  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.
