  The main objective consists in generalizing a well-known It{\^o} formula of
J. Jacod and A. Shiryaev: given a c{\`a}dl{\`a}g process S, there is an
equivalence between the fact that S is a semimartingale with given
characteristics (B^k , C, $\nu$) and a It{\^o} formula type expansion of F (S),
where F is a bounded function of class C2. This result connects weak solutions
of path-dependent SDEs and related martingale problems. We extend this to the
case when S is a weak Dirichlet process. A second aspect of the paper consists
in discussing some untreated features of stochastic calculus for finite
quadratic variation processes.
