Short proofs in extrema of spectrally one sided Lévy processes

Loïc Chaumont,Jacek Małecki

doi:10.1214/18-ecp163

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 Levy 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 [20] which gives an expression for the law of the supremum of spectrally positive Levy processes. An analogous formula is obtained for the supremum of spectrally negative Levy processes.

Highlights

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 Lévy processes
We prove through straightforward arguments based on the law of bridges and Kendall’s identity, Theorem 2.4 in [20] which gives an expression for the law of the supremum of spectrally positive Lévy processes
An analogous formula is obtained for the supremum of spectrally negative Lévy processes

Summary

Introduction

We shall see that identity (1.3) can be obtained as a direct consequence of (1.2) for Lévy processes with bounded variation and extended to the general case in a direct way. These results on first passage times will naturally lead us in Section 3 to the law of the past supremum Xt of spectrally positive Lévy processes. We show in Theorem 3.1 that a quite simple computation involving the law of the bridge of the Lévy process allows us to provide a very short proof of (1.4). As a consequence of this result, we obtain in Corollary 3.5 an integro-differential equation characterizing the entrance law of the excursion measure of the Lévy process X reflected at its infimum

Continuous ballot theorem and Kendall’s identity

The law of the extrema of spectrally one sided Lévy processes