Rough path analysis for local time of G-Brownian motion

ABSTRACT

Let $\mathcal{L}=\left\{\mathcal{L}\right(t,x),t\ge 0,x\in \mathbb{R}\}$ be the local time of a G-Brownian motion B on a sublinear expectation space $(\mathrm{\Omega },\mathcal{H},\hat{\mathbb{E}})$. In this paper, we show that the local time $\mathcal{L}$ is a rough path of roughness p quasi-surely for any 2<p<3. For every Borel function g of finite q-variation ($2\le q<3$), we establish the integral ${\int }_{\mathbb{R}}g\left(x\right)\mathcal{L}(t,\mathrm{d}x)$ as a Lyons' rough path integral. Moreover, we apply such path integrals to extend the Itô formula for a absolutely continuous function f if the derivative $f^{\prime }$ is bounded and left-continuous with a bounded q-variation ($2\le q<3)$.

COMMUNICATED BY:

• Meir Shillor

KEYWORDS:

• G-Brownian motion
• Itô's formula
• local time
• rough path

2000 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 60G20
• 60H05
• 60H30
• 60J65

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work is supported by the National Natural Science Foundation of China [No. 11171062, 11301068] and Innovation Program of Shanghai Municipal Education Commission [No. 12ZZ063], and the Fundamental Research Funds for the Central Universities [No. 15D310408].