On Wiener-Hopf Factorisation and the Distribution of Extrema for Certain Stable Processes

Abstract

It is shown that when the index $0 < \alpha < 2, \alpha \neq 1$, and the symmetry parameter $-1 \leq \beta \leq 1$ of a stable process $\{X(t); t \geq 0\}$ are such that $P\{X(1) > 0\} = l\alpha^{-1} - k$, where $l$ and $k$ are integers, Darling's integral can be evaluated. This leads to explicit formulas for a transform of the Laplace transform of $\sup_{0\leq t\leq 1}X(t)$ and the Wiener-Hopf factors of $\{X(t), t \geq 0\}$.