A Unified Framework for Utility Maximization Problems: an Orlicz Space Approach

We consider a stochastic financial incomplete market where the price processes are described by a vector valued semimartingale that is possibly non locally bounded. We face the classical problem of the utility maxi-mization from terminal wealth, with utility functions that are finite valued over (a;1), a 2 [1;1), and satisfy weak regularity assumptions. We adopt a class of trading strategies that allows for stochastic integrals that are not necessarily bounded from below. The embedding of the utility maximization problem in Orlicz spaces permits us to formulate the prob-lem in a unified way for both the cases: a 2 R or a = 1. By duality methods we prove the existence of the solutions to the primal and dual problems and show that a singular component in the pricing functionals may occur also with utility functions finite on the entire real line