Abstract
We investigate how much an investor can improve his investment
using side information without revealing to a warden observing his
wealth each day that he has access to this side information.
Specifically, we require the relative entropy between the
distributions of wealth induced by the investor's portfolio
strategy and the log-optimal portfolio to be small. We formulate
this constraint with three different degrees of strictness.
We show that the improvement in the wealth growth rate can at most
scale with the square root of the number of days n spent investing
in the stock market. If we only require 1/n times the relative
entropy to be small, it scales almost with n. We also provide the
exact scaling factors under the three constraints, as well as when
we change some of our assumptions. We consider the case where the
warden cannot observe the investor's wealth each day but every k
days and the case where his observations each day are noisy.
Furthermore, we incorporate a model from the game theory
literature to investigate the case where the investor is not fully
rational and occasionally makes errors.
Finally, we focus on price relatives that can assume a finite
number of values. We show that if a given portfolio strategy that
uses side information perfectly characterizing the stock market
induces a certain wealth distribution, there are conditions under
which a strategy utilizing imperfect side information can achieve
the same distribution.
-||- _|_ _|_ / __|__ Stefan M. Moser 
[-] --__|__ /__\ /__ Senior Scientist, ETH Zurich, Switzerland
_|_ -- --|- _ / / Adjunct Professor, National Yang Ming Chiao Tung University, Taiwan
/ \ [] \| |_| / \/ Web: https://moser-isi.ethz.ch/
Last modified: Mon Nov 10 15:19:51 UTC 2025