# A Characterization of the Finiteness of Perpetual Integrals of Levy Processes

## TL;DR

This paper establishes a criterion for determining when perpetual integrals of certain Levy processes are almost surely finite, extending previous results to more general cases without local times.

## Contribution

It provides a new, simplified criterion for the finiteness of perpetual integrals of Levy processes, addressing an open problem in the field.

## Key findings

- Derived a general criterion for finiteness of perpetual integrals.
- Showed the criterion simplifies when the process has a local time.
- Demonstrated the criterion cannot be reduced in the general case.

## Abstract

We derive a criterium for the almost sure finiteness of perpetual integrals of \LL processes for a class of real functions including all continuous functions and for general one-dimensional L\'evy processes that drifts to plus infinity. This generalizes previous work of D\"oring and Kyprianou, who considered L\'evy processes having a local time, leaving the general case as an open problem. It turns out, that the criterium in the general situation simplifies significantly in the situation, where the process has a local time, but we also demonstrate that in general our criterium can not be reduced. This answers an open problem posed in \cite{doring}.