SPDE limit of the global fluctuations in rank-based models

TL;DR

This paper demonstrates that in large rank-based diffusion models, the fluctuations of empirical distributions converge to a linear SPDE, linking particle system behavior to a porous medium PDE and enabling market analysis.

Contribution

It establishes the SPDE limit for fluctuations in rank-based models and connects it to the hydrodynamic limit described by the porous medium PDE, providing new analytical tools.

Findings

Fluctuations converge to a linear parabolic SPDE with additive noise.

Quantitative propagation of chaos estimates are derived.

The results facilitate analysis of large equity markets.

Abstract

We consider systems of diffusion processes ("particles") interacting through their ranks (also referred to as "rank-based models" in the mathematical finance literature). We show that, as the number of particles becomes large, the process of fluctuations of the empirical cumulative distribution functions converges to the solution of a linear parabolic SPDE with additive noise. The coefficients in the limiting SPDE are determined by the hydrodynamic limit of the particle system which, in turn, can be described by the porous medium PDE. The result opens the door to a thorough investigation of large equity markets and investment therein. In the course of the proof we also derive quantitative propagation of chaos estimates for the particle system.