  In this paper we explain how the notion of ''weak Dirichlet process'' is the
suitable generalization of the one of semimartingale with jumps. For such a
process we provide a unique decomposition which is new also for
semimartingales: in particular we introduce ''characteristics'' for weak
Dirichlet processes. We also introduce a weak concept (in law) of finite
quadratic variation. We investigate a set of new useful chain rules and we
discuss a general framework of (possibly path-dependent with jumps) martingale
problems with a set of examples of SDEs with jumps driven by a distributional
drift.
