The excess growth rate, defined as the gap in Jensen's inequality for the logarithm, is a fundamental functional in portfolio theory. In this paper, we present a mathematical study motivated by information theory. We begin by establishing its properties and showing that it has rich connections with information theoretic concepts such as 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.

翻译：超额增长率，定义为对数函数在詹森不等式中的间隙，是投资组合理论中的一个基本泛函。本文受信息论启发，对其展开数学研究。我们首先建立了该泛函的性质，并揭示其与信息论概念（如亥姆霍兹自由能、L. Campbell的平均码长测度以及大偏差理论）存在丰富的联系。主要结果包括超额增长率的三个公理化特征定理，分别基于：(i) 相对熵、(ii) 詹森不等式的间隙，以及(iii) 推广布雷格曼散度的对数散度。此外，我们研究了超额增长率的最大化问题，并将其与增长最优投资组合进行了比较。本文的研究结果不仅为超额增长率的重要性提供了理论依据，而且在信息论与量化金融之间建立了新的联系。