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 R\'{e}nyi 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.

翻译：我们从信息论视角研究超额增长率——投资组合理论中一个基本的对数泛函。研究表明，超额增长率可与Rényi熵、交叉熵、亥姆霍兹自由能、L. Campbell的平均码长度量以及大偏差理论建立联系。主要成果包括基于以下三方面的超额增长率公理化刻画定理：(i)相对熵，(ii) Jensen不等式间隙，(iii) 推广Bregman散度的对数散度。此外，我们探讨了超额增长率最大化问题，并将其与最优增长投资组合进行比较。这些结果不仅为超额增长率的重要性提供了理论依据，更在信息论与量化金融之间建立了新的联系。