The distribution of the supremum for spectrally asymmetric L\'evy processes

TL;DR

This paper derives formulas for the probability that the supremum of a spectrally positive Lévy process exceeds a threshold, generalizing Takbacs formulas, and also finds the joint distribution of the infimum and the value at a time for spectrally negative Lévy processes.

Contribution

It provides new formulas for supremum probabilities of spectrally asymmetric Lévy processes and the joint distribution of infimum and process value, extending classical results.

Findings

Formulas for supremum probabilities of spectrally positive Lévy processes.

Joint distribution of infimum and process value for spectrally negative Lévy processes.

Generalizations of Takbacs formulas for processes with infinite variation.

Abstract

In this article we derive formulas for the probability $P(\sup_{t\leq T} X(t)>u)$ $T>0$ and $P(\sup_{t<\infty} X(t)>u)$ where $X$ is a spectrally positive L\'evy process with infinite variation. The formulas are generalizations of the well-known Tak\'acs formulas for stochastic processes with non-negative and interchangeable increments. Moreover, we find the joint distribution of $\inf_{t\leq T} Y(t)$ and $Y(T)$ where $Y$ is a spectrally negative L\'evy process.