Empirical Log-Optimal Portfolio Selections: A Survey

This chapter provides a survey of discrete-time, multi-period, sequential investment strategies for financial markets. With memoryless assumption about the underlying process generating the asset prices, the best rebalancing is the log-optimal portfolio, which achieves the maximum asymptotic average growth rate. We show some examples (Kelly game, horse racing, St. Petersburg game) illustrating the surprising possibilities for rebalancing. Semi-log-optimal portfolio selection, as a small computational-complexity alternative of the log-optimal portfolio selection, is studied, both theoretically and empirically. For generalized dynamic portfolio selection, when asset prices are generated by a stationary and ergodic process, universally-consistent empirical methods are demonstrated. The empirical performance of the methods is illustrated for NYSE data.