  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).
