2009/01/14 by José E. Figueroa‐López, Jose E. Figueroa-Lopez, Jin Ma +2
Decision Sciences · Economics, Econometrics and Finance · Social Sciences · #Computational Finance (q-fin.CP) #FOS: Economics and business #Insurance, Mortality, Demography, Risk Management #Portfolio Management (q-fin.PM) #Risk and Portfolio Optimization #Stochastic processes and financial applications #q-fin.CP #q-fin.PM
paper · pdf · doi:10.48550/arxiv.0901.2070
arxiv created 2009/01/14 · openalex publication_date 2009/01/14 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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) ΔSt/St-=ζt ϑ(ΔZt) 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.