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,∞), a ∈ [−∞,∞), and satisfy weak regularity assumptions. We adopt a class of trading strategies that allows for non locally bounded stochastic integrals. The embedding of the utility maximization problem in Orlicz spaces permits to formulate the problem in a unified way for both the cases: a ∈ R or a = −∞. By duality methods we prove the existence of the optimal 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. Key words: utility maximization – non locally bounded semimartingale – incomplete market – σ-martingale measure – singular functional – Orlicz space – convex dualit