Characteristics and It{\^o}'s formula for weak Dirichlet processes: an equivalence result

TL;DR

This paper extends Itô's formula to weak Dirichlet processes, establishing an equivalence with semimartingale characteristics and connecting weak solutions of path-dependent SDEs with martingale problems.

Contribution

It generalizes the classical Itô formula to weak Dirichlet processes and explores stochastic calculus for finite quadratic variation processes.

Findings

Established equivalence between weak Dirichlet processes and semimartingale characteristics.

Extended Itô's formula to weak Dirichlet processes.

Discussed features of stochastic calculus for finite quadratic variation processes.

Abstract

The main objective consists in generalizing a well-known It{\^o} formula of J. Jacod and A. Shiryaev: given a c{\`a}dl{\`a}g process S, there is an equivalence between the fact that S is a semimartingale with given characteristics (B^k , C, $\nu$) and a It{\^o} formula type expansion of F (S), where F is a bounded function of class C2. This result connects weak solutions of path-dependent SDEs and related martingale problems. We extend this to the case when S is a weak Dirichlet process. A second aspect of the paper consists in discussing some untreated features of stochastic calculus for finite quadratic variation processes.