### 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.

Original language | English (US) |
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Pages (from-to) | 1042-1069 |

Number of pages | 28 |

Journal | Annals of Probability |

Volume | 46 |

Issue number | 2 |

DOIs | |

State | Published - Mar 1 2018 |

### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Keywords

- Central limit theorems
- Fluctuations in interacting particle systems
- Gaussian random fields
- 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

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## Cite this

*Annals of Probability*,

*46*(2), 1042-1069. https://doi.org/10.1214/17-AOP1200