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Cover's celebrated theorem states that the long run yield of a properlychosen "universal" portfolio is as good as the long run yield of the bestretrospectively chosen constant rebalanced portfolio. The "universality"pertains to the fact that this result is model-free, i.e., not dependent on anunderlying stochastic process. We extend Cover's theorem to the setting ofstochastic portfolio theory as initiated by R. Fernholz: the rebalancing ruleneed not to be constant anymore but may depend on the present state of thestock market. This model-free result is complemented by a comparison with thelog-optimal numeraire portfolio when fixing a stochastic model of the stockmarket. Roughly speaking, under appropriate assumptions, the optimal long runyield coincides for the three approaches mentioned in the title of this paper.We present our results in discrete and continuous time.

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