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

Ce manuscrit étudie l’approximation des solutions des équations différentielles stochastiques progressifs et rétrogrades (EDSPRs) et des applications numériques au marché des émissions de carbone, et l’approximation de couverture faible.

Mots clé: EDSPRs, Algorithme SGD, Deep learning, Grande dimension, Schémas de Splitting, particules stochastiques, Transport optimal.

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.

Abstract: The present thesis deals with stochastic control methods applied to the resolution of problems arising in the field of quantitative finance, such as portfolio selection and resolution of non linear PDEs associated to the construction of investment strategies and the pricing of derivative products. It is divided into three parts.

In the first part, we solve a mean variance portfolio selection problem where the portfolio is penalized by a distance between the wealth invested in each of its assets and the composition of a reference portfolio with fixed weights. The optimal control and value function are obtained in closed form and an analogue of the efficient frontier formula is obtained in the limit where the penalisation tends to zero. The robustness of this allocation is tested on simulated market prices with parameter misspecification.

The second part deals with the delayed control of stochastic differential equations. We solve a simple linear quadratic stochastic control problem where the control appears both in the drift and diffusion part of the state SDE and is affected by a delay. The expressions of the optimal control and value function are obtained in terms of the solution of a system of coupled Riccati PDEs for which the existence and uniqueness of a solution is proven, provided that a condition, combining the time horizon, the delay, the drift and the volatility of the state SDE is satisfied. A deep learning method is used to solve the system Riccati PDEs in the context of Markovitz portfolio selection with execution delay.

In the third part, three methods based on deep learning are defined in order to solve fully non linear PDEs with convex Hamiltonian. These methods use the stochastic representation form of the PDE, whose optimal control is approximated numerically, in order to obtain three different estimators of the PDE solution based on regression or pathwise versions of the martingale representation and its differential relation. The solution and its derivatives are then computed simultaneously. We further leverage our methods to design algorithms for solving families of PDEs with parametric terminal condition by means of DeepOnet neural networks.

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 investigate a model of simple-random walk paths in a random environment that has two competing features: an attractive one towards the highest values of a random potential, and a self-repellent one in the spirit of the well-known weakly self-avoiding random walk. We tune the strength of the second effect such that they both contribute on the same scale as the time variable tends to infinity. In this talk I will discuss our results on the identification of (1) the logarithmic asymptotics of the partition function, and (2) of the path behaviour that gives the overwhelming contribution to the partition function. This is joint work with Wolfgang König, Nicolas Pétrélis and Renato Soares dos Santos.

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.

Résumé: Nous étudions les propriétés ergodiques des horosphères sur certaines classes de variétés sans points conjugués. Notre but est de généraliser plusieurs résultats déjà connus pour les variétés à courbure négative. Nous prouvons que, pour une grande classe de variétés de rang 1 à courbure négative ou nulle, certaines horosphères sont équidistribuées sous l'action du flot géodésique vers la mesure de Bowen-Margulis. Dans les surfaces à courbure négative ou nulle, on définit un flot horocyclique sur l'ensemble des horocycles contenant un vecteur de rang 1 récurrent sous l'action du flot géodésique et nous montrons que ce flot horocyclique a une unique mesure de probabilité invariante. Enfin, nous montrons que tout flot horocyclique sur une surface compacte de genre supérieur sans points conjugués et avec fibrés de Green continus est uniquement ergodique. Notre approche s'appuie sur des méthodes spécifiques aux flots géodésiques telles que le bord à l'infini et la construction de la mesure de Bowen-Margulis via la théorie de Patterson-Sullivan. L'ingrédient principal du théorème d'équidistribution est le mélange de la mesure de Bowen-Margulis. Concernant les flots horocycliques, nos résultats sont obtenus grâce à la définition d'une paramétrisation similaire à celle utilisée par B. Marcus en courbure négative.

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Abstract: We study the ergodic properties of the horospheres on certain classes of manifolds without conjugate points. Our goal is to generalize several results already known for negatively curved manifolds. We prove that, for a large class of nonpositively curved rank 1 manifolds, certain horospheres are equidistributed under the action of the geodesic flow towards the Bowen-Margulis measure. In the case of nonflat nonpositively curved surfaces, we define a horocyclic flow on the set of horocycles containing a rank 1 vector that is recurrent under the action of the geodesic flow and we prove that this horocyclic flow has a unique invariant probability measure. Finally, we show that any horocyclic flow on a compact higher genus surface without conjugate points and with continuous Green bundles is uniquely ergodic. Our approach is based on methods specific to geodesic flows such as the boundary at infinity and the construction of the Bowen-Margulis measure via the Patterson-Sullivan theory. The main ingredient in the equidistribution theorem is the mixing of the Bowen-Margulis measure. Regarding the horocyclic flows, our results are obtained thanks to the definition of a uniformly expanding parametrization similar to the one used by B. Marcus in negative curvature.