Integration with respect to local time and Ito's formula for smooth nondegenerate martingales

Abstract

We show an It\^ o's formula for nondegenerate Brownian martingales $X_t={\int }_0^t{u}_{s}d{W}_s$ 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.