A decomposition for additive functional of Levy processes

TL;DR

This paper introduces a decomposition method for additive functionals of one-dimensional symmetric Lévy processes, extending understanding of their limit laws and moment calculations under specific conditions.

Contribution

It provides a novel decomposition framework for functionals of symmetric Lévy processes and computes their moments, advancing theoretical understanding in stochastic process analysis.

Findings

Decomposition formula for additive functionals of Lévy processes

Explicit moment calculations for the decomposition

Extension of limit law results for occupation times

Abstract

Motivated by the recent results of Nualart and Xu \cite{Nualart} concerning limits laws for occupation times of one dimensional symmetric stable processes, this paper proves a decomposition for functionals of one dimensional symmetric L\'evy processes under certain conditions on the characteristic exponent and computes the moments of the decomposition.