State-dependent utility maximization in Le19 evy markets

We revisit Merton’s portfolio optimization problem under bounded state-dependent utility functions, in a market driven by a Lévy process Z extending results by Karatzas et. al. [8] and Kunita [11]. The problem is solved using a dual variational problem as it is customarily done for nonMarkovian 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 [2]. 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 [13]. In the case that (i) the Lévy measure ν of Z is atomic with a finite number of atoms or that (ii) ∆St/St− = ζtθ(∆Zt) for a process ζ and a deterministic function θ, we explicitly characterize the admissible trading strategies and show that the dual solution is a risk-neutral local martingale. AMS 2000 subject classifications: Primary: 93E20, 60G51, secondary: 62P05.