Thomas Cover
The natural mathematics of growth optimality
Optimizing the growth rate of investment is considered by some
to be a controversial investment goal, perhaps because it is an
asymptotic criterion or perhaps because its implementation requires
maximizing the expected logarithm of wealth together with 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. 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. Many
of these properties have their counterparts in similar expressions in
information theory.