Probabilities and Statistic seminar

We introduce a unified stochastic framework for modeling multiscale financial volatility based on the Log Stationary Fractional Brownian Motion (Log S-fBM) model. This construction provides a continuous interpolation between multifractal volatility regimes and rough volatility dynamics, thereby capturing key empirical features observed in financial time series. We develop a statistically robust Generalized Method of Moments (GMM) estimation procedure within the small intermittency regime. Empirical findings indicate that market indices exhibit pronounced roughness, whereas individual assets display dynamics closer to the multifractal limit which is reproduced by the Nester Stationary fractional Factor model we proposed. The framework of the Log S-fBM is further extended to a multivariate setting, enabling the joint modeling of correlated assets through a multidimensional Log S-fBM structure. This extension preserves marginal properties while incorporating cross-asset dependencies, providing a coherent explanation for the observed discrepancy between index-level and single-asset volatility behavior. In addition, we propose an efficient simulation methodology for Volterra-type processes based on Random Fourier Features (RFF) approximations of the kernel with a particular focus on the S-fBM kernel. This approach yields improved numerical stability and computational efficiency, supported by theoretical error bounds and empirical validation. Overall, the proposed framework offers a consistent and tractable approach to linking rough volatility, multifractal scaling, and factor-based structures, with both theoretical and practical implications for financial modeling.

In this talk, we discuss the question of Gibbs point processes in R^d with pairwise interactions that are not integrable at infinity. A standard example is the Riesz potential of the form g(x)=1/|x|^s where s<d. This setting has a long history, notably because the case s=d-2 corresponds to the classical Coulomb potential, which arises from electrostatic theory. We will first address the existence of the process in the infinite volume regime when a neutralizing background is introduced (this model is known as Jellium in theoretical physics). Subsequently, we will discuss the rigidity of such point processes, specifically hyper-uniformity and number rigidity. We will provide a state-of-the-art review and present numerous conjectures and open problems.

We define a wide class of Markovian load balancing queueing networks, including classic networks studied in the lively literature on the subject. Each network has identical single-server infinite-buffer queues and implements a load balancing policy to allocate each task at its arrival and possibly reallocate it at service completions. The purpose of the policy is to optimize server utilization under constraints such as limited information, real-time decision taking, and network topology. The queue length process is not necessarily exchangeable. The invariant law is in general not known even up to normalizing constant. We provide perfect simulation methods in view of Monte Carlo estimation of quantities of interest in equilibrium, for instance for performance evaluation. In this infinite multi-dimensional state space, we use an unusual preorder defining an order up to permutation of the coordinates, define a coupling in which networks in this class are dominated by the network with uniform routing, and implement dominated coupling from the past methods.

[The talk will be in French, but slides will be in English.]

Planning, titres et résumés ici.

Chemical reactions network inside cells have been extensively studied in order to better understand various biological phenomena. The majority of experimental studies are performed with cells that are part of a growing population. This population context is rarely taken into account even if selection between cells (due for example to growth) takes places within the studied system.
In this talk, I will represent such systems as continuous-time Markov chains. The measure-valued Markov process of the cell population will take into account the chemical reactions inside the cells as well as reactions between cells. By conditioning on non-absorption, we derive an equation for the expected population distribution within a growing population.
This extension of the Chemical Master Equation provides us a new framework to study cell population dynamics. I will present theoretical results on long-term behaviour of the population (stationary distribution, growth rate of the population) and an application of this framework to experimental data.

The increasing availability of temporal and geo-coded data underscores the importance of spatio-temporal statistical modelling in tackling complex issues across various real-world settings. In the first part of this talk, we will briefly showcase novel spatio-temporal statistical methods developed to model various types of data defined both in space and time (e.g., time-series, point patterns, lattice data, geostatistical data, etc.), with a focus on applications in environmental and public health domains. In the second part, we will (I) delve into the modelling of trajectory (or path) data and (II) explore the details of a statistical method for addressing spatially varying preferential sampling when modelling geostatistical data. Specifically, we will account for preferential sampling by including a spatially varying coefficient that describes the dependence strength between the process that models the sampling locations and the corresponding latent field. We achieve this by approximating the preferentiality component with a set of basis functions, with the corresponding coefficients estimated using the integrated nested Laplace approximation (INLA) method. This approach allows for efficient inference with a low computation burden.

The push-forward operation enables one to redistribute a probability measure through a deterministic map. It plays a key role in statistics and optimization: many learning problems (notably from optimal transport, generative modeling, and algorithmic fairness) include constraints or penalties framed as push-forward conditions on the model. However, the literature lacks general theoretical insights on the (non)convexity of such constraints and its consequences on the associated learning problems. The presented work aims at filling this gap. In a first part, we provide a range of sufficient and necessary conditions for the (non)convexity of two sets of functions: the maps transporting one probability measure to another; the maps inducing equal output distributions across distinct probability measures. This highlights that for most probability measures, these push-forward constraints are not convex. In a second time, we show how this result implies critical limitations on the design of convex optimization problems for learning generative models or group-fair predictors. This work will hopefully help researchers and practitioners have a better understanding of the critical impact of push-forward conditions onto convexity.