  We consider the terminal wealth utility maximization problem from the point
of view of a portfolio manager who is paid by an incentive scheme, which is
given as a convex function $g$ of the terminal wealth. The manager's own
utility function $U$ is assumed to be smooth and strictly concave, however the
resulting utility function $U \circ g$ fails to be concave. As a consequence,
the problem considered here does not fit into the classical portfolio
optimization theory. Using duality theory, we prove wealth-independent
existence and uniqueness of the optimal portfolio in general (incomplete)
semimartingale markets as long as the unique optimizer of the dual problem has
a continuous law. In many cases, this existence and uniqueness result is
independent of the incentive scheme and depends only on the structure of the
set of equivalent local martingale measures. As examples, we discuss (complete)
one-dimensional models as well as (incomplete) lognormal mixture and popular
stochastic volatility models. We also provide a detailed analysis of the case
where the unique optimizer of the dual problem does not have a continuous law,
leading to optimization problems whose solvability by duality methods depends
on the initial wealth of the investor.
