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 Itô formula for Dirichlet processes is obtained.
"Stochastic calculus for symmetric Markov processes." Ann. Probab. 36 (3) 931 - 970, May 2008. https://doi.org/10.1214/07-AOP347