Rough path properties for local time of symmetric $\hat{I}\pm$ stable process

Abstract

In this paper, we first prove that the local time associated with symmetric $\hat{I}\pm$-stable processes is of bounded $p$-variation for any $p>\frac{2}{\hat{I}\pm -1}$ partly based on Barlow's estimation of the modulus of the local time of such processes.\,\,The fact that the local time is of bounded $p$-variation for any $p>\frac{2}{\hat{I}\pm -1}$ enables us to define the integral of the local time ${\int }_{-\infty }^{\infty }{▽}_{-}^{\hat{I}\pm -1}f(x)d_x{L}_t^x$ as a Young integral for less smooth functions being of bounded $q$-varition with $1\le q<\frac{2}{3-\hat{I}\pm }$. When $q\ge \frac{2}{3-\hat{I}\pm }$, Young's integration theory is no longer applicable. However, rough path theory is useful in this case. The main purpose of this paper is to establish a rough path theory for the integration with respect to the local times of symmetric $\hat{I}\pm$-stable processes for $\frac{2}{3-\hat{I}\pm }\le q<4$.