Weight functions and log-optimal investment portfolios

TL;DR

This paper extends the analysis of log-optimal investment portfolios by incorporating outcome-dependent weights, revealing that the optimal strategy is proportional betting and independent of the weight function's form under certain conditions.

Contribution

It demonstrates that the optimal investment strategy remains proportional betting regardless of the weight function, under specific conditions, expanding prior results with different assumptions.

Findings

Logarithmic growth rate is a supermartingale under certain conditions.

Optimal strategy is proportional betting, independent of weight function form.

Existence of optimal strategy aligns with previous general results.

Abstract

Following the paper by Algoet--Cover (1988), we analyse log-optimal portfolios where return evaluation includes `weights' of different outcomes. The results are twofold: (A) under certain conditions, logarithmic growth rate is a supermartingale, and (B) the optimal (martingale) investment strategy is a proportional betting; it does not depend on the form of the weight function, although the optimal rate does. The existence of an optimal investment strategy has been established earlier in a great generality by Kramkov--Schachermayer (2003) although our underlying assumptions are different.