Séminaire
Séminaire Analyse-Probabilités
REBJOCK Quentin (EPFL)
Le 12/11/2024
De 10:30 à 11:30
Titre : Quadratic Łojasiewicz loss landscapes and application to trust-region algorithms
Résumé : Classical optimization algorithms can see their local convergence rates deteriorate when the Hessian at the optimum is singular. The latter is inescapable when the optima are non-isolated. Yet, several algorithms behave perfectly nicely even when optima form a continuum (e.g., due to overparameterization). This has been explained under various structural assumptions, including the quadratic Łojasiewicz inequality. I will present its relationships to other standard conditions in the optimization literature, and the strong properties it implies on the landscape of the cost function. Then, I will discuss applications to trust-region methods, for which we can secure local super-linear rates.
Salle : A711
Séminaire
Séminaire des jeunes chercheurs
OMRANI Amal (CEREMADE)
Le 14/11/2024
De 17:00 à 18:00
Titre : Discrete-Time Asset Pricing: From No-Arbitrage to Absence of Instantaneous Profit.
Résumé : Giving a fair price for a financial asset is a central question in economics and finance. The selling price should be sufficient to establish a hedging strategy for a given asset. In a discrete-time framework, this problem is traditionally solved under the No-Arbitrage (NA) assumption, which assumes rational market participants with uniform access to information. NA is equivalent to the existence of at least one risk-neutral probability measure under which the asset’s price process becomes a martingale, facilitating the calculation of the super-hedging price. However, this assumption can be questionable, as it relies on idealized rationality and information uniformity. Additionally, identifying risk-neutral measures or determining the distribution of the asset’s payoff can be computationally challenging. To address these limitations, a weaker assumption known as the Absence of Instantaneous Profit (AIP) has been proposed. AIP requires only that prices remain finite, which is more realistic and flexible in many settings. In 2022, Emmanuel Lépinette and Laurence Carassus introduced the AIP condition and proposed a dynamic pricing approach within a discrete, frictionless framework using convex duality rather than martingale measures. In this talk, I will discuss the transition from NA to AIP and present the mathematical resolution under each assumption.
Salle : A707
Séminaire
Séminaire Analyse-Probabilités
BROFFERIO Sara (Université Paris Est Créteil)
Le 19/11/2024
De 10:30 à 11:30
Titre : Local stability for affine recursion in higher dimensions
Résumé : Consider a stochastic dynamical system that is a Markov process $X_n^x$ on $\mathbb{R}^d$ obtained by applying some i.i.d. random transformations $F_n$ to a starting point $x \in \mathbb{R}^d$: X_n^x = F_n(X_{n-1}^x) = F_n \cdots F_1(x), \quad X_0^x = x. A crucial question is whether such systems are sensitive to perturbations of the starting point $x$, in particular, whether the distance between two trajectories $X_n^x$ and $X_n^y$ starting from two different points $x \neq y$ converges to zero. We are particularly interested in systems that are not necessarily globally stable but exhibit local stability, meaning that the distance $|X_n^x - X_n^y|$ converges to zero when the processes are observed within a compact window $K \subset \mathbb{R}^d$: |X_n^x - X_n^y| 1_K(X_n^x) \to 0 \text{ as } n \to +\infty. While such phenomena have been highlighted for several examples of stochastic dynamical systems in dimension $d = 1$ in critical cases between contraction and dilation, they are less understood in higher dimensions. The goal of this talk is to present some results on the local stability properties for the random affine recursion $X_n^x$ induced by the action of the random walk on the (semi)group of affine transformations of the Euclidean space $\mathbb{R}^d$, which is given by X_n^x = A_n X_{n-1}^x + B_n, where $(A_n, B_n) \in M(d) \times \mathbb{R}^d$ is an i.i.d. sequence in the critical case where the Lyapunov exponent of the matrices is zero.
Salle : A711
Séminaire
Séminaire Analyse-Probabilités
ERTZBISCHOFF Lucas (Ceremade - Université Paris Dauphine PSL)
Le 26/11/2024
De 10:30 à 11:30
Titre : A journey into the realm of stratified fluids
Résumé : I will propose a gentle introduction to the mathematical analysis of inviscid stratified fluids, of importance in oceanography. The focus will be on the so-called Euler-Boussinesq system, that is a widely used toy model. I will mainly discuss some of the challenges arising in two asymptotic regimes for these equations: the long-time behaviour and the hydrostatic limit. I will try to highlight some links with dispersive equations, kinetic theory, spectral theory, etc...
Salle : A711
Colloque
Colloquium du CEREMADE
BUCUR Dorin (Université de Savoie)
Le 03/12/2024
De 15:30 à 16:30
Titre : A free boundary approach of spectral optimization problems
Résumé : Why all the drums are round! This assertion is mathematically rephrased as follows: among all membranes with prescribed area, the one producing the lowest fundamental frequency has a circular shape. But what about the shapes of membranes with higher extremal frequencies? The general question in which are are interested concerns the relationship between the geometry of the domain and the spectrum of a differential operator defined on this domain. In this talk I will focus on the Neumann eigenvalues of the Laplace operator on domains of Euclidean space and on spheres. Together with a discussion about existence/non existence of optimal geometries and possible relaxation on densities, I will give some numerical approximations of the best geometries and densities. A particular attention will be given to the lowest two non trivial eigenvalues for which a full answer is given in any dimension of the Euclidean space. A surprising phenomenon occurs on spheres: while a complete answer can be given for the second eigenvalue, for the first one an unexplained phenomenon occurs. The results presented in this talk have been obtained in series of collaborations with R. Laugesen, A. Henrot, E. Martinet, M. Nahon and E. Oudet.
Salle : D207
Séminaire
Séminaire Analyse-Probabilités
BORDENAVE Charles (Institut de Mathématiques de Marseille)
Le 10/12/2024
De 10:30 à 11:30
Titre : TBA
Résumé :
Salle : A711