Lorenzo CROISSANT - Diffusion limit control of high-frequency pure-jump processes

Pure jump processes with a large number of jumps per unit of time are frequently encountered in modern industrial systems (e.g. financial markets, online advertising auctions, server scheduling…). These applications require sequential decision making, which is mathematically formalized as a control problem. Unfortunately, the non-local nature of jumps leads the control problem to analytical difficulties and computations infeasible in practice. Leveraging the diffusion limit regime, which avoids these difficulties, we present in this talk a characterization of the convergence to this limit, and some interesting perspectives that it opens.

Grégoire SZYMANSKI - Rough volatility and optimal estimation of the Hurst parameter

The goal of this talk is to present the estimation of the Hurst parameter in rough stochastic volatility model. First, we will recall what rough volatility is and why it is used in finance. In these models, we observe a stochastic diffusion driven by a Brownian motion. The volatility is a hidden Fractional Brownian motion. Moreover, the diffusion is only observed at discrete time. We will explain how this problem relates to the somewhat easier observations of the underlying fractional Brownian motion at discrete time polluted by an additive noise, in the same spirit as [3]. Considering the observations $\eta W^H_{i/n} + \eps^n_i$, we build an estimator i/n i based on quadratic variations with convergence rate $n^{1/(4H+2)} as in [3]. We also prove that this rate is minimax, using an adequate wavelet-based construction of the fractional Brownian motion in the second case, see [4]. Finally, we will discuss how this estimator can be generalised to the multiplicative noise model and to a non-parametric setting as in [1].

References: [1] Chong, C.; Hoffmann, M.; Liu, Y.; Rosenbaum, M.; Szymanski, G. (2022) - Statistical inference for rough volatility: central limit theorems - arXiv:2210.01216 [2] Chong, C.; Hoffmann, M.; Liu, Y.; Rosenbaum, M.; Szymanski, G. (2022) - Statistical inference for rough volatility: Minimax theory - arXiv:2210.01214 [3] Fukasawa, M.; Takabatake, T.; Westphal, R. (2019) - Is Volatility Rough ? - arXiv:1905.04852 [4] Szymanski, G. (2022) - Optimal estimation of the rough Hurst parameter in additive noise - arXiv:2205.13035

Le processus Lambda-Fleming-Viot à “infinité de parents” est un modèle d'expansion de population en espace continu, en 2 dimensions, dans lequel les régions non colonisées sont considérées comme étant remplies d'individus de type “inexistant” (ou fantôme…). Ce modèle peut être vu comme une version en espace continu du modèle de croissance d'Eden et il est doté d'un processus dual qui permet de retracer l'origine “généalogique” d'un échantillon d'individus pris dans la population actuelle. Dans cet exposé, on s'intéressera aux propriétés de croissance de la région occupée par des individus “réels”. A l'aide d'un modèle auxiliaire très simple, on discutera également de la manière dont les fluctuations au front rendent la vitesse d'expansion plus rapide que celle prédite par des estimées de premier moment. Ce travail a été réalisé en collaboration avec Apolline Louvet (Université de Bath).

We describe an efficient algorithm to compute solutions for the general two-player Blotto game on n battlefields with heterogeneous values. While explicit constructions for such solutions have been limited to specific, largely symmetric or homogeneous, setups, this algorithmic resolution covers the most general situation to date: value-asymmetric game with asymmetric budget. The proposed algorithm rests on recent theoretical advances regarding Sinkhorn iterations for matrix and tensor scaling. An important case which had been out of reach of previous attempts is that of heterogeneous but symmetric battlefield values with asymmetric budget. In this case, the Blotto game is constant-sum so optimal solutions exist, and our algorithm samples from an ε-optimal solution in time O~(n2+ε−4), independently of budgets and battlefield values. In the case of asymmetric values where optimal solutions need not exist but Nash equilibria do, our algorithm samples from an ε-Nash equilibrium with similar complexity but where implicit constants depend on various parameters of the game such as battlefield values.

In this talk we introduce a new class of kinetic models, which overcome the standard assumption in kinetic transport theory that collision processes happen instantaneously. On the level of the underlining stochastic processes this results in replacing the jump-process, which are defining the collisions, with continuous stochastic processes.

We investigate a kinetic model with non-instantaneous binary alignment collisions between particles. The collisions are described by a transport process in the joint state space of a pair of particles, where the states of the particles approach their midpoint. For two spatially homogeneous models with deterministic or stochastic collision times existence and uniqueness of solutions, the long time behavior, and the instantaneous limit are considered, where the latter leads to standard kinetic models of Boltzmann type.

Reference: L. Kanzler, C. Schmeiser, V. Tora, Two kinetic models for non-instantaneous binary alignment collisions, arXiv:2203.15711

Lorsque le degré moyen est au moins logarithmique en le nombre de sommets, les valeurs propres extrêmes de matrices de Wigner sparse collent au support de la loi du semi-cercle. Nous montrons que dans ce régime de sparsité, les grandes déviations de la plus grande valeur propre sont dominées par l'émergence d'une clique de taille sous-entropique munie de hauts poids ou bien d'un sommet de haut degré. De façon intéressante, la fonction de taux est discontinue en la valeur typique, ce qui reflète le fait que les déviations sont générées par des perturbations de rang fini. Ceci est un travail en collaboration avec Anirban Basak.

Graph alignment is a generic unsupervised machine learning objective with many applications, including de-anonymization of social network data. In this talk we shall consider alignment of correlated Erdös-Rényi random graphs. We shall describe recent results on information-theoretic limits to feasibility of alignment. We shall also describe polynomial-time algorithms for such alignment and sufficient conditions under which they succeed. These latter results are obtained by analysing a hypothesis testing problem of determining whether two Galton-Watson random trees are correlated or independent. The discrepancy between the informational and computational feasibility conditions for graph alignment suggests that the alignment problem displays a so-called “hard phase”, an intriguing phenomenon that has been observed in several other high-dimensional statistical learning tasks.

We consider a deterministic optimal control problem, in a finite horizon context, and propose deep neural network approximations for Bellman's dynamic programming principle, corresponding also to some first-order Hamilton-Jacobi-Bellman (HJB) equations. This work follows the work of Huré, Pham, Bachouch, and Langrené (SIAM J. Numer. Anal., vol. 59 (1), 2021, pp. 525-557) where algorithms are proposed in a stochastic context. However, we need to develop an entirely new approach in order to deal with the propagation of errors in the deterministic setting, where no diffusion is present in the dynamics. Our analysis gives precise error estimates in an average norm. The algorithms are then illustrated on several academic numerical examples related to front propagations models in the presence of obstacle constraints (modelized by an optimal maximum running cost problem) showing the relevance of the approach for average dimensions (e.g. from $2$ to $8$), in particular in situations where the value functions is non-smooth. This is a joint work with Xavier Warin and Averil Prost.