Cover's universal portfolio, stochastic portfolio theory, and the numeraire portfolio

Cover's celebrated theorem states that the long-run yield of a properly chosen "universal" portfolio is almost as good as that of the best retrospectively chosen constant rebalanced portfolio. The "universality" refers to the fact that this result is model-free, that is, not dependent on an underlying stochastic process. We extend Cover's theorem to the setting of stochastic portfolio theory: the market portfolio is taken as the numeraire, and the rebalancing rule need not be constant anymore but may depend on the current state of the stock market. By fixing a stochastic model of the stock market this model-free result is complemented by a comparison with the numeraire portfolio. Roughly speaking, under appropriate assumptions the asymptotic growth rate coincides for the three approaches mentioned in the title of this paper. We present results in both discrete and continuous time.