International Journal of Stochastic Analysis
Volume 2010 (2010), Article ID 236587, 27 pages
Research Article

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

1Department of Statistics, Purdue University, West Lafayette, IN 47906, USA
2Department of Mathematics, University of Southern California, Los Angeles, CA 90089, USA

Received 18 August 2009; Accepted 28 January 2010

Academic Editor: Vo V. Anh

Copyright © 2010 José E. Figueroa-López and Jin Ma. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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.