  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 deals with a class of discrete-time models. A continuous-time
extension is a topic of an ongoing study.

Probability