Publications

2023

• Goal-oriented sensitivity analysis of hyperparameters in deep learning
  
  • Novello Paul
  • Poëtte Gaël
  • Lugato David
  • Congedo Pietro Marco
  Journal of Scientific Computing, Springer Verlag, 2023, 94 (3), pp.45. Tackling new machine learning problems with neural networks always means optimizing numerous hyperparameters that define their structure and strongly impact their performances. In this work, we study the use of goal-oriented sensitivity analysis, based on the Hilbert-Schmidt Independence Criterion (HSIC), for hyperparameter analysis and optimization. Hyperparameters live in spaces that are often complex and awkward. They can be of different natures (categorical, discrete, boolean, continuous), interact, and have inter-dependencies. All this makes it non-trivial to perform classical sensitivity analysis. We alleviate these difficulties to obtain a robust analysis index that is able to quantify hyperparameters’ relative impact on a neural network’s final error. This valuable tool allows us to better understand hyperparameters and to make hyperparameter optimization more interpretable. We illustrate the benefits of this knowledge in the context of hyperparameter optimization and derive an HSIC-based optimization algorithm that we apply on MNIST and Cifar, classical machine learning data sets, but also on the approximation of Runge function and Bateman equations solution, of interest for scientific machine learning. This method yields competitive and cost-effective neural networks. (10.1007/s10915-022-02083-4)
  DOI : 10.1007/s10915-022-02083-4
• Numerical shape optimization among convex sets
  
  • Bogosel Beniamin
  Applied Mathematics and Optimization, Springer Verlag (Germany), 2023, 87 (1), pp.1. This article proposes a new discrete framework for approximating solutions to shape optimization problems under convexity constraints. The numerical method, based on the support function or the gauge function, is guaranteed to generate discrete convex shapes and is easily implementable using standard optimization software. The framework can handle various objective functions ranging from geometric quantities to functionals depending on partial differential equations. Width or diameter constraints are handled using the support function. Functionals depending on a convex body and its polar body can be handled using a unified framework. (10.1007/s00245-022-09920-w)
  DOI : 10.1007/s00245-022-09920-w
• Robust prediction interval estimation for Gaussian processes by cross-validation method
  
  • Acharki Naoufal
  • Bertoncello Antoine
  • Garnier Josselin
  Computational Statistics and Data Analysis, Elsevier, 2023, 178, pp.107597. Probabilistic regression models typically use the Maximum Likelihood Estimation or Cross-Validation to fit parameters. These methods can give an advantage to the solutions that fit observations on average, but they do not pay attention to the coverage and the width of Prediction Intervals. A robust two-step approach is used to address the problem of adjusting and calibrating Prediction Intervals for Gaussian Processes Regression. First, the covariance hyperparameters are determined by a standard Cross-Validation or Maximum Likelihood Estimation method. A Leave-One-Out Coverage Probability is introduced as a metric to adjust the covariance hyperparameters and assess the optimal type II Coverage Probability to a nominal level. Then a relaxation method is applied to choose the hyperparameters that minimize the Wasserstein distance between the Gaussian distribution with the initial hyperparameters (obtained by Cross-Validation or Maximum Likelihood Estimation) and the proposed Gaussian distribution with the hyperparameters that achieve the desired Coverage Probability. The method gives Prediction Intervals with appropriate coverage probabilities and small widths. (10.1016/j.csda.2022.107597)
  DOI : 10.1016/j.csda.2022.107597
• Development and Theoretical Analysis of the Algorithms for Optimal Control and Reinforcement Learning
  
  • Kaledin Maksim
  , 2023. In this PhD dissertation, we address the problems of optimal stopping and learning in Markov decision processes used in reinforcement learning (RL). In the first direction, we derive complexity estimates for the algorithm called Weighted Stochastic Mesh (WSM) and give a new method for comparing the complexity of optimal stopping algorithms with the semi tractability index. We show that WSM is optimal with respect to this criterion when the commonly used regression methods are much mess effectiveFor reinforcement learning, we give a non-asymptotic convergence analysis of a stochastic approximation scheme with two time scales - gradient TD - under assumptions of "martingale increment" noise - buffer replay - and of "Markov noise" (when learning is done along a single run). We obtain upper bounds that are rate-optimal by constructing an error expansion method that provides accurate control of the remainders terms.We also present a new algorithm for variance reduction in policy gradient schemes. The proposed approach is based on minimising an estimator for the empirical variance of the weighted rewards. We establish theoretical and practical gains over the classical actor-critic (A2C) method. (10.70675/d7fbe111z5bacz4419zb392z7adbeab6f5f9)
  DOI : 10.70675/d7fbe111z5bacz4419zb392z7adbeab6f5f9
• Truncation errors and modified equations for the lattice Boltzmann method via the corresponding Finite Difference schemes
  
  • Bellotti Thomas
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2023, 57, pp.1225–1255. Lattice Boltzmann schemes are efficient numerical methods to solve a broad range of problems under the form of conservation laws. However, they suffer from a chronic lack of clear theoretical foundations. In particular, the consistency analysis and the derivation of the modified equations are still open issues. This has prevented, until today, to have an analogous of the Lax equivalence theorem for Lattice Boltzmann schemes. We propose a rigorous consistency study and the derivation of the modified equations for any lattice Boltzmann scheme under acoustic and diffusive scalings. This is done by passing from a kinetic (lattice Boltzmann) to a macroscopic (Finite Difference) point of view at a fully discrete level in order to eliminate the non-conserved moments relaxing away from the equilibrium. We rewrite the lattice Boltzmann scheme as a multi-step Finite Difference scheme on the conserved variables, as introduced in our previous contribution. We then perform the usual analyses for Finite Difference by exploiting its precise characterization using matrices of Finite Difference operators. Though we present the derivation of the modified equations until second-order under acoustic scaling, we provide all the elements to extend it to higher orders, since the kinetic-macroscopic connection is conducted at the fully discrete level. Finally, we show that our strategy yields, in a more rigorous setting, the same results as previous works in the literature. (10.1051/m2an/2023008)
  DOI : 10.1051/m2an/2023008
• STATISTICAL INFERENCE FOR ROUGH VOLATILITY: MINIMAX THEORY
  
  • Chong Carsten
  • Hoffmann Marc
  • Liu Yanghui
  • Szymanski Grégoire
  • Rosenbaum Mathieu
  , 2023. Rough volatility models have gained considerable interest in the quantitative finance community in recent years. In this paradigm, the volatility of the asset price is driven by a fractional Brownian motion with a small value for the Hurst parameter H. In this work, we provide a rigorous statistical analysis of these models. To do so, we establish minimax lower bounds for parameter estimation and design procedures based on wavelets attaining them. We notably obtain an optimal speed of convergence of n −1/(4H+2) for estimating H based on n sampled data, extending results known only for the easier case H > 1/2 so far. We therefore establish that the parameters of rough volatility models can be inferred with optimal accuracy in all regimes.
• Optimal incentives in a limit order book: a SPDE control approach
  
  • Baldacci Bastien
  • Bergault Philippe
  , 2023. With the fragmentation of electronic markets, exchanges are now competing in order to attract trading activity on their platform. Consequently, they developed several regulatory tools to control liquidity provision / consumption on their liquidity pool. In this paper, we study the problem of an exchange using incentives in order to increase market liquidity. We model the limit order book as the solution of a stochastic partial differential equation (SPDE) as in [12]. The incentives proposed to the market participants are functions of the time and the distance of their limit order to the mid-price. We formulate the control problem of the exchange who wishes to modify the shape of the order book by increasing the volume at specific limits. Due to the particular nature of the SPDE control problem, we are able to characterize the solution with a classic Feynman-Kac representation theorem. Moreover, when studying the asymptotic behavior of the solution, a specific penalty function enables the exchange to obtain closed-form incentives at each limit of the order book. We study numerically the form of the incentives and their impact on the shape of the order book, and analyze the sensitivity of the incentives to the market parameters.
• Automated Market Makers: Mean-Variance Analysis of LPs Payoffs and Design of Pricing Functions
  
  • Bergault Philippe
  • Bertucci Louis
  • Bouba David
  • Guéant Olivier
  , 2023. We analyze the performance of Liquidity Providers (LPs) providing liquidity to different types of Automated Market Makers (AMMs). This analysis is carried out using a mean / standard deviation viewpoint \`a la Markowitz, though based on the PnL of LPs compared to that of agents holding coins outside of AMMs. We show that LPs tend to perform poorly in a wide variety of CFMMs under realistic market conditions. We then explore an alternative AMM design in which an oracle feeds the current market exchange rate to the AMM which then quotes a bid/ask spread. This allows us to define an efficient frontier for the performance of LPs in an idealized world with perfect information and to show that the smart use of oracles greatly improves LPs' risk / return profile, even in the case of a lagged oracle.
• A Formal Disproof of Hirsch Conjecture
  
  • Allamigeon Xavier
  • Canu Quentin
  • Strub Pierre-Yves
  , 2023, pp.17-29. The purpose of this paper is the formal verification of a counterexample of Santos et al. to the so-called Hirsch Conjecture on the diameter of polytopes (bounded convex polyhedra). In contrast with the pen-and-paper proof, our approach is entirely computational: we have implemented in Coq and proved correct an algorithm that explicitly computes, within the proof assistant, vertex-edge graphs of polytopes as well as their diameter. The originality of this certificate-based algorithm is to achieve a tradeoff between simplicity and efficiency. Simplicity is crucial in obtaining the proof of correctness of the algorithm. This proof splits into the correctness of an abstract algorithm stated over proof-oriented data types and the correspondence with a low-level implementation over computation-oriented data types. A special effort has been made to reduce the algorithm to a small sequence of elementary operations (e.g. matrix multiplications, basic routines on graphs), in order to make the derivation of the correctness of the low-level implementation more transparent. Efficiency allows us to scale up to polytopes with a challenging combinatorics. For instance, we formally check the two counterexamples to Hirsch conjecture due to Matschke, Santos and Weibel, respectively 20- and 23-dimensional polytopes with 36425 and 73224 vertices involving rational coefficients with up to 20 digits. We also illustrate the performance of the method by computing the list of vertices or the diameter of well-known classes of polytopes, such as (polars of) cyclic polytopes involved in McMullen's Upper Bound Theorem. (10.1145/3573105.3575678)
  DOI : 10.1145/3573105.3575678
• State and parameter learning with PARIS particle Gibbs
  
  • Cardoso Gabriel Victorino
  • Janati Yazid
  • Le Corff Sylvain
  • Moulines Éric
  • Olsson Jimmy
  , 2023. Non-linear state-space models, also known as general hidden Markov models, are ubiquitous in statistical machine learning, being the most classical generative models for serial data and sequences in general. The particle-based, rapid incremental smoother (PARIS) is a sequential Monte Carlo (SMC) technique allowing for efficient online approximation of expectations of additive functionals under the smoothing distribution in these models. Such expectations appear naturally in several learning contexts, such as likelihood estimation (MLE) and Markov score climbing (MSC). PARIS has linear computational complexity, limited memory requirements and comes with non-asymptotic bounds, convergence results and stability guarantees. Still, being based on selfnormalised importance sampling, the PARIS estimator is biased. Our first contribution is to design a novel additive smoothing algorithm, the Parisian particle Gibbs (PPG) sampler, which can be viewed as a PARIS algorithm driven by conditional SMC moves, resulting in bias-reduced estimates of the targeted quantities. We substantiate the PPG algorithm with theoretical results, including new bounds on bias and variance as well as deviation inequalities. Our second contribution is to apply PPG in a learning framework, covering MLE and MSC as special examples. In this context, we establish, under standard assumptions, non-asymptotic bounds highlighting the value of bias reduction and the implicit Rao-Blackwellization of PPG. These are the first non-asymptotic results of this kind in this setting. We illustrate our theoretical results with numerical experiments supporting our claims. (10.48550/arXiv.2301.00900)
  DOI : 10.48550/arXiv.2301.00900
• Time reversal of spinal processes for linear and non-linear branching processes near stationarity
  
  • Henry Benoît
  • Méléard Sylvie
  • Chi Tran Viet
  Electronic Journal of Probability, Institute of Mathematical Statistics (IMS), 2023, 28 (none). We consider a stochastic individual-based population model with competition, trait-structure affecting reproduction and survival, and changing environment. The changes of traits are described by jump processes, and the dynamics can be approximated in large population by a non-linear PDE with a non-local mutation operator. Using the fact that this PDE admits a non-trivial stationary solution, we can approximate the non-linear stochastic population process by a linear birth-death process where the interactions are frozen, as long as the population remains close to this equilibrium. This allows us to derive, when the population is large, the equation satisfied by the ancestral lineage of an individual uniformly sampled at a fixed time $T$, which is the path constituted of the traits of the ancestors of this individual in past times $t\leq T$. This process is a time inhomogeneous Markov process, but we show that the time reversal of this process possesses a very simple structure (e.g. time-homogeneous and independent of $T$). This extends recent results where the authors studied a similar model with a Laplacian operator but where the methods essentially relied on the Gaussian nature of the mutations. (10.1214/23-EJP911)
  DOI : 10.1214/23-EJP911
• Non-asymptotic convergence bounds for Sinkhorn iterates and their gradients: a coupling approach
  
  • Greco Giacomo
  • Noble Maxence
  • Conforti Giovanni
  • Durmus Alain
  , 2023. Computational optimal transport (OT) has recently emerged as a powerful framework with applications in various fields. In this paper we focus on a relaxation of the original OT problem, the entropic OT problem, which allows to implement efficient and practical algorithmic solutions, even in high dimensional settings. This formulation, also known as the Schrödinger Bridge problem, notably connects with Stochastic Optimal Control (SOC) and can be solved with the popular Sinkhorn algorithm. In the case of discrete-state spaces, this algorithm is known to have exponential convergence; however, achieving a similar rate of convergence in a more general setting is still an active area of research. In this work, we analyze the convergence of the Sinkhorn algorithm for probability measures defined on the d-dimensional torus , that admit densities with respect to the Haar measure. In particular, we prove pointwise exponential convergence of Sinkhorn iterates and their gradient. Our proof relies on the connection between these iterates and the evolution along the Hamilton-Jacobi-Bellman equations of value functions obtained from SOC-problems. Our approach is novel in that it is purely probabilistic and relies on coupling by reflection techniques for controlled diffusions on the torus.
• Tree-Based Diffusion Schrödinger Bridge with Applications to Wasserstein Barycenters
  
  • Noble Maxence
  • de Bortoli Valentin
  • Doucet Arnaud
  • Durmus Alain
  , 2023. Multi-marginal Optimal Transport (mOT), a generalization of OT, aims at minimizing the integral of a cost function with respect to a distribution with some prescribed marginals. In this paper, we consider an entropic version of mOT with a tree-structured quadratic cost, i.e., a function that can be written as a sum of pairwise cost functions between the nodes of a tree. To address this problem, we develop Tree-based Diffusion Schrödinger Bridge (TreeDSB), an extension of the Diffusion Schrödinger Bridge (DSB) algorithm. TreeDSB corresponds to a dynamic and continuous state-space counterpart of the multi-marginal Sinkhorn algorithm. A notable use case of our methodology is to compute Wasserstein barycenters which can be recast as the solution of a mOT problem on a star-shaped tree. We demonstrate that our methodology can be applied in high-dimensional settings such as image interpolation and Bayesian fusion.
• Error estimates for finite differences approximations of the total variation
  
  • Caillaud Corentin
  • Chambolle Antonin
  IMA Journal of Numerical Analysis, Oxford University Press (OUP), 2023, 43 (2), pp.692--736. We present a convergence rate analysis of the Rudin-Osher-Fatemi (ROF) denoising problem for two different discretizations of the total variation. The first discretization is the well-known isotropic total variation that suffers from a blurring effect in a special diagonal direction. We prove that in the setting corresponding to this direction, the discrete ROF energy converges to the continuous one in O(h^2/3). The second total variation is based on Raviart-Thomas fields and achieves a O(h) convergence rate for the same quantity under some standard hypotheses. (10.1093/imanum/drac001)
  DOI : 10.1093/imanum/drac001
• Design patterns of hierarchies for order structures
  
  • Allamigeon Xavier
  • Canu Quentin
  • Cohen Cyril
  • Sakaguchi Kazuhiko
  • Strub Pierre-Yves
  , 2023. Using order structures in a proof assistant naturally raises the problem of working with multiple instances of a same structure over a common type of elements. This goes against the main design pattern of hierarchies used for instance in Coq's MathComp or Lean's mathlib libraries, where types are canonically associated to at most one instance and instances share a common overloaded syntax. We present new design patterns to leverage these issues, and apply them to the formalization of order structures in the MathComp library. A common idea in these patterns is underloading, i.e., a disambiguation of operators on a common type. In addition, our design patterns include a way to deal with duality in order structures in a convenient way. We hence formalize a large hierarchy which includes partial orders, semilattices, lattices as well as many variants. We finally pay a special attention to order substructures. We introduce a new kind of structure called prelattice. They are abstractions of semilattices, and allow us to deal with finite lattices and their sublattices within a common signature. As an application, we report on significant simplifications of the formalization of the face lattices of polyhedra in the Coq-Polyhedra library.
• An optimal control-based numerical method for scalar transmission problems with sign-changing coefficients
  
  • Ciarlet Patrick
  • Lassounon David
  • Rihani Mahran
  SIAM Journal on Numerical Analysis, Society for Industrial and Applied Mathematics, 2023, 61 (3), pp.1316-1339. In this work, we present a new numerical method for solving the scalar transmission problem with sign-changing coefficients. In electromagnetism, such a transmission problem can occur if the domain of interest is made of a classical dielectric material and a metal or a metamaterial, with for instance an electric permittivity that is strictly negative in the metal or metamaterial. The method is based on an optimal control reformulation of the problem. Contrary to other existing approaches, the convergence of this method is proved without any restrictive condition. In particular, no condition is imposed on the a priori regularity of the solution to the problem, and no condition is imposed on the meshes, other than that they fit with the interface between the two media. Our results are illustrated by some (2D) numerical experiments. (10.1137/22M1495998)
  DOI : 10.1137/22M1495998
• A Robust Low-Voltage-Ride-Through Strategy for Grid-Forming Converters Based on Reactive Power Synchronization
  
  • Deng Han
  • Qi Yang
  • Fang Jingyang
  • Tang Yi
  • Debusschere Vincent
  IEEE Transactions on Power Electronics, Institute of Electrical and Electronics Engineers, 2023, 38 (1), pp.346--357. (10.1109/TPEL.2022.3204912)
  DOI : 10.1109/TPEL.2022.3204912
• The nonlocal isoperimetric problem for polygons: Hardy-Littlewood and Riesz inequalities
  
  • Bogosel Beniamin
  • Bucur Dorin
  • Fragalà Ilaria
  Mathematische Annalen, Springer Verlag, 2023. Given a non-increasing and radially symmetric kernel in $L ^ 1 _{\rm loc} (\Bbb{R} ^ 2 ; \Bbb{R} _+)$, we investigate counterparts of the classical Hardy-Littlewood and Riesz inequalities when the class of admissible domains is the family of polygons with given area and $N$ sides. The latter corresponds to study the polygonal isoperimetric problem in nonlocal version. We prove that, for every $N \geq 3$, the regular $N$-gon is optimal for Hardy-Littlewood inequality. Things go differently for Riesz inequality: while for $N = 3$ and $N = 4$ it is known that the regular triangle and the square are optimal, for $N\geq 5$ we prove that symmetry or symmetry breaking may occur (i.e.\ the regular $N$-gon may be optimal or not), depending on the value of $N$ and on the choice of the kernel. (10.1007/s00208-023-02683-x)
  DOI : 10.1007/s00208-023-02683-x
• Stochastic Approximation Beyond Gradient for Signal Processing and Machine Learning
  
  • Dieuleveut Aymeric
  • Fort Gersende
  • Moulines Eric
  • Wai Hoi-To
  IEEE Transactions on Signal Processing, Institute of Electrical and Electronics Engineers, 2023, 71, pp.3117-3148. Stochastic approximation (SA) is a classical algorithm that has had since the early days a huge impact on signal processing, and nowadays on machine learning, due to the necessity to deal with a large amount of data observed with uncertainties. An exemplar special case of SA pertains to the popular stochastic (sub)gradient algorithm which is the working horse behind many important applications. A lesser-known fact is that the SA scheme also extends to non-stochastic-gradient algorithms such as compressed stochastic gradient, stochastic expectation-maximization, and a number of reinforcement learning algorithms. The aim of this article is to overview and introduce the non-stochastic-gradient perspectives of SA to the signal processing and machine learning audiences through presenting a design guideline of SA algorithms backed by theories. Our central theme is to propose a general framework that unifies existing theories of SA, including its non-asymptotic and asymptotic convergence results, and demonstrate their applications on popular non-stochastic-gradient algorithms. We build our analysis framework based on classes of Lyapunov functions that satisfy a variety of mild conditions. We draw connections between non-stochastic-gradient algorithms and scenarios when the Lyapunov function is smooth, convex, or strongly convex. Using the said framework, we illustrate the convergence properties of the non-stochastic-gradient algorithms using concrete examples. Extensions to the emerging variance reduction techniques for improved sample complexity will also be discussed. (10.1109/TSP.2023.3301121)
  DOI : 10.1109/TSP.2023.3301121
• Well posedness and stochastic derivation of a diffusion-growth-fragmentation equation in a chemostat
  
  • Tchouanti Josué
  Stochastics and Partial Differential Equations: Analysis and Computations, Springer US, 2023. We study the existence and uniqueness of the solution of a non-linear coupled system constituted of a degenerate diffusion-growth-fragmentation equation and a differential equation, resulting from the modeling of bacterial growth in a chemostat. This system is derived, in a large population approximation, from a stochastic individual-based model where each individual is characterized by a non-negative real valued trait described by a diffusion. Two uniqueness results are highlighted. They differ in their hypotheses related to the influence of the resource on individual trait dynamics, the main difficulty being the non-linearity due to this dependence and the degeneracy of the diffusion coefficient. Further we show that the semi-group of the stochastic trait dynamics admits a density by probabilistic arguments, that allows the measure solution of the diffusiongrowth-fragmentation equation to be a function with a certain Besov regularity. (10.1007/s40072-023-00288-8)
  DOI : 10.1007/s40072-023-00288-8
• A Posteriori Validation of Generalized Polynomial Chaos Expansions
  
  • Breden Maxime
  SIAM Journal on Applied Dynamical Systems, Society for Industrial and Applied Mathematics, 2023, 22 (2), pp.765-801. Generalized polynomial chaos (gPC) expansions are a powerful tool for studying differential equations with random coefficients, allowing, in particular, one to efficiently approximate random invariant sets associated with such equations. In this work, we use ideas from validated numerics in order to obtain rigorous a posteriori error estimates together with existence results about gPC expansions of random invariant sets. This approach also provides a new framework for conducting validated continuation, i.e., for rigorously computing isolated branches of solutions in parameter-dependent systems, which generalizes in a straightforward way to multiparameter continuation. We illustrate the proposed methodology by rigorously computing random invariant periodic orbits in the Lorenz system, as well as branches and 2 dimensional manifolds of steady states of the Swift–Hohenberg equation. (10.1137/22M1493197)
  DOI : 10.1137/22M1493197
• Cross impact in derivative markets
  
  • Tomas Mehdi
  • Mastromatteo Iacopo
  • Benzaquen Michael
  Wilmott Journal, Wiley, 2023, 123, pp.16–28. We introduce a linear cross-impact framework in a setting in which the price of some given financial instruments (derivatives) is a deterministic function of one or more, possibly tradeable, stochastic factors (underlying). We show that a particular cross-impact model, the multivariate Kyle model, prevents arbitrage and aggregates (potentially non-stationary) traded order flows on derivatives into (roughly stationary) liquidity pools aggregating order flows traded on both derivatives and underlying. Using E-Mini futures and options along with VIX futures, we provide empirical evidence that the price formation process from order flows on derivatives is driven by cross-impact and confirm that the simple Kyle cross-impact model is successful at capturing parsimoniously such empirical phenomenology. Our framework may be used in practice for estimating execution costs, in particular hedging costs.
• Filling the gap between individual-based evolutionary models and Hamilton-Jacobi equations
  
  • Champagnat Nicolas
  • Méléard Sylvie
  • Mirrahimi Sepideh
  • Chi Tran Viet
  Journal de l'École polytechnique — Mathématiques, École polytechnique, 2023, 10, pp.1247-1275. We consider a stochastic model for the evolution of a discrete population structured by a trait with values on a finite grid of the torus, and with mutation and selection. Traits are vertically inherited unless a mutation occurs, and influence the birth and death rates. We focus on a parameter scaling where population is large, individual mutations are small but not rare, and the grid mesh for the trait values is much smaller than the size of mutation steps. When considering the evolution of the population in a long time scale, the contribution of small sub-populations may strongly influence the dynamics. Our main result quantifies the asymptotic dynamics of sub-population sizes on a logarithmic scale. We establish that under the parameter scaling the logarithm of the stochastic population size process, conveniently normalized, converges to the unique viscosity solution of a Hamilton-Jacobi equation. Such Hamilton-Jacobi equations have already been derived from parabolic integro-differential equations and have been widely developed in the study of adaptation of quantitative traits. Our work provides a justification of this framework directly from a stochastic individual based model, leading to a better understanding of the results obtained within this approach. The proof makes use of almost sure maximum principles and careful controls of the martingale parts. (10.5802/jep.244)
  DOI : 10.5802/jep.244
• A growth-fragmentation-isolation process on random recursive trees and contact tracing
  
  • Bansaye Vincent
  • Gu Chenlin
  • Yuan Linglong
  The Annals of Applied Probability, Institute of Mathematical Statistics (IMS), 2023, 33 (6B), pp.5233-5278. We consider a random process on recursive trees, with three types of events. Vertices give birth at a constant rate (growth), each edge may be removed independently (fragmentation of the tree) and clusters (or trees) are frozen with a rate proportional to their sizes (isolation of connected component). A phase transition occurs when the isolation is able to stop the growthfragmentation process and cause extinction. When the process survives, the number of clusters increases exponentially and we prove that the normalised empirical measure of clusters a.s. converges to a limit law on recursive trees. We exploit the branching structure associated to the size of clusters, which is inherited from the splitting property of random recursive trees. This work is motivated by the control of epidemics and contact tracing where clusters correspond to trees of infected individuals that can be identified and isolated. We complement this work by providing results on the Malthusian exponent to describe the effect of control policies on epidemics. (10.1214/23-AAP1947)
  DOI : 10.1214/23-AAP1947
• Mean field game of mutual holding with defaultable agents, and systemic risk
  
  • Djete Mao Fabrice
  • Guo Gaoyue
  • Touzi Nizar
  SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2023, 64 (1), pp.24-58. We introduce the possibility of default in the mean field game of mutual holding of Djete and Touzi [11]. This is modeled by introducing absorption at the origin of the equity process. We provide an explicit solution of this mean field game. Moreover, we provide a particle system approximation, and we derive an autonomous equation for the time evolution of the default probability, or equivalently the law of the hitting time of the origin by the equity process. The systemic risk is thus described by the evolution of the default probability. (10.1137/24M1658668)
  DOI : 10.1137/24M1658668