Some functional analytic tools for utility maximization
The aim of the paper is to extend the application of convex duality methods to the problem of maximizing expected utility from terminal wealth. More precisely, we restrict attention to a dual characterization of the value function of this problem and to a static setting. A general scheme to solve this problem is proposed. In the case where the utility function is finite on $\R$, we use the approach, suggested by Biagini and Frittelli, based on using an Orlicz space constructed from an investor's utility function. We reduce the original problem to an optimization problem in this space in a nontrivial way, which allows us to weaken essentially assumptions on the model. We also study the problem of utility maximization with random endowment considered by Cvitani\'c, Schachermayer, and Wang. Using the space $\psi L^\infty$ with a weight function $\psi$ constructed from a random endowment permits us to consider unbounded random endowments. Another important contribution is that in both problems under consideration we provide versions of the dual problem that are free of singular functionals.