From rank-based models with common noise to pathwise entropy solutions of SPDEs

Paper: arXiv 2406.07286

Authors: Mykhaylo Shkolnikov, Lane Chun Yeung

Abstract

We study the mean field limit of a rank-based model with common noise, which arises as an extension to models for the market capitalization of firms in stochastic portfolio theory. We show that, under certain conditions on the drift and diffusion coefficients, the empirical cumulative distribution function converges to the solution of a stochastic PDE. A key step in the proof, which is of independent interest, is to show that any solution to an associated martingale problem is also a pathwise entropy solution to the stochastic PDE, a notion introduced in a recent series of papers [32, 33, 19, 16, 17].

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 1.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper presents a rigorous theoretical treatment of rank-based models with common noise in stochastic portfolio theory using advanced SPDE techniques. While mathematically complex and novel in establishing pathwise entropy solutions for martingale problems, it is entirely theoretical without empirical backtests or market data analysis.

Research Flowchart

  flowchart TD
    A[Research Goal: Derive Mean Field Limit of Rank-Based Model with Common Noise] --> B[Inputs: Stochastic Portfolio Theory, Empirical CDF & Interacting Particle System]
    B --> C[Formulate Associated Martingale Problem]
    C --> D[Prove Solution Uniqueness via Pathwise Entropy Solutions Framework]
    D --> E[Key Outcome: Convergence of Empirical CDF to Unique Solution of SPDE]