Two-sided exit problem for a Spectrally Negative $\alpha$-Stable Ornstein-Uhlenbeck Process and the Wright's generalized hypergeometric functions

Abstract

The Laplace transform of the first exit time from a finite interval by a regular spectrally negative $\alpha$-stable Ornstein-Uhlenbeck process is provided in terms of the Wright's generalized hypergeometric function. The Laplace transform of first passage times is also derived for some related processes such as the process killed when it enters the negative half line and the process conditioned to stay positive. The law of the maximum of the associated bridges is computed in terms of the $q$-resolvent density. As a byproduct, we deduce some interesting analytical properties for some Wright's generalized hypergeometric functions.