Weighted entropy and optimal portfolios for risk-averse Kelly investments
Following a series of works on capital growth investment, we analyse log-optimal portfolios where
the return evaluation includes ‘weights’ of different outcomes. The results are twofold: (A) under
certain conditions, the logarithmic growth rate leads to a supermartingale, and (B) the optimal
(martingale) investment strategy is a proportional betting. We focus on properties of the optimal
portfolios and discuss a number of simple examples extending the well-known Kelly betting scheme.
An important restriction is that the investment does not exceed the current capital value and
allows the trader to cover the worst possible losses.
The paper covers a class of discrete-time models. A continuous-time extension will be a topic of
a forthcoming study