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

In this article we get simple explicit formulas for $\text{Exp}{\sup }_{s\le t}X(s)$ 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 $\text{Prob}({\sup }_{s\le t}{Z}_{\alpha }(s)\ge u)=\alpha \text{Prob}(Z_{\alpha }(t)\ge u)$ for $u\ge 0$ where $Z_{\alpha }$ is a spectrally negative L'evy 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 L'evy process which follows easily from the elementary Seals formula.