Brownian motion and Lévy processes are semimartingale and Itô calculus is based on the theory of semimartingales. However it is sometimes important to consider processes which are not semimartingales, but for instance Dirichlet or weak Dirichlet processes. The aim of this talk consists in mentioning recent developments about stochastic calculus  for jump processes via the method of regularization. A weak Dirichlet process with respect to a given underlying filtration is the sum of a local martingale and a process such that for every continuous local martingale. We introduce the notion of special weak Dirichlet process; whenever such a process is a semimartingale, then it is a special semimartingale. We will provide conditions on a function and on an adapted cadlag process such that is a special weak Dirichlet  process. Two applications will be discussed.

 

(a) The existence of a solution to a (strong) solution of a BSDEs with distributional driver, with underlyng Brownian filtration (with Elena Issoglio, Leeds).

(b) Consider the case a BSDE driven by a random measure: a solution is a triplet where is a random field. The function is deterministic. If has some minimal regulartiy, the calculus will allow to link to (with Elena Bandini, Milano Bicocca).