Convergence rates for rank-based models with applications to portfolio theory

TL;DR

This paper establishes convergence rates for rank-based stochastic models and applies these results to analyze fluctuations and performance metrics in equity markets within stochastic portfolio theory.

Contribution

It provides explicit convergence rates for rank-based diffusions and Brownian motions, along with quantitative bounds for market weight fluctuations and portfolio performance comparisons.

Findings

Derived convergence rates using Lyapunov functions.

Obtained bounds for fluctuations of market weights and occupation times.

Compared performance of different portfolio strategies.

Abstract

We determine rates of convergence of rank-based interacting diffusions and semimartingale reflecting Brownian motions to equilibrium. Convergence rate for the total variation metric is derived using Lyapunov functions. Sharp fluctuations of additive functionals are obtained using Transportation Cost-Information inequalities for Markov processes. We work out various applications to the rank-based abstract equity markets used in Stochastic Portfolio Theory. For example, we produce quantitative bounds, including constants, for fluctuations of market weights and occupation times of various ranks for individual coordinates. Another important application is the comparison of performance between symmetric functionally generated portfolios and the market portfolio. This produces estimates of probabilities of "beating the market".