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 supermartingales due to Föllmer and Kramkov (1997). As a key step we prove a closure property for integrals with respect to a fixed Poisson random measure, extending a result by Mémin (1980). In the case where either the Lévy measure ν of Z has finite number of atoms or ΔSt/St−=ζtϑ(ΔZt) 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.