Thomas Cover - The Natural Mathematics Arising in Information Theory and Investment
Monday, October 20, 2008
Optimizing the growth rate of investment is considered a controversial investment goal, perhaps because it is an asymptotic criterion or perhaps because its implementation requires maximizing the expected logarithm of wealth and its implicit suggestion of log utility. Whatever the reason, we shall reverse the argument by focusing on the natural mathematics of the solution rather than the appropriateness of the question. Maybe graceful mathematics is an indication of the right approach.
We find that growth optimality is characterized by expected ratio optimality, by competitive one-shot optimality, by Martingale processes and an associated asymptotic equipartition theorem. It also yields Black Scholes option pricing as a special case and leads naturally to so called universal portfolios that perform as well to first order in the exponent as the best constant rebalanced portfolio in hindsight. Finally we will relate the quantities arising in investment to their counterpart quantities in information theory. 1