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

Abstract

In this article we get simple formulas for $E\sup_{s\leq t}X(s)$ where $X$ is a spectrally positive or negative Lévy 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 $P(\sup_{s\leq t}Z_{\alpha}(s)\geq u)=\alpha\,P(Z_{\alpha}(t)\geq u)$ for $u\geq 0$ where $Z_{\alpha}$ is a spectrally negative $\alpha$-stable Lévy 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 Lévy process which follows easily from the elementary Seal's formula.