Optimal portfolios in Lévy markets under state-dependent bounded utility functions

Motivated by the so-called shortfall risk minimization problem,we consider Merton's portfolio optimization problem in a non-Markovian market driven by a Lévy process, with a bounded state-dependent utility function. Following the usual dual variational approach, we show that the domain of the dual problem enjoys an explicit ‘parametrization,’ built on a multiplicative optional decomposition for nonnegative supermartingalesdue to Föllmer and Kramkov (1997). As a key step we prove a closure propertyfor integrals with respect to a fixed Poisson random measure, extending a result by Mémin (1980). In the case where either the Lévy measure v of Z has finite number of atoms or ΔS^t/S^t−= ζ^tθ(ΔZ^t) for a process ζ and a deterministic function θ, we characterize explicitly the admissible trading strategies and show that the dual solution is a risk-neutral local martingale.