We study the on-line portfolio and the stochastic portfolio investment algorithms and test them with historical data sets. With regard to the stochastic portfolio we develop an optimal formula to manage the portfolio with daily trading in terms of the weights that are assigned to the different stocks in the portfolio. The implementation of the optimal stochastic portfolio depends on good estimation of the parameters that are in our case drifts and volatilities. We present some procedures to estimate the parameters dynamically. The problem of estimating drifts is inherently very hard as the noise (volatility) overwhelms the drifts. Volatilities are easier to estimate than the drifts and we can take advantage of the unique properties of the Brownian motion process to get pretty good estimates taking into account the decreasing effects of older financial data. Then we apply Karush-Kuhn-Tucker Theorem to get the weights of the optimal stochastic portfolio using the estimators. Finally we compare the results of the stochastic portfolio to that of the on-line portfolio using real stock data that now is widely available. In some cases the results achieved by the stochastic portfolio on real historical data are stunning.
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