It{\^o}-Dupire's formula for C^{0,1}-functionals of c{\`a}dl{\`a}g weak Dirichlet processes

TL;DR

This paper extends the Itô-Dupire formula to càdlàg weak Dirichlet processes, providing conditions for functional transformations to preserve the process type without relying on smooth approximations, using a small-jumps argument.

Contribution

It introduces a novel extension of the Itô-Dupire formula to càdlàg weak Dirichlet processes, avoiding smooth approximation methods.

Findings

Provides sufficient conditions for functional transformations to preserve weak Dirichlet processes.

Uses a small-jumps cutting argument instead of smooth approximation.

Extends previous results to non-Markovian, path-dependent cases.

Abstract

We extend to c{\`a}dl{\`a}g weak Dirichlet processes the C^{0,1}-functional It{\^o}-Dupire's formula of Bouchard, Loeper and Tan (2021). In particular, we provide sufficient conditions under which a C^{0,1}-functional transformation of a special weak Dirichlet process remains a special weak Dirichlet process. As opposed to Bandini and Russo (2018) who considered the Markovian setting, our approach is not based on the approximation of the functional by smooth ones, which turns out not to be available in the pathdependent case. We simply use a small-jumps cutting argument.