  We consider the value function originating from an expected utility
maximization problem with finite fuel constraint and show its close relation to
a nonlinear parabolic degenerated Hamilton-Jacobi-Bellman (HJB) equation with
singularity. On one hand, we give a so-called verification argument based on
the dynamic programming principle, which allows us to derive conditions under
which a classical solution of the HJB equation coincides with our value
function (provided that it is smooth enough). On the other hand, we establish a
comparison principle, which allows us to characterize our value function as the
unique viscosity solution of the HJB equation.
