Weak Dirichlet processes with jumps

TL;DR

This paper extends stochastic calculus via regularization to jump processes, analyzing weak Dirichlet processes with jumps and providing a chain rule expansion for functions of such processes.

Contribution

It systematically develops the stochastic calculus for jump processes and offers a chain rule for weak Dirichlet processes with jumps, expanding previous continuous-process results.

Findings

Provides a chain rule expansion for functions of weak Dirichlet processes with jumps.

Analyzes the structure of weak Dirichlet processes in the jump setting.

Extends stochastic calculus via regularization to processes with jumps.

Abstract

This paper develops systematically the 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 $N$. Given a function $u:[0,T] \times \mathbb{R} \to \mathbb{R}$, which is of class $C^{0,1}$ (or sometimes less), we provide a chain rule type expansion for $u(t,X_t)$ which stands in applications for a chain It\^o type rule.