Two-sided exit problem for a spectrally negative α-stable ornstein-uhlenbeck pro-cess and the wrightșs generalized hyper-geometric functions

Abstract

The Laplace transform of the first exit time from a finite interval by a spectrally negative α- stable Ornstein-Uhlenbeck process (1 < α ≤ 2) is provided in terms of the Wright’s generalized hypergeometric function 2Ψ1. 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. © 2007 Applied Probability Trust.