Universal portfolios with side information

We present a sequential investment algorithm, the μ-weighted universal portfolio with side information, which achieves, to first order in the exponent, the same wealth as the best side-information dependent investment strategy (the best state-constant rebalanced portfolio) determined in hindsight from observed market and side-information outcomes. This is an individual sequence result which shows the difference between the exponential growth wealth of the best state-constant rebalanced portfolio and the universal portfolio with side information is uniformly less than (d/(2n))log (n+1)+(k/n)log 2 for every stock market and side-information sequence and for all time n. Here d=k(m-1) is the number of degrees of freedom in the state-constant rebalanced portfolio with k states of side information and m stocks. The proof of this result establishes a close connection between universal investment and universal data compression