Laboratoire de Mathématiques Raphaël Salem

UMR 6085 CNRS - Université de Rouen Normandie

La conjecture de Sarnak dit que tout système dynamique déterministe $(X,T)$ est disjoint (au sens arithmétique) de la fonction de Möbius $\mu$: $$\lim_{N\to\infty}\frac1N\sum_{n\leq N}f(T^nx)\mu(n)=0$$ pour toute fonction continue $f$ et tout $x\in X$. Les systèmes rigides sont déterministes, mais la rigidité peut être définie soit de façon topologique, soit métrique en utilisant les systèmes dynamiques métriques $(X,\nu,T)$ où $\nu$ parcurt l'ensemble des mesures $T$-invariantes.

In this talk we study high dimensional ergodic diffusion models in nonparametric setting on the basis of discrete data, when the diffusion coefficients are unknown. For this problem, by using efficient sequential point-wise estimators we construct a model selection procedure and then we show sharp oracle inequalities, i.e. the inequalities in which the main term coefficient is closed to one. This means that the proposed sequential model selection procedure is optimal in this sense.

In this introductory talk, I will present a general limiting
theory for the spectrum of large networks. The models I will consider
are quite general, but they share a common feature : all of them are
studied in their very sparse regime where the number of connections has
the same order as the number of nodes (Erdös-Rényi with fixed mean
degree, regular graphs, uniform trees, uniform triangulations,
preferential attachments). The spectrum of such networks is notoriously

Financial markets are characterized by continuous upward or downward fluctuations in prices, caused by the vast amount of information they receive. A strong price instability historically and cyclically caused strong market collapses that prompted investors to control the risk related to the excessive fluctuation of the prices in order to prevent significant portfolio losses.

We introduce our method $\sigma$-antithetic multilevel Monte Carlo for multi-dimensional stochastic differential equations driven by Brownian motions with drifts. Here $\sigma$ is a specific permutation of order $m$, with $m\in\mathbb{N}\backslash\{0,1\}$. Following the spirit of Giles and Szpruch in [b], we consider the Milstein scheme without Lévy area. Our aim is to prove the stable convergence for the $\sigma$-antithetic multilevel error between two consecutive levels. To do so, we chose the triangular array approach using the limit theorem of Jacod [c].