Tanaka formula for strictly stable processes

TL;DR

This paper extends the Tanaka formula to strictly stable processes with index between 1 and 2, including spectrally positive and negative cases, using a novel approach within Itô's calculus.

Contribution

It introduces a new method for establishing the Tanaka formula for strictly stable processes, differing from previous potential theory techniques.

Findings

Derived Tanaka formula for strictly stable processes with index in (1,2)

Established existence of local times for these processes

Applied Itô's calculus framework to stable processes

Abstract

For symmetric L\'evy processes, if the local times exist, the Tanaka formula has already constructed via the techniques in the potential theory by Salminen and Yor (2007). In this paper, we study the Tanaka formula for arbitrary strictly stable processes with index $\alpha \in (1,2)$ including spectrally positive and negative cases in a framework of It\^o's stochastic calculus. Our approach to the existence of local times for such processes is different from Bertoin (1996).