A potential theoretic approach to Tanaka formula for asymmetric L\'evy processes

TL;DR

This paper develops a potential theoretic approach to derive the Tanaka formula for asymmetric Lévy processes, extending previous methods and providing new examples and invariant functions.

Contribution

It introduces a potential theoretic framework for the Tanaka formula in asymmetric Lévy processes, generalizing prior results and including new examples and invariant functions.

Findings

Derived Tanaka formula for asymmetric Lévy processes using potential theory

Provided examples for key processes demonstrating the approach

Generalized previous results to asymmetric cases

Abstract

In this paper, we shall introduce the Tanaka formula from viewpoint of the Doob-Meyer decomposition. For symmetric L\'evy processes, if the local time exists, Salminen and Yor (2007) obtained the Tanaka formula by using the potential theoretic techniques. On the other hand, for asymmetric stable processes with index $\alpha \in (1,2)$, we studied the Tanaka formula by using It\^o's stochastic calculus and the Fourier analysis. In this paper, we study the Tanaka formula for asymmetric L\'evy processes via the potential theoretic approach. We give several examples for important processes. Our approach also gives the invariant excessive function with respect to the killed process in the case of asymmetric L\'evy processes and it generalized the result in Yano (2013).