  For a spectrally negative positive self-similar Markov process with an a.s.
finite overall supremum we provide, in tractable detail, a kind of conditional
Wiener-Hopf factorization at the maximum of the absorption time at zero, the
conditioning being on the overall supremum and the jump at the overall
supremum. In a companion result the Laplace transform of said absorption time
(on the event that the process does not go above a given level) is identified
under no other assumptions (such as the process admitting a recurrent extension
and/or hitting zero continuously), generalizing some existent results in the
literature.
