# Séminaire des Doctorants

# Séminaire des doctorants du CMAP et du CMLS

**Équipe d'organisation : **Margherita Castellano, Armand Gissler, Emmanuel Kammerer, Grégoire Pacreau, Matthias Rakotomalala, Nathan Sauldubois, Emmanouil Sfendourakis.

## 2023-2024

**Mercredi 19 Juin (15h-16h): Leila Bassou (CMAP) **

**Mean Field Game of Mutual Holding with common noise**

**Mean Field Game of Mutual Holding with common noise**

We consider the Mean Field Game of Mutual Holding introduced in the paper of Djete & Touzi (2020), within a framework where equity value dynamics are affected by common noise. The problem formulation uncovers a No-Arbitrage (NA) condition that is necessary for the existence of equilibria and streamlines their investigation. The presentation is structured into two parts: The first part is dedicated to the one-period model. We explicitly characterize the NA condition and the mean field equilibria related to a mean-variance criterion. In the second part, we extend the study to a continuous-time setting. Here, we use a weak notion of the NA condition, under which the representative agent's optimization step is reduced to a standard portfolio optimization problem with random endowment. This is a joint work with Mao Fabrice Djete and Nizar Touzi.

**Mercredi 4 Juin (15h-16h): Jean Pachebat (CMAP) **

**Matching Heavy tailed distributions with Generative models: Theory and experiments**

**Matching Heavy tailed distributions with Generative models: Theory and experiments**

Examining extreme events is a critical concern across various fields such as economics, engineering, and life sciences, with wide-ranging applications like ac- tuarial and financial risks, communication network reliability, and aircraft safety. Extreme events play a crucial role also in the context of climate change, with the occurrence of more and more severe weather events, or in the context of cyber- security with the increasing number of cyber-attacks of private companies or public entities. Generative modeling, a Machine Learning framework, aims at reproducing sample from a generative process, may it be from known historical samples (a dataset) or a data density known up to a constant. In this talk, I will present ongoing work on designing Generative Models that match heavy tailed distributions. This work specifically focuses on reproducing the dependence structure in the extremes of a target signal, i.e. regions far from the origin. This presentation will include a theoretical analysis of the problem and illustrations on numerical simulations.

**Mercredi 17 mai (15h-16h): Guillaume Broux-Quemerais (LMM, Le Mans Université)**

**Deep learning scheme for forward utilities using ergodic BSDEs**

**Deep learning scheme for forward utilities using ergodic BSDEs**

In this work, we develop a probabilistic numerical method for a class of forward utilities in a stochastic factor model. For this purpose, we use the representation of dynamic utilities using the ergodic Backward Stochastic Differential Equations (eBSDEs) introduced by Liang and Zariphopoulou. We establish a connection between the solution of the ergodic BSDE and the solution of an associated BSDE with random terminal time, defined as the hitting time of the positive recurrent stochastic factor. The viewpoint based on BSDEs with random horizon yields a new characterization of the ergodic cost, which is a part of the solution of the eBSDEs. In particular, for a certain class of eBSDEs with quadratic generator, the Cole-Hopf transformation leads to a semi-explicit representation of the solution as well as a new expression of the ergodic cost. The latter can be estimated with Monte Carlo methods. We also propose two new deep learning numerical schemes for eBSDEs, where the ergodic cost is optimized according either to a global loss function at the random horizon or to the aggregation of local loss functions. Finally, we present numerical results for different examples of eBSDEs and forward utilities together with the associated investment strategies.

**Mercredi 24 avril (15h-16h): Orso Forghieri (CMAP)**

**State Abstraction discovery in Model-Based Reinforcement Learning**

**State Abstraction discovery in Model-Based Reinforcement Learning**

In Reinforcement Learning, we embody an agent that evolves within a given environment. At each time step, we take an action and receive a reward along with the next state we visit. In model-based RL, we assume exact knowledge of the state transitions probabilities and subsequent rewards, aiming to determine the optimal policy to maximize the expected sum of future rewards. However, solving the environment model (a Markov Decision Process) suffers from high dimensionality when using traditional dynamic programming method. Hierarchical Reinforcement Learning introduces state and action abstraction as a divide-and-conquer approach on the state and action spaces, mitigating this issue. We focus here on spatial abstraction (division of the state space), maximizing a discounted sum of rewards in an infinite horizon context. As explicit methods for building useful spatial abstractions of models are rare, we present an algorithm that combines Approximate Dynamic Programming with state space disaggregation. Our approach involves aggregations of similar states explaining the underlying MDP structure and ensures convergence to the optimal value. We base our convergence proof on a bound that estimates the quality of any piecewise constant value function approximation based on the related aggregation quality.

**Mercredi 10 avril (15h-16h): Armand Gissler (CMAP)**

**Convergence analysis of evolution strategies with covariance matrix adaptation**

**Convergence analysis of evolution strategies with covariance matrix adaptation**

In optimization, we seek to identify the global minimum (or maximum) of an objective function. However this task may become difficult, particularly when dealing with nonconvex, multimodal, nondifferentiable functions. I will present evolutionary strategy (ES) algorithms, designed to solve such problems, with a particular focus on ES with covariance matrix adaptation (CMA-ES). Despite empirical evidences suggesting that it converges to the solution of many optimization problems---including highly ill-conditioned problems for which second-order information seems to be learnt---mathematical proofs validating these observations have been lacking. I will present a successful approach to prove convergence of CMA-ES, which relies on the analysis of a normalized stochastic process.

**Mercredi 20 mars (15h-16h): Adriano Prade (CMAP)**

**Fractional Laplacian operator and regularity of nonlocal PDEs**

**Fractional Laplacian operator and regularity of nonlocal PDEs**

After the breakthrough paper by Caffarelli and Silvestre in 2007, the study of fractional Laplacian and more general nonlocal operators has gained increasing popularity, from both an analytical and a probabilistic point of view. The purpose of the seminar is to present such a class of operators, starting from basic notions and then focusing on the PDEs’ theory developing from them. First, the formula for the square root of the Laplacian (-∆)^½ is provided, together with a few immediate remarks and the motivation behind its name. Then, after introducing the definition of fractional Laplacian (-∆)^s, we give an overview of its main properties, highlighting some similarities and differences with the classical Laplacian (-∆). The second part of the talk is entirely devoted to nonlocal PDEs, reserving particular attention to some regularity issues. We begin by dealing with the possible notions of solutions and next various results available in the literature are outlined. Finally, after introducing the class of Reifenberg flat sets, we present the problem of boundary Hölder regularity of some nonlocal PDEs on this kind of sets, sketching some possible solution strategies if time permits.

**Mercredi 6 mars (15h-16h): Jules Delemotte (CMAP)**

**Evaluating the Skew-Stickiness Ratio in stochastic and rough volatility models**

**Evaluating the Skew-Stickiness Ratio in stochastic and rough volatility models**

After an introduction to option pricing, we will study the dynamic properties of some classes of stochastic and rough volatility models (including well-known classical examples with their "rough volatility counterpart": the 2-factor Bergomi model, the rough Bergomi model, Heston and rough Heston). For dynamic properties, we intend the dynamics of option implied volatilities, as induced by the model. For some of the recently introduced models (notably rough volatility models), quite some effort in the literature has been concentrated on the analysis of their static properties such as their calibration power or the term structure of ATM skews but, to the best of our knowledge, their dynamic properties have received only little attention so far. One specific indicator of joint spot-price and implied volatility dynamics is the Skew-Stickiness Ratio (SSR), introduced by Bergomi [Bergomi, Smile dynamics IV, Risk 2009] and related to classical smile dynamic regimes (namely, sticky-strike and sticky-delta). We evaluate different estimators of the model SSR -- mainly Monte Carlo based -- and compare the results with the empirical market SSR for some large stock indices, which sheds light on the interest of using a certain modeling choice with respect to another. With a view on explicit approximation formulas, we build on the celebrated Bergomi-Guyon expansion for ATM implied volatilities and skews so to obtain explicit expansions of the model SSR, for which we analyse the accuracy with respect to our Monte Carlo benchmark.

**Mercredi 28 février (15h-16h): Charles Meynard (CMAP)**

**Noise through an additional variable for mean field games master equation on finite state space**

**Noise through an additional variable for mean field games master equation on finite state space**

This paper provides a mathematical study of the well-posedness of master equation on finite state space involving terms modelling common noise. In this setting, the solution of the master equation depends on an additional variable modelling the value of a stochastic process impacting all players. Using technique from viscosity solutions, we give sufficient conditions for the existence of a Lipschitz continuous solution on any time interval. Under some structural assumptions, we are even able to treat cases in which the dynamics of this stochastic process depend on the state of the game.

**Mercredi 14 février (15h-16h): Antoine Van Biesbroeck (CMAP)**

**Construction de priors de référence en inférence bayésienne, application à l’estimation de courbes de fragilité sismique**

**Construction de priors de référence en inférence bayésienne, application à l’estimation de courbes de fragilité sismique**

Les courbes de fragilité sismique quantifient la probabilité de défaillance d’une structure mécanique conditionnellement à une Mesure d'Intensité (IM) d’un signal sismique. L'estimation des courbes de fragilité s'effectue à partir de données constituées de paires signal sismique (entrée fonctionnelle) et défaillance de la structure (sortie binaire). Les méthodes paramétriques traditionnelles sont souvent mises en défaut en raison du nombre limité de données disponibles dans ce contexte. L’approche bayésienne permet un apprentissage efficace des paramètres d’intérêt, mais reste tributaire du choix du prior, pour lequel toute subjectivité est à proscrire. Nous proposons l’appui de la théorie des priors de référence, que nous enrichissons d’une généralisation de la définition d’information mutuelle, pour justifier l’implémentation du prior de Jeffreys dans notre cadre d’étude. Nos résultats démontrent la supériorité et la robustesse de ce dernier comparé à la littérature, tout en soulignant la sensibilité des estimations au choix du prior et l’impact de la corrélation de l’IM sur la dégénérescence des courbes de fragilité. Nos conclusions mettent en avant l’importance d’une construction objective du prior dans les études bayésiennes.

**Mercredi 25 janvier (15h-16h): Emmanouil Sfendourakis (CMAP)**

**Understanding the worst-kept secret of high-frequency trading**

**Understanding the worst-kept secret of high-frequency trading**

After an introduction to optimal control, we present the stakes of market-making problems and their adaptation to the mechanism of limit order books, which is the way most securities are traded in electronic markets. The volume imbalance is often considered as a reliable indicator for predicting future price movements. In this study, we confirm this statement by analyzing an optimal control problem in which a market maker controls volumes in the limit order book of a large-tick stock and quotes prices at a half-tick distance from the mid-price. We model the mid-price, which is not a controlled variable, using uncertainty zones. The market maker has information about the underlying efficient price and consequently of the probability of a price jump in the future. By using this information , it is optimal for the market maker to create imbalances which are predictive of price movements. The value function of the market maker's control problem can be understood as a family of functions, indexed by the level of the market maker's inventory, solving a coupled system of PDEs. We show existence and uniqueness of smooth solutions for this coupled system of equations. In the case of a continuous inventory, we also prove the uniqueness of the market maker's optimal control policy.

**Mercredi 6 décembre (15h-16h): Pierre Mackowiak (CMAP)**

**Équation de Gross-Pitaevskii avec potentiel bruit blanc**

**Équation de Gross-Pitaevskii avec potentiel bruit blanc**

L'équation de Gross-Pitaevskii avec potentiel bruit blanc est une équation de Schrödinger non-linéaire qui comporte un terme de bruit multiplicatif. Elle apparaît comme un modèle jouet dans l'étude des condensats de Bose-Einstein en milieu inhomogène. Je présenterai les résultats que j'ai obtenus en dimension 1 et 2 concernant l'existence et l'unicité de solutions. En dimension 1, le bruit est suffisamment régulier pour obtenir des solutions locales par point fixe. En dimension 2, le bruit est singulier et une procédure de renormalisation est nécessaire pour donner un sens à l'équation. J'expliquerai cette renormalisation, puis je donnerai une idée de la preuve d'existence globale pour les non-linéarités cubiques.

**Mercredi 22 novembre (15h-16h): Grégoire Szymanski (CMAP)**

**The two square root laws of Market Impact and the role of sophisticated market participants**

**The two square root laws of Market Impact and the role of sophisticated market participants**

In this study, we revisit the Hawkes order flow model for market impact initially introduced by Jaisson in 2015, which assumes linear market impact for individual orders and a price process modeled as a martingale. Our approach extends this model by introducing sophisticated market participants capable of reshaping the volatility profile. The strength of our methodology lies in two primary aspects: firstly, it relies on minimal or no additional assumptions, and secondly, it yields closed-form expressions for market impact. Notably, our analysis leads to the recovery of two well-known square root laws. Specifically, for a fixed duration, market impact scales proportionally to $\sqrt{\gamma}$, where $\gamma$ denotes the participation rate. Additionally, for a given order size, market impact adheres to a power-law behavior with respect to the total duration.

**Mercredi 8 novembre (15h-16h): Roberta Flenghi (CERMICS, Ecole des Ponts)**

**Central limit theorem for the stratified selection mechanism**

**Central limit theorem for the stratified selection mechanism**

The stratified resampling mechanism is one of the resampling schemes commonly used in the resampling steps of particle filters. In the present paper, we prove a central limit theorem for this mechanism under the assumption that the initial positions are independent and identically distributed and the weights proportional to a positive function of the positions such that the image of their common distribution by this function has a non zero component absolutely continuous with respect to the Lebesgue measure. This result relies on the convergence in distribution of the fractional part of partial sums of the normalized weights to some random variable uniformly distributed on [0,1], which is established in a companion paper by overcoming the difficulty raised by the coupling through the normalization. Under the conjecture that a similar convergence in distribution remains valid at the next steps of a particle filter which alternates selections according to the stratified resampling mechanism and mutations according to Markov kernels, we provide an inductive formula for the asymptotic variance of the resampled population after n steps. We perform numerical experiments which support the validity of this formula.

**Mercredi 25 octobre (15h-16h): Mohamed Gharafi (CMAP)**

**Algorithmes mono-objectifs assistés par modèles de substitution pour résoudre des problèmes d'optimisation multi-objectifs**

**Algorithmes mono-objectifs assistés par modèles de substitution pour résoudre des problèmes d'optimisation multi-objectifs**

Multiobjective optimization (MO) problems involve balancing multiple, often conflicting objectives, resulting in diverse, incomparable solutions. Solving these problems typically demands costly algorithms due to the challenge of approximating the Pareto set, which may be infinite. Especially when dealing with expensive black-box functions, minimizing function evaluations is crucial.

Sofomore is a framework that leverages existing high-performance single-objective algorithms, such as CMA-ES, to solve MO problem. By construction, It allows easily the extension of single-objective algorithms to costly setups through surrogate-assisted single-objective variants. In this presentation, we propose novel algorithms within the Sofomore framework, tailored for MO problems with expensive functions.

**Mercredi 11 octobre (15h-16h): Songbo Wang (CMAP)**

**Uniform-in-time propagation of chaos for mean field Langevin dynamics**

**Uniform-in-time propagation of chaos for mean field Langevin dynamics**

We study the mean field Langevin (MFL) dynamics and the associated

-particle system under the functional convexity of the energy. We establish the particle system’s uniform-in- exponential convergence, and, by combining this with standard finite-time propagation of chaos, we obtain a uniform-in-time propagation of chaos result. If time permits, I will also talk about -convergence of MFL and hypocoercive systems.