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

TL;DR

This paper establishes the existence of quadratic covariation and local time integrals for G-Brownian motion, extending Itô's formula to C^1 functions through a sublinear Bouleau-Yor identity.

Contribution

It introduces a sublinear Bouleau-Yor identity for G-Brownian motion and extends Itô's formula to C^1 functions using local time integrals.

Findings

Existence of quadratic covariation <f(B), B>_t.

Existence of local time integral (x) al L(dx,t).

A sublinear Bouleau-Yor identity for G-Brownian motion.

Abstract

Let ${\mathscr L}$ 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 $\int_{\mathbb R}f(x){\mathscr L}(dx,t)$. Moreover, a sublinear version of the Bouleau-Yor identity $$ \int_{\mathbb R}f(x){\mathscr L}(dx,t)=-<f(B),B>_{t} $$ is showed to hold under some suitable conditions. These allow us to write the It\^o's formula for $C^1$-functions.