On exit time of stable processes

We study the exit time τ=τ(0,∞)τ=τ(0,∞) for 1-dimensional strictly stable processes and express its Laplace transform at tαtα as the Laplace transform of a positive random variable with explicit density. Consequently, ττ satisfies some multiplicative convolution relations. For some stable processes, e.g. for the symmetric 23-stable process, explicit formulas for the Laplace transform and the density of ττ are obtained as an application.