  We consider the problem of utility maximization for small traders on
incomplete financial markets. As opposed to most of the papers dealing with
this subject, the investors' trading strategies we allow underly constraints
described by closed, but not necessarily convex, sets. The final wealths
obtained by trading under these constraints are identified as stochastic
processes which usually are supermartingales, and even martingales for
particular strategies. These strategies are seen to be optimal, and the
corresponding value functions determined simply by the initial values of the
supermartingales. We separately treat the cases of exponential, power and
logarithmic utility.
