Log-Optimal Portfolio-Selection Strategies with Proportional Transaction Costs

Abstract

Discrete time growth optimal investment in stock markets with proportional transactions costs is considered. The market process is a sequence of daily relative prices (called returns), and it is modelled by a first order Markov process. Assuming that the distribution of the market process is known, we show sequential investment strategies such that, in the long run, the growth rate on trajectories achieves the maximum with probability 1. Investment with consumption and with fixed transaction cost where the cost depends on the number of the shares involved in the transaction is also analyzed. IntroductionThe purpose of this chapter is to investigate sequential investment strategies for financial markets such that the strategies are allowed to use information collected from the past of the market and determine, at the beginning of a trading period, a portfolio, that is, a way to distribute their current capital among the available assets. The goal of the investor is to maximize his wealth on the long run. If there is no transaction cost then the only assumption used in the mathematical analysis is that the daily price relatives form a stationary and ergodic process. Under this assumption the best strategy 117