  This paper is devoted to present a method of proving verification theorems
for stochastic optimal control of finite dimensional diffusion processes
without control in the diffusion term. The value function is assumed to be
continuous in time and once differentiable in the space variable ($C^{0,1}$)
instead of once differentiable in time and twice in space ($C^{1,2}$), like in
the classical results. The results are obtained using a time dependent
Fukushima - Dirichlet decomposition proved in a companion paper by the same
authors using stochastic calculus via regularization. Applications, examples
and comparison with other similar results are also given.
