Short proofs in extrema of spectrally one sided L\'evy processes

TL;DR

This paper presents concise proofs of key theorems related to spectrally one-sided Lévy processes, extending classical results and deriving new formulas for the supremum distribution.

Contribution

It introduces simplified proofs of the ballot theorem and Kendall's identity, extends the ballot theorem to processes with negative jumps, and derives formulas for the supremum distribution of spectrally negative Lévy processes.

Findings

Simplified proofs of the continuous-time ballot theorem and Kendall's identity.

Extension of the ballot theorem to processes with negative jumps.

Derived formulas for the law of the supremum of spectrally negative Lévy processes.

Abstract

We provide short and simple proofs of the continuous time ballot theorem for processes with cyclically interchangeable increments and Kendall's identity for spectrally positive L\'evy processes. We obtain the later result as a direct consequence of the former. The ballot theorem is extended to processes having possible negative jumps. Then we prove through straightforward arguments based on the law of bridges and Kendall's identity, Theorem 2.4 in \cite{mpp} which gives an expression for the law of the supremum of spectrally positive L\'evy processes. An analogous formula is obtained for the supremum of spectrally negative L\'evy processes.