexity learning of continuous Markov semigroups
  
  • Kostic Vladimir
  • Lounici Karim
  • Halconruy Hélène
  • Devergne Timothée
  • Novelli Pietro
  • Pontil Massimiliano
  , 2024. Markov processes serve as a universal model for many real-world random processes. This paper presents a data-driven approach for learning these models through the spectral decomposition of the infinitesimal generator (IG) of the Markov semigroup. The unbounded nature of IGs complicates traditional methods such as vector-valued regression and Hilbert-Schmidt operator analysis. Existing techniques, including physics-informed kernel regression, are computationally expensive and limited in scope, with no recovery guarantees for transfer operator methods when the time-lag is small. We propose a novel method that leverages the IG's resolvent, characterized by the Laplace transform of transfer operators. This approach is robust to time-lag variations, ensuring accurate eigenvalue learning even for small time-lags. Our statistical analysis applies to a broader class of Markov processes than current methods while reducing computational complexity from quadratic to linear in the state dimension. Finally, we illustrate the behaviour of our method in two experiments. (10.48550/arXiv.2410.14477)
  DOI : 10.48550/arXiv.2410.14477
• Any2Graph: Deep End-To-End Supervised Graph Prediction With An Optimal Transport Loss
  
  • Krzakala Paul
  • Yang Junjie
  • Flamary Rémi
  • d'Alché-Buc Florence
  • Laclau Charlotte
  • Labeau Matthieu
  , 2024. We propose Any2graph, a generic framework for end-to-end Supervised Graph Prediction (SGP) i.e. a deep learning model that predicts an entire graph for any kind of input. The framework is built on a novel Optimal Transport loss, the Partially-Masked Fused Gromov-Wasserstein, that exhibits all necessary properties (permutation invariance, differentiability and scalability) and is designed to handle any-sized graphs. Numerical experiments showcase the versatility of the approach that outperform existing competitors on a novel challenging synthetic dataset and a variety of real-world tasks such as map construction from satellite image (Sat2Graph) or molecule prediction from fingerprint (Fingerprint2Graph). (10.48550/arXiv.2402.12269)
  DOI : 10.48550/arXiv.2402.12269
• Aligning Embeddings and Geometric Random Graphs: Informational Results and Computational Approaches for the Procrustes-Wasserstein Problem
  
  • Even Mathieu
  • Ganassali Luca
  • Maier Jakob
  • Massoulié Laurent
  , 2024. The Procrustes-Wasserstein problem consists in matching two high-dimensional point clouds in an unsupervised setting, and has many applications in natural language processing and computer vision. We consider a planted model with two datasets X, Y that consist of n datapoints in R^d , where Y is a noisy version of X, up to an orthogonal transformation and a relabeling of the data points. This setting is related to the graph alignment problem in geometric models. In this work, we focus on the euclidean transport cost between the point clouds as a measure of performance for the alignment. We first establish information-theoretic results, in the high (d ≫ log n) and low (d ≪ log n) dimensional regimes. We then study computational aspects and propose the 'Ping-Pong algorithm', alternatively estimating the orthogonal transformation and the relabeling, initialized via a Franke-Wolfe convex relaxation. We give sufficient conditions for the method to retrieve the planted signal after one single step. We provide experimental results to compare the proposed approach with the state-of-the-art method of Grave et al. [2019].
• Renewal theorems in a periodic environment
  
  • Cormier Quentin
  , 2024. We study a renewal problem within a periodic environment, departing from the classical renewal theory by relaxing the assumption of independent and identically distributed inter-arrival times. Instead, the conditional distribution of the next arrival time, given the current one, is governed by a periodic kernel, denoted as $H$. The periodicity property of $H$ is expressed as $\mathbb{P}(T_{k+1} > t ~ |~ T_k) = H(t, T_k)$, where $H(t+T,s+T) = H(t, s)$. For a fixed time $t$, we define $N_t$ as the count of events occurring up to time $t$. The focus is on two temporal aspects: $Y_t$, the time elapsed since the last event, and $X_t$, the time until the next event occurs, given by $Y_t = t - T_{N_t}$ and $X_t = T_{N_{t}+1} - t$. The study explores the long-term behavior of the distributions of $X_t$ and $Y_t$.
• Statistical modeling for financial applications : rough volatility, market impact, and Hawkes processes
  
  • Szymanski Grégoire
  , 2024. This thesis is divided into three parts, addressing various issues ranging from statistical inference to financial modeling.In the first part, we study various problems of statistical inference in the presence of observational noise. In particular, we analyze the estimation of the Hurst parameter in fractional models under discrete noisy observations. When the noise consists of i.i.d. variables, we show that the Hurst parameter can be estimated with a convergence rate of order n^{1/(4H+2)}. We also establish a minimax theory demonstrating the optimality of this estimator. Different types of noise are considered, leading to different convergence rates. We then focus on the problem of estimating the invariant measure of an ergodic process from noisy observations. More precisely, we show that when the observations are at high frequency, we can estimate the invariant measure at a polynomial rate, which strongly contrasts with the case of low-frequency observations where the estimation rate is logarithmic.The second part is devoted to the estimation of the Hurst parameter H in fractional stochastic volatility models. This study proceeds in three steps. First, we show how to estimate H in parametric models by extending the previously mentioned minimax theory, which leads to the same convergence rate n^{1/(4H+2)} as before. Then, we apply this method to semi-parametric models commonly used in practice, including the rough Heston and rough Bergomi models as special cases. We finally demonstrate the robustness of this estimation procedure in the presence of microstructure noise, as well as when the price and volatility processes exhibit jumps. This ensures that this estimator is applicable in practice when using high-frequency data. This theoretical study is accompanied by a numerical analysis based on simulated data and is then applied to real data consisting of high-frequency observations of the S&P 500 over a four-year period.The last part of this thesis explores three issues related to financial microstructure. First, we examine the scaling limits of multidimensional Hawkes processes and analyze their behavior in mean-field-type regimes. We show a propagation of chaos result: when the Hawkes processes approach their critical regime, the global state of the system converges to a stochastic Volterra integral equation, and each particle becomes independent conditional on this mean state. Next, we study price impact models for large meta-orders. Specifically, we show how price models based on Hawkes processes reproduce various stylized facts concerning market impact. We highlight a power-law type dependence with respect to the traded volume. We also show that the market impact is proportional to the participation rate for small orders and proportional to its square root for larger orders. Finally, we enrich these price models to include passive orders, that is, transactions carried out via limit orders. (10.70675/daa96348z6abbz4738zadb3zaebd2f215f51)
  DOI : 10.70675/daa96348z6abbz4738zadb3zaebd2f215f51
• Polynomial Volterra processes
  
  • Abi Jaber Eduardo
  • Cuchiero Christa
  • Pelizzari Luca
  • Pulido Sergio
  • Svaluto-Ferro Sara
  Electronic Journal of Probability, Institute of Mathematical Statistics (IMS), 2024, 29, pp.1-37. We study the class of continuous polynomial Volterra processes, which we define as solutions to stochastic Volterra equations driven by a continuous semimartingale with affine drift and quadratic diffusion matrix in the state of the Volterra process. To demonstrate the versatility of possible state spaces within our framework, we construct polynomial Volterra processes on the unit ball. This construction is based on a stochastic invariance principle for stochastic Volterra equations with possibly singular kernels. Similarly to classical polynomial processes, polynomial Volterra processes allow for tractable expressions of the moments in terms of the unique solution to a system of deterministic integral equations, which reduce to a system of ODEs in the classical case. By applying this observation to the moments of the finite-dimensional distributions we derive a uniqueness result for polynomial Volterra processes. Moreover, we prove that the moments are polynomials with respect to the initial condition, another crucial property shared by classical polynomial processes. The corresponding coefficients can be interpreted as a deterministic dual process and solve integral equations dual to those verified by the moments themselves. Additionally, we obtain a representation of the moments in terms of a pure jump process with killing, which corresponds to another non-deterministic dual process.
• How does sailor morphology affect Olympic windfoil performance?
  
  • Hochhausen Martin
  • Sacher Matthieu
  • Augier Benoît
  • Nicolas Hugo
  • Bot Patrick
  • Hauville Frédéric
  • Clouet Yves
  , 2024. The Olympic windfoiling class consists of a windsurfing board that flies above the water with a hydrofoil. The sail is held by the sailor whose weight and position are key in the balance with aerodynamic and hydrodynamic forces. Therefore, understanding performance relative to sailor morphology is crucial. Windfoil performance is complex to model due to factors like aero-hydrodynamics, cavitation [1], and/or free surface effects [2] and fluid-structure interactions [3]. To assess the performance of a windfoil, the use of a Velocity Prediction Program (VPP) is a conventional approach [4] for sailing boats. This work presents the development of a 5DOF VPP tailored to windfoiling, optimizing sail angle, sailor position, and kinematic orientation. A simplified biomechanical model of the sailor is included to evaluate the impact of the sailor’s morphology on performance. The numerical flow model uses the Vortex Lattice Method (VLM), without considering structural effects, to minimize calculation costs. The results show that the VPP converges to different optimal positions and orientations for sailors with different morphologies. This work is part of the project ”Du Carbone à l’Or Olympique” and is funded by the Agence Nationale de La Recherche (ANR) through grant n°ANR-19- STHP-0002.