On exit time of stable processes

Abstract

We study the exit time $\tau ={\tau }_{(0,\infty )}$ for 1-dimensional strictly stable processes and express its Laplace transform at $t^{\alpha }$ as the Laplace transform of a positive random variable with explicit density. Consequently, $\tau$ satisfies some multiplicative convolution relations. For some stable processes, e.g. for the symmetric $\frac{2}{3}$-stable process, explicit formulas for the Laplace transform and the density of $\tau$ are obtained as an application.