The Natural Mathematics that Arise in Investment and Information Theory
Dr. Thomas Cover
Thursday, October 16, 2008
2:15 PM – Salvatori Computer Science Center (SAL-101) Lecture
3:15 PM – Salvatori Computer Science Center (SAL-101) Reception
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.
Thomas Cover, the K.T. Li Professor of Engineering at Stanford, does research in information theory, communication theory and statistics, and is the coauthor of the textbook, Elements of Information Theory. He was Lab Director of the Information Systems Laboratory in Electrical Engineering from 1989 to 1996. He has been the contract statistician for the California State Lottery and a consultant to AT&T Laboratories and IBM. He received the 1990 Claude E. Shannon Award in information theory and has also received the IEEE Neural Network Council’s Pioneer Award in 1993 for his work on the capacity of neural nets. He received the 1997 IEEE Richard M. Hamming medal for contributions to information, communication theory and statistics and is a member of the National Academy of Engineering and the American Academy of Arts and Sciences. He is currently working on network information theory and the interplay between information theory and investment.