Local Time Rough Path for Lévy Processes

Chunrong Feng (Shanghai Jiaotong University)
Huaizhong Zhao (Loughborough University)

Abstract


In this paper, we will prove that the local time of a Lévy process is a rough path of roughness $p$ a.s. for any $2 < p < 3$ under some condition for the Lévy measure. This is a new class of rough path processes. 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 $f^\prime_{-}$ exists and is of finite $q$-variation when $1\leq q<3$, for both continuous semi-martingales and a class of Lévy processes.

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Pages: 452-483

Publication Date: April 29, 2010

DOI: 10.1214/EJP.v15-770

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