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 of has finite number of atoms or for a process and a
deterministic function , we characterize explicitly the admissible trading
strategies and show that the dual solution is a risk-neutral local martingale.