Strong solutions of stochastic equations with rank-based coefficients

TL;DR

This paper investigates stochastic differential equations where coefficients depend on particle ranks, establishing strong solutions and collision conditions, with improvements for finite and infinite particle systems.

Contribution

It provides new conditions for the existence, uniqueness, and collision avoidance in rank-based stochastic systems, including the first for countably infinite particles.

Findings

Strong solutions exist until three particles collide.

Improved conditions prevent triple collisions in finite systems.

First collision avoidance condition established for infinite systems.

Abstract

We study finite and countably infinite systems of stochastic differential equations, in which the drift and diffusion coefficients of each component (particle) are determined by its rank in the vector of all components of the solution. We show that strong existence and uniqueness hold until the first time three particles collide. Motivated by this result, we improve significantly the existing conditions for the absence of such triple collisions in the case of finite-dimensional systems, and provide the first condition of this type for systems with a countable infinity of particles.