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≤ q <3), we
establish the integral
∫-∞∞g(x)dLtx
as a
Young integral when 1≤ q<2 and a Lyons' rough path integral
when 2≤ q<3. We therefore apply these path integrals to extend
the Tanaka-Meyer formula for a continuous function f if f'- exists and is of finite q-variation when 1≤ q<3 , for
both continuous semi-martingales and a class of Lévy processes.
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