The equivalence of optimal market gain and minimax regret universal portfolios

Suppose one is given a set of joint distributions {p_θ (x)} over stock market outcomes x∈R^m. We ask what actions θ should an investor take, and with what frequency, to maximize the investor´s advantage over the market. In another question we ask for the universal portfolio minimizing the maximum regret in the growth rate of wealth. We argue, in parallel with Gallager´s (1968) proof for the equivalence of channel capacity and minimum redundancy in data compression, that both problems have the same answer min_b max_θ ∫p_θlnb_θ^tx/b^t x