Utility maximization problem in the case of unbounded endowment

We consider a problem of maximizing expected utility with an utility function finite on $\mathbb{R}_+$ and with an unbounded random endowment in an abstract model of financial market. We formulate a dual problem to the primal one and prove duality relations between them. In addition, we study necessary conditions to the existence of solutions to the primal problem. Finally, we reduce the dual problem to a form more convenient for practice.