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

In this article we get simple formulas for IE sups≤tX(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 IP(sups≤t Zα(s) ≥ u) = α IP(Zα(t) ≥ u) for u ≥ 0 where Zα is a spectrally negative α-stable Lévy process with 1 < α ≤ 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.