Rough path properties for local time of symmetric [formula omitted] stable process

Abstract

In this paper, we first prove that the local time associated with symmetric α-stable processes is of bounded p-variation for any p>2α−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>2α−1 enables us to define the integral of the local time ∫−∞∞▿−α−1f(x)dxLtx as a Young integral for less smooth functions being of bounded q-variation with 1≤q<23−α. When q≥23−α, Young’s integration theory is no longer applicable.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 α-stable processes for 23−α≤q<4.