  We study optimal investment in a financial market having a finite number of
assets from a signal processing perspective. We investigate how an investor
should distribute capital over these assets and when he should reallocate the
distribution of the funds over these assets to maximize the cumulative wealth
over any investment period. In particular, we introduce a portfolio selection
algorithm that maximizes the expected cumulative wealth in i.i.d. two-asset
discrete-time markets where the market levies proportional transaction costs in
buying and selling stocks. We achieve this using "threshold rebalanced
portfolios", where trading occurs only if the portfolio breaches certain
thresholds. Under the assumption that the relative price sequences have
log-normal distribution from the Black-Scholes model, we evaluate the expected
wealth under proportional transaction costs and find the threshold rebalanced
portfolio that achieves the maximal expected cumulative wealth over any
investment period. Our derivations can be readily extended to markets having
more than two stocks, where these extensions are pointed out in the paper. As
predicted from our derivations, we significantly improve the achieved wealth
over portfolio selection algorithms from the literature on historical data
sets.
