## The Annals of Probability

- Ann. Probab.
- Volume 46, Number 2 (2018), 1042-1069.

### SPDE limit of the global fluctuations in rank-based models

Praveen Kolli and Mykhaylo Shkolnikov

#### 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.

#### Article information

**Source**

Ann. Probab., Volume 46, Number 2 (2018), 1042-1069.

**Dates**

Received: August 2016

Revised: May 2017

First available in Project Euclid: 9 March 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1520586275

**Digital Object Identifier**

doi:10.1214/17-AOP1200

**Mathematical Reviews number (MathSciNet)**

MR3773380

**Zentralblatt MATH identifier**

06864079

**Subjects**

Primary: 60H15: Stochastic partial differential equations [See also 35R60] 82C22: Interacting particle systems [See also 60K35] 91G80: Financial applications of other theories (stochastic control, calculus of variations, PDE, SPDE, dynamical systems)

**Keywords**

Central limit theorems Gaussian random fields fluctuations in interacting particle systems large equity markets mean field interaction porous medium equation quantitative propagation of chaos estimates rank-based models stochastic partial differential equations stochastic portfolio theory Wasserstein distances

#### Citation

Kolli, Praveen; Shkolnikov, Mykhaylo. SPDE limit of the global fluctuations in rank-based models. Ann. Probab. 46 (2018), no. 2, 1042--1069. doi:10.1214/17-AOP1200. https://projecteuclid.org/euclid.aop/1520586275