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

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