Extended It\^{o} calculus for symmetric Markov processes

Chen, Fitzsimmons, Kuwae and Zhang (Ann. Probab. 36 (2008) 931-970) have established an Ito formula consisting in the development of F(u(X)) for a symmetric Markov process X, a function u in the Dirichlet space of X and any $\mathcal{C}^2$-function F. We give here an extension of this formula for u locally in the Dirichlet space of X and F admitting a locally bounded Radon-Nikodym derivative. This formula has some analogies with various extended Ito formulas for semi-martingales using the local time stochastic calculus. But here the part of the local time is played by a process $(\Gamma^a_t,a\in \mathbb{R},t \geq 0)$ defined thanks to Nakao's operator (Z. Wahrsch. Verw. Gebiete 68 (1985) 557-578).