SPDE LIMIT OF THE GLOBAL FLUCTUATIONS IN RANK-BASED MODELS

• 成果类型: Article
• 署名作者: Kolli, Praveen; Shkolnikov, Mykhaylo
• 署名单位: Carnegie Mellon University; Princeton University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/17-AOP1200
• 发表日期: 2018
• 页码: 1042-1069
• 关键词: large numbers; DIFFUSIONS; propagation; DYNAMICS; systems; LAW

摘要

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.