A temporal factorization at the maximum for certain positive self-similar Markov processes

For a spectrally negative self-similarMarkov process on [0, infinity) 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 this 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 existing results in the literature.