Occupation times of general L\'evy processes

TL;DR

This paper derives compact formulas for the Laplace transform of the joint distribution of a general Lévy process and its occupation times, providing insights useful for stochastic process analysis and financial applications.

Contribution

It introduces a novel approach to obtain explicit formulas for occupation times of Lévy processes, extending understanding beyond compound Poisson processes.

Findings

Formulas for the Laplace transform of joint distribution of Lévy process and occupation times

Clear demonstration of key quantities for calculating occupation times

Potential applications in finance and stochastic process analysis

Abstract

For an arbitrary L\'evy process $X$ which is not a compound Poisson process, we are interested in its occupation times. We use a quite novel and useful approach to derive formulas for the Laplace transform of the joint distribution of $X$ and its occupation times. Our formulas are compact, and more importantly, the forms of the formulas clearly demonstrate the essential quantities for the calculation of occupation times of $X$. It is believed that our results are important not only for the study of stochastic processes, but also for financial applications.