Variation and Rough Path Properties of Local Times of L\'evy Processes

TL;DR

This paper investigates the variation and rough path properties of local times of Lévy processes, establishing conditions under which they exhibit finite p-variation and can be integrated using Young or Lyons' rough path integrals, extending the Tanaka-Meyer formula.

Contribution

It provides new conditions for the p-variation and rough path properties of Lévy process local times and extends stochastic calculus tools to these processes.

Findings

Local time of Lévy processes has finite p-variation for p>2 under certain conditions.

Local time can be a rough path of roughness p for 2<p<3.

Extension of Tanaka-Meyer formula using Young and Lyons integrals.

Abstract

In this paper, we will prove that the local time of a L\'evy process is of finite $p$-variation in the space variable in the classical sense, a.s. for any $p>2$, $t\geq 0$, if the L\'evy measure satisfies $\int_{R\setminus \{0\}}(|y|^{3\over 2}\wedge 1)n(dy)<\infty$, and is a rough path of roughness $p$ a.s. for any $2<p<3$ under a slightly stronger condition for the L\'evy measure. Then for any function $g$ of finite $q$-variation ($1\leq q <3$), we establish the integral $\int_{-\infty}^{\infty}g(x)dL_t^x$ as a Young integral when $1\leq q<2$ and a Lyons' rough path integral when $2\leq q<3$. 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 $1\leq q<3$, for both continuous semi-martingales and a class of L\'evy processes.