Stochastic calculus for symmetric Markov processes

TL;DR

This paper develops a stochastic calculus framework for symmetric Markov processes using time-reversal, introducing stochastic integrals for zero-energy additive functionals and deriving an Itô formula for Dirichlet processes.

Contribution

It extends stochastic calculus to symmetric Markov processes by defining integrals for zero-energy additive functionals and establishing an Itô formula, building on Nakao's earlier work.

Findings

Defined stochastic integrals for zero-energy additive functionals

Derived properties of these stochastic integrals

Established an Itô formula for Dirichlet processes

Abstract

Using time-reversal, we introduce a stochastic integral for zero-energy additive functionals of symmetric Markov processes, extending earlier work of S. Nakao. Various properties of such stochastic integrals are discussed and an It\^{o} formula for Dirichlet processes is obtained.