  The motivation of this paper is to prove verification theorems for stochastic
optimal control of finite dimensional diffusion processes without control in
the diffusion term, in the case that 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. For this purpose, the replacement tool of the It\^{o}
formula will be the Fukushima-Dirichlet decomposition for weak Dirichlet
processes. Given a fixed filtration, a weak Dirichlet process is the sum of a
local martingale $M$ plus an adapted process $A$ which is orthogonal, in the
sense of covariation, to any continuous local martingale. The mentioned
decomposition states that a $C^{0,1}$ function of a weak Dirichlet process with
finite quadratic variation is again a weak Dirichlet process. That result is
established in this paper and it is applied to the strong solution of a Cauchy
problem with final condition. Applications to the proof of verification
theorems will be addressed in a companion paper.
