Characteristics and It{ô}'s formula for weak Dirichlet processes: an equivalence result

Abstract

The main objective consists in generalizing a well-known It{ô} formula of J. Jacod and A. Shiryaev: given a c{à }dl{à }g process S, there is an equivalence between the fact that S is a semimartingale with given characteristics (B^k , C, $\hat{I}½$) and a It{ô} 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.