Optimal Portfolios in Lévy Markets under State-Dependent Bounded Utility Functions

[摘要] Motivated by the so-called shortfall risk minimization problem,we consider Merton's portfolio optimization problem in a non-Markovianmarket driven by a Lévy process, with a bounded state-dependent utilityfunction. Following the usualdual variational approach, we show that thedomain of thedual problemenjoys an explicit “parametrization,” built onamultiplicative optional decompositionfor nonnegative supermartingalesdue to Föllmer and Kramkov (1997). As a key step we prove aclosure propertyfor integrals with respect to a fixed Poisson random measure, extending aresult by Mémin (1980). In the case where either the Lévy measureνofZhas finite number of atoms orΔSt/St−=ζtϑ(ΔZt)for a processζand adeterministic functionϑ, we characterize explicitly the admissible tradingstrategies and show that the dual solution is a risk-neutral local martingale.

[学科分类] 应用数学