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

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 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 $\Prob(\sup_{s\leq t}Z_α(s)\geq u)=α\Prob(Z_α(t)\geq u)$ for $u\ge 0$ where $Z_{\alpha }$ is a spectrally negative Lévy process with $1<\alpha \le 2$ 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 Seals formula.