Integral with respect to the $G$-Brownian local time

Abstract

Let ${\mathscr L}$ be the local time of $G$-Brownian motion $B$. In this paper, we prove the existence of the quadratic covariation $ _{t}$ and the integral $\int_{\mathbb R}f(x){\mathscr L}(dx,t)$. Moreover, a sublinear version of the Bouleau-Yor identity $$ \int_{\mathbb R}f(x){\mathscr L}(dx,t)=- _{t} $$ is showed to hold under some suitable conditions. These allow us to write the Itô's formula for $C^1$-functions.