Long time behaviour and mean-field limit of Atlas models

Abstract

Abstract. This article reviews a few basic features of systems of one-dimensional diffusions with rankbased characteristics. Such systems arise in particular in the modelling of financial markets, where they go by the name of Atlas models. We mostly describe their long time and large scale behaviour, and lay a particular emphasis on the case of mean-field interactions. We finally present an application of the reviewed results to the modelling of capital distribution in systems with a large number of agents.Résumé. Cet article présente quelques propriétés basiques des systèmes de diffusions en une dimension, interagissant à travers leur rang. De tels systèmes apparaissent en particulier dans la modélisation des marchés financiers, où ils portent le nom de modèles d'Atlas. Nous décrivons principalement leur comportement en temps long et à grande échelle, et nous nous intéressons particulièrement au cas d'interactions de champ moyen. Nous décrivons enfin une application des résultats exposés à la modélisation de la distribution du capital dans des systèmes avec un grand nombre d'agents.

Equilibrium large deviations for mean-field systems with translation invariance

“…The inequality (42) shows that F is bounded from below on P(R d ), which proves the statement (ii) of Lemma 2.10. We may now complete the proof of Lemma 4.1.…”

“…This particle system serves as a model for large equity markets, and is also related to the probabilistic interpretation of nonlinear scalar conservation laws [26,42].…”

“…(8), which ensures that the energy functional W defined by (9) is not identically equal to +∞. It is known [42] that the condition (18) ∀u ∈ (0, 1),…”