Variation and Rough Path Properties of Local Times of Lévy Processes

Abstract

In this paper, we will prove that the local time of a Lévy process is of finite $p$-variation in the space variable in the classical sense, a.s. for any $p>2$, $t\ge 0$, if the Lévy measure satisfies ${\int }_{R\setminus \{0\}}(|y{|}^{\frac{3}{2}}\wedge 1)n(dy)<\infty$, and is a rough path of roughness $p$ a.s. for any $2<p<3$ under a slightly stronger condition for the Lévy measure. Then for any function $g$ of finite $q$-variation ($1\le q<3$), we establish the integral ${\int }_{-\infty }^{\infty }g(x)d{L}_t^x$ as a Young integral when $1\le q<2$ and a Lyons' rough path integral when $2\le q<3$. We therefore apply these path integrals to extend the Tanaka-Meyer formula for a continuous function $f$ if ${\nabla }^{-}f$ exists and is of finite $q$-variation when $1\le q<3$, for both continuous semi-martingales and a class of Lévy processes.