Stochastic integration with respect to additive functionals of zero quadratic variation

TL;DR

This paper develops a stochastic integral framework for additive functionals of zero quadratic variation associated with Markov processes, extending Itô's formula to a broader class of processes.

Contribution

It introduces a new stochastic integral with respect to additive functionals of zero quadratic variation for Markov processes linked to non-symmetric Dirichlet forms.

Findings

Established an Itô formula for processes involving these additive functionals.

Defined a stochastic integral for a class of zero quadratic variation additive functionals.

Extended stochastic calculus tools to non-symmetric Dirichlet form settings.

Abstract

We consider a Markov process $X$ associated to a nonnecessarily symmetric Dirichlet form $\mathcal{E}$. We define a stochastic integral with respect to a class of additive functionals of zero quadratic variation and then we obtain an It\^{o} formula for the process $u(X)$, when $u$ is locally in the domain of $\mathcal{E}$.