Utility maximization with a given pricing measure when the utility is not necessarily concave

We study the problem of maximizing expected utility from terminal wealth for a not necessarily concave utility function $$U$$ and for a budget set given by one fixed pricing measure. We prove the existence and several fundamental properties of a maximizer. We analyze the (not necessarily concave) value function (indirect utility) $$u(x,U)$$ . In particular, we show that the concave envelope of $$u(x,U)$$ is the value function $$u(x,U_c)$$ of the utility maximization problem for the concave envelope $$U_c$$ of the utility function $$U$$ . The two value functions are shown to coincide if the underlying probability space is atomless. This allows us to characterize the maximizers for several model classes explicitl