  The aim of this paper is to investigate the impact of rebalancing frequency
and transaction costs on the log-optimal portfolio, which is a portfolio that
maximizes the expected logarithmic growth rate of an investor's wealth. We
prove that the frequency-dependent log-optimal portfolio problem with costs is
equivalent to a concave program and provide a version of the dominance theorem
with costs to determine when an investor should invest all available funds in a
particular asset. Then, we show that transaction costs may cause a bankruptcy
issue for the frequency-dependent log-optimal portfolio. To address this issue,
we approximate the problem to obtain a quadratic concave program and derive
necessary and sufficient optimality conditions. Additionally, we prove a
version of the two-fund theorem, which states that any convex combination of
two optimal weights from the optimality conditions is still optimal. We test
our proposed methods using both intraday and daily price data. Finally, we
extend our empirical studies to an online trading scenario by implementing a
sliding window approach. This approach enables us to solve a sequence of
concave programs rather than a potentially computational complex stochastic
dynamic programming problem.
