We present a new Fukushima type decomposition in the framework of semi-Dirichlet forms. This generalizes the result of Ma, Sun and Wang [17, Theorem 1.4] by removing the condition (S). We also extend Nakao's integral to semi-Dirichlet forms and derive Itô's formula related to it.
"Stochastic calculus for Markov processes associated with semi-Dirichlet forms." Tohoku Math. J. (2) 70 (1) 97 - 119, 2018. https://doi.org/10.2748/tmj/1520564420