Mittag-Leffler functions and stable Levy processes without negative jumps

We remark that a certain transformation of the Mittag-Leffler function alpha is completely monotone for every alpha is an element of [1,2]. Thanks to the exact expression of its Bernstein density function, we obtain an identity in law between one-sided exit times for completely asymmetric stable Levy processes. In the spectrally positive case, this identity gives an expression for the density of the running supremum which is different from the one recently obtained by Bernyk, Dalang and Peskir (2008) [1]. (C) 2010 Elsevier GmbH. All rights reserved.