State-dependent utility maximization in Lévy markets
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We revisit Merton's portfolio optimization problem under boun-ded state-dependent utility functions, in a market driven by a Lévy 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 Föllmer 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 Mémin (1980). In the case that (i) the Lévy measure of $Z$ is atomic with a finite number of atoms or that (ii) $ΔS_{t}/S_{t^{-}}=ζ_{t} \vartheta(ΔZ_{t})$ for a process $ζ$ and a deterministic function $\vartheta$, we explicitly characterize the admissible trading strategies and show that the dual solution is a risk-neutral local martingale.