Explicit formula for the supremum distribution of a spectrally negative stable process

TL;DR

This paper derives explicit formulas for the expected supremum of spectrally negative and positive Lévy processes with infinite variation, generalizing classical results for Wiener processes and using elementary probabilistic identities.

Contribution

It provides simple explicit formulas for the supremum distribution of spectrally negative Lévy processes with infinite variation, extending known results for Wiener processes.

Findings

Explicit formulas for supremum expectations of spectrally positive/negative Lévy processes.

Generalization of Wiener process supremum distribution formula.

Derivation from Kendall's identity and elementary Seals formula.

Abstract

In this article we get simple explicit formulas for $\Exp\sup_{s\leq t}X(s)$ where $X$ is a spectrally positive or negative L\'evy process with infinite variation. As a consequence we derive a generalization of the well-known formula for the supremum distribution of Wiener process that is we obtain $\Prob(\sup_{s\leq t}Z_{\alpha}(s)\geq u)=\alpha \Prob(Z_{\alpha}(t)\geq u)$ for $u\geq 0$ where $Z_{\alpha}$ is a spectrally negative L\'evy process with $1<\alpha\leq 2$ which also stems from Kendall's identity for the first crossing time. Our proof uses a formula for the supremum distribution of a spectrally positive L\'evy process which follows easily from the elementary Seals formula.