Weak Dirichlet processes with jumps

Elena Bandini, Francesco Russo

Published 2015-12-19Version 1

This paper develops systematically stochastic calculus via regularization in the case of jump processes. In particular one continues the analysis of real-valued c\`adl\`ag weak Dirichlet processes with respect to a given filtration. Such a process is the sum of a local martingale and an adapted process A such that $[N, A] = 0$, for any continuous local martingale $N$. In particular, given a function $u : [0, T ] \times \mathbb{R} \rightarrow \mathbb{R}$, which is of class C^{0,1} (or sometimes less), we provide a chain rule type expansion for X\_t = u(t, X\_t) which stands in applications for a chain It\^o type rule.