Weighted entropy and optimal portfolios for risk-averse Kelly investments

TL;DR

This paper investigates weighted entropy-based log-optimal portfolios in discrete-time models, revealing conditions for supermartingale growth and showing that proportional betting is optimal under risk-averse constraints.

Contribution

It introduces a framework for weighted entropy in portfolio optimization, extending Kelly betting to risk-averse scenarios with new properties and examples.

Findings

Logarithmic growth rate can form a supermartingale under certain conditions.

Optimal strategy is proportional betting, respecting capital and loss constraints.

Extends Kelly betting scheme with weighted outcomes and risk considerations.

Abstract

Following a series of works on capital growth investment, we analyse log-optimal portfolios where the return evaluation includes `weights' of different outcomes. The results are twofold: (A) under certain conditions, the logarithmic growth rate leads to a supermartingale, and (B) the optimal (martingale) investment strategy is a proportional betting. We focus on properties of the optimal portfolios and discuss a number of simple examples extending the well-known Kelly betting scheme. An important restriction is that the investment does not exceed the current capital value and allows the trader to cover the worst possible losses. The paper deals with a class of discrete-time models. A continuous-time extension is a topic of an ongoing study.