  We revisit Merton's portfolio optimization problem under boun-ded
state-dependent utility functions, in a market driven by a L\'evy process $Z$
extending results by Karatzas et. al. (1991) and Kunita (2003). The problem is
solved using a dual variational problem as it is customarily done for
non-Markovian models. One of the main features here is that the domain of the
dual problem enjoys an explicit "parametrization", built on a multiplicative
optional decomposition for nonnegative supermartingales due to F\"ollmer and
Kramkov (1997). As a key step in obtaining the representation result we prove a
closure property for integrals with respect to Poisson random measures, a
result of interest on its own that extends the analog property for integrals
with respect to a fixed semimartingale due to M\'emin (1980). In the case that
(i) the L\'evy measure of $Z$ is atomic with a finite number of atoms or that
(ii) $\Delta S_{t}/S_{t^{-}}=\zeta_{t} \vartheta(\Delta Z_{t})$ for a process
$\zeta$ and a deterministic function $\vartheta$, we explicitly characterize
the admissible trading strategies and show that the dual solution is a
risk-neutral local martingale.
