TY - JOUR

T1 - Spde limit of the global fluctuations in rank-based models

AU - Kolli, Praveen

AU - Shkolnikov, Mykhaylo

N1 - Funding Information:
Received August 2016; revised May 2017. 1Supported in part by NSF Grant DMS-1506290. MSC2010 subject classifications. 60H15, 82C22, 91G80. Key words and phrases. 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.

PY - 2018/3/1

Y1 - 2018/3/1

N2 - 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.

AB - 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.

KW - Central limit theorems

KW - Fluctuations in interacting particle systems

KW - Gaussian random fields

KW - Large equity markets

KW - Mean field interaction

KW - Porous medium equation

KW - Quantitative propagation of chaos estimates

KW - Rank-based models

KW - Stochastic partial differential equations

KW - Stochastic portfolio theory

KW - Wasserstein distances

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U2 - 10.1214/17-AOP1200

DO - 10.1214/17-AOP1200

M3 - Article

AN - SCOPUS:85043372791

VL - 46

SP - 1042

EP - 1069

JO - Annals of Probability

JF - Annals of Probability

SN - 0091-1798

IS - 2

ER -