Large Rank-Based Models with Common Noise

TL;DR

This paper studies large systems of rank-based Brownian particles with common noise, showing that their empirical distribution converges to a solution of a stochastic PDE related to porous medium equations, extending previous deterministic results.

Contribution

It introduces a stochastic PDE framework for rank-based particle systems with common noise, generalizing deterministic PDE models to stochastic settings.

Findings

Empirical distribution converges to a stochastic PDE solution.

The stochastic PDE resembles conservation laws with rough stochastic fluxes.

Extension of porous medium PDE results to stochastic environments.

Abstract

For large systems of Brownian particles interacting through their ranks introduced in (Banner, Fernholz, Karatzas, 2005), the empirical cumulative distribution function satisfies a porous medium PDE. However, when we introduce a common noise, the limit is no longer deterministic. Instead, we show that this limit is a solution of a stochastic PDE related to this porous medium PDE. This stochastic PDE is somewhat similar to the equations developed for conservation laws with rough stochastic fluxes (Lions, Perthame, Souganidis, 2013).