From rank-based models with common noise to pathwise entropy solutions of SPDEs

TL;DR

This paper investigates the mean field limit of rank-based models with common noise, demonstrating convergence to stochastic PDE solutions and establishing a connection with pathwise entropy solutions.

Contribution

It introduces a novel link between martingale problem solutions and pathwise entropy solutions for stochastic PDEs in the context of rank-based models.

Findings

Empirical distribution functions converge to stochastic PDE solutions.

Established the equivalence of martingale solutions and pathwise entropy solutions.

Extended the theoretical framework for models in stochastic portfolio theory.

Abstract

We study the mean field limit of a rank-based model with common noise, which arises as an extension to models for the market capitalization of firms in stochastic portfolio theory. We show that, under certain conditions on the drift and diffusion coefficients, the empirical cumulative distribution function converges to the solution of a stochastic PDE. A key step in the proof, which is of independent interest, is to show that any solution to an associated martingale problem is also a pathwise entropy solution to the stochastic PDE, a notion introduced in a recent series of papers [32, 33, 19, 16, 17].