On regularity of primal and dual dynamic value functions related to investment problem

We study regularity properties of the dynamic value functions of primal and dual problems of optimal investing for utility functions defined on the whole real line. Relations between decomposition terms of value processes of primal and dual problems and between optimal solutions of basic and conditional utility maximization problems are established. These properties are used to show that the value function satisfies a corresponding backward stochastic partial differential equation. In the case of complete markets we give conditions on the utility function when this equation admits a solution.