We show an Itô's formula for nondegenerate Brownian martingales $X_t=\int_0^t u_s \,dW_s$ and functions $F(x,t)$ with locally integrable derivatives in $t$ and $x$. We prove that one can express the additional term in Itô's s formula as an integral over space and time with respect to local time.
Xavier Bardina. Carles Rovira. "Integration with respect to local time and Itô's formula for smooth nondegenerate martingales." Publ. Mat. 54 (1) 187 - 208, 2010.