Characteristics and Itô's formula for weak Dirichlet processes: an equivalence result

The main objective consists in generalizing a well-known Itô formula of J. Jacod and A. Shiryaev: given a càdlàg process S, there is an equivalence between the fact that S is a semimartingale with given characteristics (Formula presented.) and a Itô formula type expansion of (Formula presented.), where F is a bounded function of class (Formula presented.). 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 of discussing some untreated features of stochastic calculus for finite quadratic variation processes.