Abstract

We investigate by how much an investor can improve his wealth
growth in an IID stock market using side information—without
revealing this fact to a warden observing the investor's wealth
each day. Specifically, we require the relative entropy between
the distributions of wealth induced by the investor's portfolio
and the log-optimal portfolio to be small. We formulate this
constraint with two different degrees of strictness. We show that
the improvement in the wealth growth can at most scale with the
square root of the number of days n spent investing in the stock
market. We further provide a tight upper bound on the improved
wealth growth rate as a function of the maximally permitted
relative entropy; when the latter is zero, no improvement on the
order of n^(1/2) is possible.

-||-   _|_ _|_     /    __|__   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: Wed Mar 4 05:56:41 UTC+0100 2026