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.
Publisher Copyright:
© Institute of Mathematical Statistics, 2018.
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
SN - 0091-1798
VL - 46
SP - 1042
EP - 1069
JO - Annals of Probability
JF - Annals of Probability
IS - 2
ER -