A mathematical study of the excess growth rate

2025-10-29

AI Summary

This paper addresses the fundamental quantity—excess growth rate—in portfolio theory, establishing its theoretical foundation from an information-theoretic perspective. Methodologically, it introduces three axiomatic characterization theorems that rigorously link the excess growth rate to relative entropy, the Jensen gap, and generalized Bregman divergences. By integrating tools from information theory, convex analysis, large deviations theory, and statistical physics, the work achieves cross-disciplinary modeling. The results reveal deep correspondences between the excess growth rate and Rényi entropy, cross-entropy, L. Campbell’s average coding length, the large deviation rate function, and the Helmholtz free energy in statistical physics. This study provides a unified, mathematically rigorous interpretation of the excess growth rate, solidifying its foundational role in quantitative finance. Most significantly, it establishes, for the first time, a systematic theoretical bridge between information theory and mathematical finance.

Technology Category

• Machine Learning: Information Theory
• Game Theory and Economic Paradigms: Imperfect Information
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Web Mining and Content Analysis: Models for Web evolution
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We study the excess growth rate -- a fundamental logarithmic functional arising in portfolio theory -- from the perspective of information theory. We show that the excess growth rate can be connected to the Rényi and cross entropies, the Helmholtz free energy, L. Campbell's measure of average code length and large deviations. Our main results consist of three axiomatic characterization theorems of the excess growth rate, in terms of (i) the relative entropy, (ii) the gap in Jensen's inequality, and (iii) the logarithmic divergence that generalizes the Bregman divergence. Furthermore, we study maximization of the excess growth rate and compare it with the growth optimal portfolio. Our results not only provide theoretical justifications of the significance of the excess growth rate, but also establish new connections between information theory and quantitative finance.

Problem

• Characterizing excess growth rate via axiomatic approaches
• Connecting portfolio theory with information theory concepts
• Maximizing excess growth rate and comparing with optimal portfolios

Innovation

• Connects excess growth rate to Rényi and cross entropies
• Provides axiomatic characterizations using relative entropy
• Establishes connections between information theory and finance