Extinction time of non-Markovian self-similar processes, persistence, annihilation of jumps and the Fréchet distribution

We start by providing an explicit characterization and analytical properties, including the persistence phenomena, of the distribution of the extinction time T of a class of non- Markovian self-similar stochastic processes with two-sided jumps that we introduce as a stochastic time-change of Markovian self-similar processes. For a suitably chosen timechange, we observe, for classes with two-sided jumps, the following surprising facts. On the one hand, all the T's within a class have the same law which we identify in a simple form for all classes and reduces, in the spectrally positive case, to the Fréchet distribution. On the other hand, each of its distribution corresponds to the law of an extinction time of a single Markov process without positive jumps, leaving the interpretation that the time-change has annihilated the effect of positive jumps. The example of the non-Markovian processes associated to Lévy stable processes is detailed.