This paper is in progress. The summary below previews what it will cover; the working paper and a less-technical write-up will be published here when ready.
Log returns are analytically convenient. They’re also the wrong units for the decision an investor is actually making.
Optimization in log-return space tells the investor how to allocate capital across the log-return process — but investors allocate by purchasing shares of securities with cash, in quantities denominated in gross returns. This paper traces the practical consequences of that mismatch for the well-known growth-optimal allocation rule, and derives a closed-form mapping from optimal log-return allocations to the gross-return allocations an investor can actually implement. The result reconciles the textbook log-return formulation with what practitioners actually trade.